{"messages": [{"content": "ABCDE is a regular 5-gon with center G and circumradius 6. Let P be the reflection of point B across point G. Let F be the perpendicular bisector of segment DA. Line Y passes through point B parallel to line F. Point Q is the reflection of point D across line Y. Line X is the reflection of line F across line Y. Let K be the reflection of line X across line F. Point O is the reflection of point P across line K. What is the length of segment QO? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.53", "dataset": "MATH"}
{"messages": [{"content": "ACDEFGHIJK is a regular 10-gon with center B and circumradius 3. Line Y passes through points E and H. Let N be the midpoint of segment GD. Line R is the reflection of line Y across point N. Point L is the intersection of line R and segment IH. Find the distance between points B and L. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.86", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center X and circumradius 12. Point K is obtained by rotating point X by angle pi/3 about point F. Let I be the midpoint of segment KG. Point J is the midpoint of segment EI. Let L be the line through points E and J. Let W be the midpoint of segment KX. Let U be the line through points G and W. Lines L and U intersect at point H. Point V is the midpoint of segment XD. Find the length of segment HV. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.55", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJK is a regular 11-gon with center R and circumradius 9. Let W be the reflection of point B across point D. Point P is the midpoint of segment AR. Point V is the midpoint of segment IP. Point Y is the reflection of point V across point R. What is the distance between points Y and W? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.04", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center O and circumradius 6. Line Y is the angle bisector of angle EHG. Line M passes through point D parallel to line Y. Point Z is the intersection of segment CH and line M. Find the distance between points Z and A. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.39", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center Z and circumradius 10. Let M be the circumcircle of polygon ABCDE. Point Q is the reflection of point Z across point D. Let R be the midpoint of segment BQ. Let I be the inversion of point R in circle M. What is the distance between points R and I? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.42", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center K and circumradius 10. Point Q is the reflection of point F across point B. Let L be the rotation of point I by angle 5pi/12 around point A. Let Y be the circumcircle of polygon ABCDEFGHI. Point V is the inversion of point Q with respect to circle Y. Let J be the perpendicular bisector of segment FL. Line Z passes through point F parallel to line J. Line S passes through point V parallel to line Z. Line S and segment FL intersect at point R. What is the distance between points R and F? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.11", "dataset": "MATH"}
{"messages": [{"content": "BCDEFGHI is a regular octagon with center A and circumradius 10. Point L is the midpoint of segment EG. Line M is the angle bisector of angle DGL. Point K is the reflection of point D across line M. Point J is the reflection of point K across point H. Point U is the midpoint of segment JL. What is the distance between points U and C? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.85", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center M and circumradius 1. Point K is the reflection of point D across segment BC. Point I is the midpoint of segment KG. Point X is the midpoint of segment KC. Find the length of segment IX. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.97", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDFGHIJKL be a regular 11-gon with center E and circumradius 4. Let S be the circumcircle of polygon ABCDFGHIJKL. Line Z is the angle bisector of angle CKG. Line N is the perpendicular bisector of segment EJ. Let Y be the midpoint of segment AC. Let T be the reflection of line N across line Z. Circle X is the reflection of circle S across line T. Let M be the inversion of point Y in circle X. Find the length of segment MG. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.71", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center T and circumradius 4. Point X is the midpoint of segment DE. Let Q be the rotation of point D by angle pi/12 around point A. Let M be the angle bisector of angle TXQ. Let O be the reflection of line M through point F. Point I is the reflection of point B across line O. Point V is the intersection of segment XI and line O. What is the length of segment VA? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.87", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center R and circumradius 11. Let Z be the circumcircle of polygon ABCDEFGHIJKL. Let Q be the midpoint of segment IL. Let P be the inversion of point Q in circle Z. What is the distance between points Q and P? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.78", "dataset": "MATH"}
{"messages": [{"content": "ACDEF is a regular 5-gon with center B and circumradius 2. Line J is the angle bisector of angle DBF. Let V be the reflection of line J through point A. Point T is the reflection of point C across point F. Line X passes through points T and C. Point M is the reflection of point C across line V. Point L is the reflection of point D across line X. Find the length of segment LM. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.33", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center S and circumradius 9. Point Y is obtained by rotating point A by angle pi/6 about point F. Line Q is the angle bisector of angle DLK. Let T be the reflection of line Q through point K. Point M is the reflection of point F across line T. Point O is the midpoint of segment MA. Find the length of segment OY. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.29", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center R and circumradius 5. Line Z is the angle bisector of angle BCF. Line S passes through point R perpendicular to line Z. Point V is the reflection of point F across line S. Point X is the midpoint of segment RD. Point I is the midpoint of segment VX. What is the distance between points I and X? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.48", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center Q and circumradius 1. Let R be the perpendicular bisector of segment QE. Line T is the reflection of line R across point A. Line S is the reflection of line T across point J. Line S and segment DJ intersect at point X. What is the length of segment XC? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.24", "dataset": "MATH"}
{"messages": [{"content": "BCDEFG is a regular hexagon with center A and circumradius 2. Point L is the midpoint of segment BF. Line N is the perpendicular bisector of segment LE. Point I is the reflection of point B across line N. Find the length of segment AI. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.09", "dataset": "MATH"}
{"messages": [{"content": "ACDEFGHI is a regular octagon with center B and circumradius 9. Let K be the midpoint of segment HI. Point O is the reflection of point K across point G. Point L is the midpoint of segment BI. Point T is the midpoint of segment LD. Find the length of segment OT. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19.22", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center W and circumradius 4. Let X be the rotation of point G by angle 7pi/12 around point B. Let U be the line through point I parallel to segment FL. Let T be the circumcircle of polygon ABCDEFGHIJKL. Line Z passes through point A parallel to segment FL. Let V be the reflection of line U through point W. Let M be the reflection of circle T across line V. Circle R is the reflection of circle M across line Z. Point Y is the inversion of point X with respect to circle R. What is the length of segment YA? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.67", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJLM is a regular dodecagon with center K and circumradius 3. Point Z is the reflection of point M across point H. Let R be the line through points L and H. Let Q be the midpoint of segment ZA. Point N is the reflection of point G across line R. Find the length of segment NQ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.26", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center M and circumradius 9. Line Y is the angle bisector of angle LAD. Let W be the perpendicular bisector of segment CE. Let X be the reflection of line Y across line W. Line V passes through point M parallel to line Y. Point O is the intersection of lines V and X. Find the length of segment DO. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.15", "dataset": "MATH"}
{"messages": [{"content": "ABDEFGHI is a regular octagon with center C and circumradius 5. Let M be the perpendicular bisector of segment AF. Point N is the reflection of point F across point E. Point V is the reflection of point N across line M. Find the length of segment CV. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.37", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKM is a regular dodecagon with center L and circumradius 8. Point U is the reflection of point C across segment MG. Segments GJ and IL intersect at point Q. Let X be the perpendicular bisector of segment AL. Point O is the reflection of point Q across line X. What is the distance between points O and U? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.86", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center N and circumradius 8. Line G passes through points A and C. Let V be the line through point D parallel to line G. Line Y is the reflection of line G across line V. Let U be the line through points B and N. Let S be the reflection of line Y through point N. Point R is the intersection of lines S and U. Find the measure of angle DNR, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.51", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center T and circumradius 9. Point M is the midpoint of segment HB. Point X is the midpoint of segment GE. Line P passes through point X perpendicular to segment DH. Line L is the reflection of line P across point E. Point Q is the reflection of point M across line L. Find the length of segment QF. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32.86", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center X and circumradius 4. Let Y be the line through points B and K. Line Z is the reflection of line Y across point C. Point U is the intersection of line Z and segment HB. What is the distance between points U and X? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.07", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center Y and circumradius 6. Point H is the midpoint of segment YE. Point N is obtained by rotating point H by angle pi/2 about point A. Point V is the midpoint of segment NB. Let P be the midpoint of segment DN. Point R is the reflection of point V across point C. Find the distance between points R and P. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.14", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center I and circumradius 4. Point J is the reflection of point D across point C. Let R be the circumcircle of polygon ABCDEF. Point V is the inversion of point J with respect to circle R. Let H be the reflection of point I across point C. Find the measure of angle HBV, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.05", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center U and circumradius 11. Line W passes through points C and B. Point N is the reflection of point G across segment BF. Point L is the reflection of point A across line W. Find the length of segment LN. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26.75", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center J and circumradius 10. Let U be the rotation of point C by angle 11pi/12 around point F. Point X is the midpoint of segment DB. Point W is the midpoint of segment FU. What is the length of segment WX? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24.94", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center R and circumradius 12. Let N be the rotation of point D by angle 2pi/3 around point F. Point V is the reflection of point K across segment ND. Let S be the angle bisector of angle LVK. Point O is the reflection of point L across line S. What is the distance between points I and O? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.7", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGHIJ is a regular 9-gon with center F and circumradius 12. Line V passes through points H and C. Line S is the angle bisector of angle FJB. Line M is the reflection of line S across line V. Point P is the reflection of point F across line M. Let Q be the reflection of point P across point G. Find the length of segment AQ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35.9", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center W and circumradius 7. Point O is the midpoint of segment WD. Point K is the reflection of point O across segment WA. Let Z be the perpendicular bisector of segment WA. Point V is the reflection of point C across line Z. Find the length of segment VK. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.4", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center Z and circumradius 11. Let L be the circumcircle of polygon ABCDEFGHIJ. Circle Q is the reflection of circle L across point I. Point X is the inversion of point G with respect to circle Q. Point M is the reflection of point B across point H. What is the length of segment XM? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21.02", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center P and circumradius 12. Let R be the circumcircle of polygon ABCD. Line Z is the angle bisector of angle DAP. Line N is the perpendicular bisector of segment BP. Let I be the reflection of circle R across line Z. Let H be the reflection of circle I across line N. Let S be the perpendicular bisector of segment AP. Point T is the inversion of point C with respect to circle H. Point L is the reflection of point T across line S. What is the distance between points T and L? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.74", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center T and circumradius 6. Point U is obtained by rotating point H by angle pi/6 about point B. Let W be the angle bisector of angle UDJ. Let Z be the perpendicular bisector of segment DC. Let N be the reflection of line W across line Z. Let P be the line through point U parallel to line N. Point X is the reflection of point K across line P. What is the distance between points D and X? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.07", "dataset": "MATH"}
{"messages": [{"content": "BCDEFG is a regular hexagon with center A and circumradius 10. Line H passes through points D and A. Let L be the reflection of line H through point C. Let W be the reflection of line L across line H. Let O be the reflection of point E across point B. Point Y is the reflection of point O across line W. Find the distance between points E and Y. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "55.68", "dataset": "MATH"}
{"messages": [{"content": "ABDEFGHI is a regular octagon with center C and circumradius 8. Point Y is the reflection of point H across point D. Let K be the midpoint of segment FD. Point Z is the midpoint of segment GY. Let Q be the reflection of point A across point Z. What is the length of segment KQ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.29", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center P and circumradius 7. Line I is the angle bisector of angle PDH. Line M passes through points E and C. Point V is the reflection of point F across line M. Point N is the reflection of point V across line I. What is the distance between points N and V? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCEFGHIJKLM be a regular dodecagon with center D and circumradius 8. Line Y is the perpendicular bisector of segment LK. Line X is the reflection of line Y across point A. Let R be the reflection of line X across line Y. Point Z is the reflection of point E across line X. Point S is the reflection of point F across line X. Line Q is the angle bisector of angle ZMS. Line P is the reflection of line R across line Q. Let V be the line through points I and C. Lines P and V intersect at point T. Find the length of segment ST. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "59.85", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center M and circumradius 12. Let R be the reflection of point C across point F. Let Q be the midpoint of segment GF. Let P be the line through points Q and J. Segment DI and line P intersect at point Y. What is the length of segment YR? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30.11", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center O and circumradius 12. Point N is the reflection of point D across point B. Let Y be the perpendicular bisector of segment CA. Point M is the reflection of point D across segment AN. Point K is the reflection of point M across line Y. Let J be the midpoint of segment MK. Find the distance between points J and A. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.24", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center T and circumradius 10. Let Q be the midpoint of segment LA. Let O be the line through point F parallel to segment GK. Point V is the reflection of point Q across line O. Find the length of segment VL. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.59", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center R and circumradius 5. Let O be the perpendicular bisector of segment LR. Line U is the reflection of line O across point C. Let M be the circumcircle of polygon ABCDEFGHIJKL. Let N be the reflection of circle M across line U. Point X is the inversion of point J with respect to circle N. Find the length of segment XG. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.11", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center S and circumradius 3. Let W be the circumcircle of polygon ABCDEFGHIJKL. Let Z be the angle bisector of angle HCD. Circle Y is the reflection of circle W across line Z. Line T is the perpendicular bisector of segment SK. Let P be the rotation of point D by angle 2pi/3 around point F. Circle N is the reflection of circle Y across line T. Point X is the midpoint of segment PJ. Let U be the inversion of point X in circle N. Point M is the reflection of point U across line Z. Find the distance between points M and J. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.28", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDFGHIJ be a regular 9-gon with center E and circumradius 10. Let N be the rotation of point A by angle 11pi/12 around point J. Point L is the reflection of point B across point N. Let S be the angle bisector of angle GCJ. Point Y is the reflection of point L across line S. Point Z is the reflection of point H across point Y. Find the length of segment ZI. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42.57", "dataset": "MATH"}
{"messages": [{"content": "EM = 8. ABCDFGHIJKL is a regular 11-gon with center M and circumradius 7. Line O is the perpendicular bisector of segment JI. Point Z is the midpoint of segment GC. Line T passes through point D perpendicular to line O. Line V passes through point K perpendicular to line T. Point Y is the reflection of point Z across line V. Find the length of segment LY. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.16", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center I and circumradius 8. Let L be the line through points G and I. Let T be the midpoint of segment BG. Point S is the reflection of point D across point F. Let Z be the perpendicular bisector of segment BT. Let V be the line through points S and F. Let Q be the reflection of line Z across line V. Lines L and Q intersect at point N. Point M is the midpoint of segment FE. What is the length of segment MN? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.39", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center Q and circumradius 8. Let X be the reflection of point Q across point A. Let H be the angle bisector of angle QXB. Point O is the reflection of point C across line H. Let M be the midpoint of segment EO. What is the distance between points M and A? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.72", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center X and circumradius 2. Let Y be the line through points C and G. Line K passes through points H and X. Let S be the reflection of line Y through point E. Lines K and S intersect at point Q. What is the length of segment QA? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.21", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center R and circumradius 1. Let J be the line through points B and D. Segment CF and line J intersect at point W. Let N be the midpoint of segment AW. Find the length of segment FN. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.09", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGIJKLM is a regular dodecagon with center H and circumradius 5. Let Q be the circumcircle of polygon ABCDEFGIJKLM. Let X be the midpoint of segment MJ. Let S be the reflection of circle Q through point D. Point R is the reflection of point K across point X. Let Z be the inversion of point E in circle S. Point N is the reflection of point F across segment RK. What is the length of segment NZ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.29", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center Q and circumradius 9. Let Z be the circumcircle of polygon ABCD. Let O be the midpoint of segment BA. Let V be the reflection of circle Z through point B. Point W is the inversion of point O with respect to circle V. Point M is the reflection of point W across point A. What is the distance between points M and D? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.6", "dataset": "MATH"}
{"messages": [{"content": "Let BCDEFGHI be a regular octagon with center A and circumradius 3. Point U is the reflection of point C across point I. Let K be the midpoint of segment FA. Let X be the line through point K parallel to segment BE. Point P is the reflection of point H across line X. Find the length of segment UP. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.44", "dataset": "MATH"}
{"messages": [{"content": "ABCEF is a regular 5-gon with center D and circumradius 2. Let K be the rotation of point B by angle 11pi/12 around point C. Point M is the reflection of point C across point K. Point T is the reflection of point F across point E. Point O is the reflection of point M across segment DK. What is the length of segment TO? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.47", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center U and circumradius 10. Line Z passes through points D and C. Let X be the line through point E parallel to line Z. Line Q is the reflection of line X across line Z. Point G is the midpoint of segment CD. Point V is the reflection of point G across line Q. Find the length of segment VC. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.12", "dataset": "MATH"}
{"messages": [{"content": "Let ABCEFGHIJKLM be a regular dodecagon with center D and circumradius 11. Let X be the reflection of point K across point J. Let Z be the perpendicular bisector of segment IK. Line R passes through points M and X. Let Y be the reflection of line R across line Z. Point S is the reflection of point X across line Y. Find the distance between points A and S. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.2", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJK be a regular 11-gon with center Q and circumradius 8. Line N is the perpendicular bisector of segment GF. Line O is the angle bisector of angle JQA. Let P be the line through point I parallel to line O. Line Z passes through point C parallel to line P. Let L be the reflection of line Z across line N. Lines Z and L intersect at point U. What is the measure of angle UQG, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.86", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center Q and circumradius 8. Line Z is the angle bisector of angle DFE. Let L be the perpendicular bisector of segment DE. Let Y be the rotation of point F by angle pi/12 around point A. Let I be the reflection of line Z across line L. Let R be the reflection of line L across line I. Point J is the reflection of point E across point D. Line V passes through point D parallel to line R. Let G be the angle bisector of angle DJF. Let O be the line through point Y parallel to line G. Point K is the intersection of lines V and O. What is the distance between points E and K? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43.06", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center S and circumradius 9. Line T is the angle bisector of angle CAF. Point U is the reflection of point H across line T. Let Z be the circumcircle of polygon ABCDEFGH. Circle L is the reflection of circle Z across point U. Point X is the inversion of point E with respect to circle L. Find the distance between points X and A. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.2", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center Q and circumradius 9. Let M be the circumcircle of polygon ABCDEFG. Let P be the reflection of circle M through point F. Point H is the inversion of point A with respect to circle P. Point J is the inversion of point G with respect to circle P. Point T is the reflection of point J across point G. Find the length of segment TH. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19.12", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center P and circumradius 10. Point O is obtained by rotating point C by angle pi about point B. Line R passes through point C perpendicular to segment CG. Line Y passes through point E parallel to line R. Let Q be the circumcircle of polygon ABCDEFG. Let W be the inversion of point O in circle Q. Let H be the reflection of circle Q through point W. Line L passes through points D and P. Point U is the reflection of point A across point G. Let I be the reflection of line Y across line L. Circle Z is the reflection of circle H across line I. Let S be the inversion of point U in circle Z. Find the distance between points S and U. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.38", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center Y and circumradius 4. Let M be the circumcircle of polygon ABCDEFGHIJ. Circle V is the reflection of circle M across point G. Let N be the reflection of point C across point B. Point L is the inversion of point N with respect to circle V. What is the length of segment IL? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.61", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGHIJ is a regular 9-gon with center F and circumradius 4. Let U be the perpendicular bisector of segment CD. Line T is the reflection of line U across point D. Let N be the line through point I perpendicular to line T. Point Q is the reflection of point A across line N. Find the measure of angle BQH, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.94", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center W and circumradius 10. Line H is the angle bisector of angle DBW. Line Y passes through points W and A. Line N passes through point A parallel to line H. Point U is the midpoint of segment WC. Point F is the reflection of point B across line N. Point M is the reflection of point F across line Y. What is the distance between points U and M? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26.93", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center I and circumradius 8. Line P is the perpendicular bisector of segment CI. Point Q is the reflection of point D across line P. Let N be the midpoint of segment DQ. Point R is the midpoint of segment QC. Find the length of segment RN. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.06", "dataset": "MATH"}
{"messages": [{"content": "BCDEFGHIJKLM is a regular dodecagon with center A and circumradius 12. Let S be the line through points M and J. Line Z is the perpendicular bisector of segment GF. Let T be the reflection of line Z through point E. Lines S and T intersect at point Y. What is the distance between points F and Y? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34.13", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center W and circumradius 12. Point N is on segment EG. Let Y be the perpendicular bisector of segment GB. Point L is the reflection of point N across line Y. What is the measure of angle ENL, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.18", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center X and circumradius 10. Let S be the angle bisector of angle XHE. Let J be the midpoint of segment FA. Point N is the reflection of point X across line S. Point T is the midpoint of segment NJ. Find the distance between points T and C. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.39", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center G and circumradius 11. Let F be the rotation of point A by angle pi/3 around point G. Point K is the midpoint of segment DF. Point X is the reflection of point B across point K. Point O is the midpoint of segment FX. Point H is the reflection of point O across point X. What is the length of segment KH? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.71", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center W and circumradius 9. Let I be the circumcircle of polygon ABCD. Let T be the reflection of circle I through point D. Let X be the midpoint of segment WB. Point G is the inversion of point X with respect to circle T. Find the length of segment GW. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.4", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHJK be a regular 10-gon with center I and circumradius 4. Line O passes through points H and B. Let P be the line through point K parallel to segment JE. Lines P and O intersect at point S. What is the length of segment SG? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.31", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center S and circumradius 11. Let N be the perpendicular bisector of segment ID. Point M is the midpoint of segment CD. Let X be the line through point M parallel to line N. Segment ID and line X intersect at point Y. Find the distance between points Y and G. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.65", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKM be a regular dodecagon with center L and circumradius 7. Let Z be the midpoint of segment DC. Point V is the reflection of point Z across segment MF. Find the length of segment ZV. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.06", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center R and circumradius 11. Let O be the line through points A and C. Point Y is the reflection of point F across line O. Point V is the reflection of point Y across segment BC. What is the distance between points V and B? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.5", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center X and circumradius 10. Point Y is the reflection of point G across point E. Let P be the midpoint of segment XA. Point S is obtained by rotating point P by angle 3pi/4 about point F. Point M is the reflection of point E across point P. Line R passes through points M and J. Let Z be the line through point Y parallel to line R. Let V be the reflection of line Z through point D. Let N be the midpoint of segment SL. Point T is the reflection of point N across line V. What is the distance between points T and I? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.44", "dataset": "MATH"}
{"messages": [{"content": "ABCEFGHIJKLM is a regular dodecagon with center D and circumradius 2. Let X be the line through points I and G. Let Q be the angle bisector of angle KFM. Let Y be the reflection of line X across line Q. Point R is the reflection of point G across line Y. What is the distance between points M and R? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.29", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center I and circumradius 11. Let K be the circumcircle of polygon ABCDEFGH. Let Z be the reflection of circle K through point B. Point J is the midpoint of segment FA. Point Q is the inversion of point I with respect to circle Z. Point O is the midpoint of segment CJ. What is the distance between points O and Q? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.56", "dataset": "MATH"}
{"messages": [{"content": "ABCDFGHI is a regular octagon with center E and circumradius 3. Line R passes through points C and D. Line Y passes through point G parallel to segment HF. Point P is the intersection of lines R and Y. What is the length of segment HP? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.42", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center Y and circumradius 4. Line K is the perpendicular bisector of segment YF. Point I is the reflection of point C across line K. Let U be the line through points G and A. Let W be the line through point I perpendicular to line U. Point O is the reflection of point Y across line W. Find the distance between points O and Y. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.6", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center Z and circumradius 5. Line P passes through point B perpendicular to segment GE. Point M lies on line P. Let W be the reflection of point F across point D. Find the measure of angle WBM, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.4", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJK be a regular 11-gon with center T and circumradius 7. Let N be the perpendicular bisector of segment FI. Point Z is the reflection of point A across point C. Line R is the reflection of line N across point I. Line U is the perpendicular bisector of segment IJ. Line L is the reflection of line U across line R. Line P passes through points H and Z. Point Q is the intersection of lines L and P. Find the length of segment BQ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.48", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center L and circumradius 12. Let K be the circumcircle of polygon ABCDEF. Point O is the midpoint of segment FL. Let H be the inversion of point O in circle K. Point R is the reflection of point L across point C. What is the distance between points R and H? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGIJK is a regular 10-gon with center H and circumradius 10. Line V is the angle bisector of angle GEI. Let U be the midpoint of segment BD. Let P be the line through point F perpendicular to line V. Point X is the reflection of point U across line P. Find the length of segment XE. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.48", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFG be a regular hexagon with center C and circumradius 2. Point R is obtained by rotating point A by angle pi/12 about point G. Point W is obtained by rotating point B by angle pi/12 about point A. Let T be the circumcircle of polygon ABDEFG. Let O be the perpendicular bisector of segment CA. Line L passes through points R and E. Let J be the reflection of circle T across line L. Point H is the inversion of point C with respect to circle J. Point U is the reflection of point W across line O. What is the length of segment HU? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18.8", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center R and circumradius 9. Let U be the circumcircle of polygon ABCDE. Let M be the reflection of circle U through point A. Let J be the perpendicular bisector of segment RB. Let T be the reflection of circle M across line J. Point L is the midpoint of segment EA. Point S is the inversion of point A with respect to circle T. Find the distance between points S and L. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.03", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center Y and circumradius 5. Let O be the reflection of point C across point A. Let Q be the perpendicular bisector of segment OY. Line R is the reflection of line Q across point F. Let V be the line through point Y parallel to line R. Point M is the intersection of segment DB and line V. What is the distance between points O and M? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.86", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center Z and circumradius 11. Line S is the angle bisector of angle ACB. Let M be the circumcircle of polygon ABCDEF. Let U be the reflection of circle M across line S. Let N be the inversion of point F in circle U. Find the length of segment NF. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19.68", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center S and circumradius 10. Line Q passes through point C parallel to segment FJ. Point Z is the reflection of point L across line Q. Point P is the midpoint of segment EZ. What is the distance between points K and P? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.32", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center R and circumradius 10. Line Y is the perpendicular bisector of segment CB. Let W be the line through point F parallel to line Y. Point U is the intersection of line W and segment AE. Find the distance between points U and D. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.65", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center X and circumradius 12. Let N be the line through points B and E. Let J be the line through point X parallel to line N. Point I is the midpoint of segment AC. Let U be the line through point D parallel to line J. Let H be the reflection of line J across line U. Point Y is the reflection of point I across line H. Find the length of segment EY. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46.17", "dataset": "MATH"}
{"messages": [{"content": "ABCEF is a regular 5-gon with center D and circumradius 10. Point T is the reflection of point B across point D. Line N is the perpendicular bisector of segment TB. Point M is the reflection of point C across segment BA. Let Z be the line through point F perpendicular to line N. Point J is the reflection of point C across line Z. What is the measure of angle JAM, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.89", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center L and circumradius 4. Let H be the circumcircle of polygon ABCDEF. Let P be the midpoint of segment DL. Point I is the inversion of point P with respect to circle H. Point V is the midpoint of segment CD. What is the distance between points I and V? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.29", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center X and circumradius 5. Let S be the reflection of point K across point C. Let O be the angle bisector of angle SGE. Line T is the reflection of line O across point J. Point N is the reflection of point E across line T. Find the distance between points I and N. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24.18", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center L and circumradius 3. Let N be the line through point E perpendicular to segment BE. Point T is the reflection of point G across segment BE. Point X is the reflection of point T across line N. Find the length of segment LX. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.71", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDFGHIJK be a regular 10-gon with center E and circumradius 9. Let V be the reflection of point I across point B. Let T be the perpendicular bisector of segment EV. Let U be the angle bisector of angle CIV. Let Q be the reflection of line T through point I. Let Z be the reflection of line U through point C. Lines Q and Z intersect at point L. Find the distance between points L and G. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25.45", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center M and circumradius 9. Point L is the reflection of point M across point A. Point G is the reflection of point B across point F. Let N be the midpoint of segment GL. Let X be the midpoint of segment EM. What is the distance between points N and X? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.59", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center I and circumradius 9. Let T be the perpendicular bisector of segment ID. Line M is the reflection of line T across point E. Let Y be the circumcircle of polygon ABCDEFG. Let Z be the reflection of circle Y through point C. Let N be the midpoint of segment IC. Point L is the reflection of point C across segment ND. Line M and segment ND intersect at point R. Let J be the inversion of point L in circle Z. What is the distance between points R and J? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.23", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center P and circumradius 7. Point Q is the reflection of point D across point C. Let T be the reflection of point Q across point E. Let I be the midpoint of segment CQ. Point V is the reflection of point P across segment IT. Find the distance between points V and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.76", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center S and circumradius 9. Let J be the angle bisector of angle CDE. Line O is the perpendicular bisector of segment EB. Line M is the reflection of line J across line O. Point U is the reflection of point B across point C. Let Q be the line through point E parallel to segment CA. Lines Q and M intersect at point R. What is the length of segment RU? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39.71", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center N and circumradius 10. Let K be the line through points F and E. Let S be the line through point C parallel to line K. Let L be the reflection of line S across line K. Point T is the reflection of point F across line L. What is the measure of angle BFT, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.51", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGI be a regular octagon with center H and circumradius 6. Let X be the midpoint of segment IB. Let W be the circumcircle of polygon ABCDEFGI. Let P be the inversion of point X in circle W. Line Q is the perpendicular bisector of segment PE. Let J be the reflection of circle W through point P. Circle Z is the reflection of circle J across line Q. Point S is the inversion of point G with respect to circle Z. What is the length of segment DS? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.6", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center N and circumradius 12. Line I is the perpendicular bisector of segment NE. Let S be the midpoint of segment NC. Let T be the rotation of point S by angle pi/3 around point C. Line U passes through points S and D. Point R is the midpoint of segment SD. Let L be the reflection of line U across line I. Let J be the angle bisector of angle SRT. Point Y is the reflection of point B across point E. Let P be the reflection of line J across line L. Point K is the reflection of point D across line P. What is the length of segment KY? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30.46", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEFGHI be a regular octagon with center B and circumradius 10. Let X be the reflection of point F across point G. Let Z be the line through points H and X. Line S passes through point E perpendicular to line Z. Point K is the reflection of point F across line Z. Point N is the reflection of point K across line S. What is the distance between points N and H? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGI is a regular octagon with center H and circumradius 7. Point T is the reflection of point G across point I. Let L be the midpoint of segment BD. Point M is obtained by rotating point C by angle 3pi/4 about point D. Line O is the perpendicular bisector of segment MF. Point Q is the reflection of point T across line O. Let Z be the perpendicular bisector of segment EQ. Point P is the reflection of point M across line Z. What is the length of segment LP? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.35", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGHIJKLM is a regular dodecagon with center F and circumradius 1. Line R is the angle bisector of angle DHI. Point W is the reflection of point F across line R. Line S passes through points I and E. Point X is the reflection of point L across line S. What is the distance between points X and W? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.13", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJLM is a regular dodecagon with center K and circumradius 7. Let N be the line through points A and K. Line R passes through point C parallel to line N. Line V is the reflection of line R across point H. Point T is the reflection of point L across line V. Find the distance between points T and I. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.65", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEF be a regular 5-gon with center B and circumradius 6. Line T passes through points B and F. Let V be the line through points A and D. Line Z passes through point D parallel to line T. Line L is the reflection of line Z across point B. Let M be the reflection of point D across point B. Point Y is the midpoint of segment DA. Point W is the midpoint of segment MY. Point G is the intersection of lines L and V. What is the distance between points G and W? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.86", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center S and circumradius 1. Let T be the midpoint of segment BS. Let O be the perpendicular bisector of segment AT. Point X is the reflection of point B across line O. Let V be the midpoint of segment DX. Let Q be the reflection of point B across point V. Point J is the midpoint of segment FE. What is the length of segment JQ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.5", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center K and circumradius 6. Line L is the angle bisector of angle FCG. Let X be the circumcircle of polygon ABCDEFGH. Let O be the reflection of circle X through point H. Let Q be the inversion of point A in circle O. Point U is the reflection of point E across line L. Find the distance between points Q and U. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.93", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center R and circumradius 3. Let W be the reflection of point B across point R. Let K be the circumcircle of polygon ABCDE. Circle Y is the reflection of circle K across point C. Point Q is the inversion of point B with respect to circle Y. Let I be the perpendicular bisector of segment EW. Point J is the reflection of point B across line I. Find the distance between points J and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.44", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center S and circumradius 3. Let Q be the midpoint of segment BA. Point Y is the reflection of point G across point I. Let O be the line through point G parallel to segment BA. Point K is the reflection of point Y across line O. Find the distance between points K and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.49", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center R and circumradius 2. Let K be the rotation of point E by angle 2pi/3 around point G. Point Z is the midpoint of segment IB. Let P be the line through points E and Z. Line Y passes through point G parallel to line P. Point N is the reflection of point K across line Y. Find the distance between points N and K. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.05", "dataset": "MATH"}
{"messages": [{"content": "ACDEFGHIJKLM is a regular dodecagon with center B and circumradius 1. Point T is the midpoint of segment JG. Let R be the midpoint of segment KT. Line V passes through point L parallel to segment GJ. Point N is the intersection of line V and segment RD. Find the measure of angle KHN, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.05", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJK be a regular 11-gon with center Z and circumradius 9. Point P is the reflection of point K across point I. Let X be the midpoint of segment ZK. Let L be the line through points X and P. Point U is the intersection of line L and segment JZ. What is the length of segment GU? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.53", "dataset": "MATH"}
{"messages": [{"content": "Let BCDEFGHI be a regular octagon with center A and circumradius 7. Point N is the reflection of point H across point D. Line L is the angle bisector of angle ADC. Line O passes through point N parallel to line L. Point T is the reflection of point I across line O. What is the length of segment DT? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22.09", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGIJKLM is a regular dodecagon with center H and circumradius 7. Point Y is the reflection of point G across segment DI. Let V be the line through points A and G. Let X be the line through point Y parallel to segment EH. Point U is the reflection of point E across point M. Point W is the intersection of lines V and X. What is the length of segment WU? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.61", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center O and circumradius 5. Point S is the midpoint of segment AF. Let I be the midpoint of segment CD. Let G be the midpoint of segment AS. Point N is the midpoint of segment IO. Find the distance between points N and G. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.61", "dataset": "MATH"}
{"messages": [{"content": "ABDEFGHI is a regular octagon with center C and circumradius 2. Line N passes through points I and H. Line P is the perpendicular bisector of segment AC. Let S be the reflection of line P across line N. Lines S and N intersect at point J. What is the distance between points J and H? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.08", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFHIJK be a regular 10-gon with center G and circumradius 2. Let Z be the rotation of point B by angle pi/12 around point J. Line U is the angle bisector of angle EIK. Line X passes through point B parallel to line U. Point N lies on line X. Let S be the line through point N parallel to segment GZ. Let Q be the line through point A parallel to line S. Point L is the reflection of point B across line Q. What is the length of segment LD? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.18", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center W and circumradius 8. Let T be the perpendicular bisector of segment BA. Let U be the line through points D and J. Line M is the reflection of line T across line U. Point O is the reflection of point G across line M. Find the length of segment JO. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center X and circumradius 3. Point M is the midpoint of segment CD. Point I is the midpoint of segment GM. Let N be the reflection of point C across point I. What is the length of segment EN? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.95", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center L and circumradius 3. Point X is the reflection of point C across point B. Let O be the perpendicular bisector of segment DL. Line K passes through point L perpendicular to line O. Point P is the midpoint of segment AC. Point J is the reflection of point X across line K. Find the distance between points J and P. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.41", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJK is a regular 11-gon with center L and circumradius 9. Let V be the line through points H and A. Point Q is the reflection of point J across line V. Point P is the reflection of point C across line V. Point R is the reflection of point L across line V. Point M is the reflection of point R across point Q. Find the distance between points M and P. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24.84", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center N and circumradius 7. Let K be the midpoint of segment CB. Let P be the midpoint of segment CN. Let U be the reflection of point K across point B. Let W be the circumcircle of polygon ABCDEFGH. Let T be the inversion of point P in circle W. Let R be the reflection of point U across point A. Find the length of segment TR. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19.09", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center K and circumradius 8. Line P passes through points F and A. Point Y is the reflection of point H across line P. Let M be the midpoint of segment YK. What is the length of segment CM? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.93", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center N and circumradius 2. Let T be the circumcircle of polygon ABCDEFG. Point S is the reflection of point E across point D. Let X be the midpoint of segment SD. Point R is the midpoint of segment CA. Let K be the inversion of point R in circle T. Find the length of segment XK. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.88", "dataset": "MATH"}
{"messages": [{"content": "JY = 2. ABCDEFGH is a regular octagon with center Y and circumradius 3. Let I be the circumcircle of polygon ABCDEFGH. Let T be the reflection of circle I through point F. Let O be the inversion of point G in circle T. Let S be the midpoint of segment AH. Find the distance between points O and S. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.51", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGI be a regular octagon with center H and circumradius 12. Point W is the midpoint of segment CI. Let Q be the reflection of point H across point W. Let Z be the angle bisector of angle WEQ. Point K is the reflection of point A across line Z. Find the distance between points K and D. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19.66", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEFGHI be a regular octagon with center B and circumradius 3. Line Z passes through points B and A. Let O be the circumcircle of polygon ACDEFGHI. Line T passes through point I parallel to line Z. Let P be the line through point H parallel to line T. Circle R is the reflection of circle O across line P. Point W is the inversion of point F with respect to circle R. What is the distance between points G and W? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.08", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center R and circumradius 10. Let Q be the angle bisector of angle IFH. Let V be the circumcircle of polygon ABCDEFGHIJKL. Let Y be the reflection of circle V across line Q. Let O be the inversion of point R in circle Y. Find the length of segment OC. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.55", "dataset": "MATH"}
{"messages": [{"content": "ABCDF is a regular 5-gon with center E and circumradius 12. Point X is obtained by rotating point C by angle 7pi/12 about point F. Point W is the reflection of point A across point X. Let L be the reflection of point W across point B. Let J be the circumcircle of polygon ABCDF. Let P be the reflection of point L across point X. Let Z be the inversion of point P in circle J. Point K is the reflection of point F across point Z. Find the distance between points K and W. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "73.85", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJK be a regular 11-gon with center M and circumradius 2. Line P is the angle bisector of angle DCK. Let U be the midpoint of segment FK. Let Q be the reflection of point A across point H. Point X is the reflection of point Q across line P. What is the distance between points U and X? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.05", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDFGHI be a regular octagon with center E and circumradius 6. Point T is the reflection of point B across point D. Point V is obtained by rotating point T by angle 5pi/6 about point A. Let W be the rotation of point V by angle 3pi/4 around point G. Point L is the reflection of point E across segment GW. Let P be the line through points L and G. Point R is the reflection of point C across line P. Let Y be the reflection of point C across point R. Find the length of segment HY. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27.31", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center G and circumradius 11. Let M be the rotation of point B by angle pi/12 around point C. Line Z passes through point C perpendicular to segment CB. Line L passes through point D parallel to line Z. Let X be the midpoint of segment ME. Point S is the reflection of point X across line L. Find the length of segment SA. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34.63", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center L and circumradius 11. Point P is the midpoint of segment EL. Let W be the midpoint of segment DB. Point R is the midpoint of segment PA. Let X be the angle bisector of angle HFG. Point M is the reflection of point W across line X. What is the distance between points R and M? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26.79", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIKLM be a regular dodecagon with center J and circumradius 9. Let Z be the angle bisector of angle MGH. Point T is the reflection of point L across line Z. Point U is the reflection of point T across segment ME. Let V be the midpoint of segment LC. Let P be the midpoint of segment UV. Find the length of segment PA. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.79", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center Q and circumradius 7. Point H is obtained by rotating point B by angle pi about point A. Let P be the midpoint of segment DB. Point J is the midpoint of segment BP. Let Z be the perpendicular bisector of segment CH. Point N is the reflection of point J across line Z. What is the distance between points E and N? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.62", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center V and circumradius 1. Let S be the angle bisector of angle BFV. Let U be the line through point H parallel to line S. Line P passes through point D parallel to line U. Point Q is the reflection of point F across line P. What is the distance between points B and Q? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.56", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDFGHIJKLM be a regular dodecagon with center E and circumradius 3. Let O be the reflection of point H across point I. Let X be the midpoint of segment LK. Point Y is the reflection of point X across point O. Line V is the perpendicular bisector of segment LK. Point S is the reflection of point Y across line V. Find the length of segment AS. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.06", "dataset": "MATH"}
{"messages": [{"content": "BCDEFGHIJ is a regular 9-gon with center A and circumradius 1. Let V be the midpoint of segment IG. Line X passes through point D parallel to segment FA. Point M is the reflection of point V across line X. Find the length of segment CM. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.96", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center J and circumradius 8. Let U be the midpoint of segment AB. Point Z is the midpoint of segment UC. Line K is the angle bisector of angle EDA. Point O is the reflection of point U across line K. Point P is the reflection of point O across segment BZ. Find the distance between points P and G. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24.92", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGI be a regular octagon with center H and circumradius 9. Line L passes through points I and E. Point W is the reflection of point F across line L. Point P is the midpoint of segment WB. Find the length of segment PI. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.76", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center V and circumradius 8. Let O be the line through point B perpendicular to segment AB. Point H is the reflection of point V across line O. Point X is on segment AB. Find the measure of angle XBH, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.2", "dataset": "MATH"}
{"messages": [{"content": "Let ABCE be a square with center D and circumradius 7. Let I be the circumcircle of polygon ABCE. Line Z is the perpendicular bisector of segment BE. Line R passes through point B parallel to line Z. Let H be the reflection of circle I across line R. Point J is the inversion of point C with respect to circle H. Point M is the reflection of point D across segment JA. Let Y be the inversion of point A in circle H. Point Q is the reflection of point M across segment YJ. Find the length of segment QJ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.29", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKM be a regular dodecagon with center L and circumradius 11. Let Q be the reflection of point E across point D. Line P is the perpendicular bisector of segment FI. Line T passes through point J parallel to line P. Point V is the reflection of point L across line T. Let X be the midpoint of segment GV. Find the length of segment QX. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24.19", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEFGHIJKLM be a regular dodecagon with center B and circumradius 7. Let Y be the line through points K and H. Line U passes through point D parallel to line Y. Let Z be the midpoint of segment GE. Let P be the reflection of line U through point Z. Let N be the circumcircle of polygon ACDEFGHIJKLM. Circle T is the reflection of circle N across line P. Point O is the inversion of point Z with respect to circle T. What is the length of segment OB? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.25", "dataset": "MATH"}
{"messages": [{"content": "ABCDFGHIJKLM is a regular dodecagon with center E and circumradius 2. Line Y is the perpendicular bisector of segment KC. Let X be the line through point C parallel to line Y. Let P be the line through points E and J. Line Z is the reflection of line P across line X. Point Q is the reflection of point K across line Z. Let T be the midpoint of segment QH. What is the measure of angle EMT, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.52", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFG be a regular hexagon with center C and circumradius 9. Point R is the reflection of point F across point E. Let H be the reflection of point R across point C. Let X be the perpendicular bisector of segment HC. Let Z be the line through point E parallel to line X. Point N is the intersection of segment RA and line Z. Find the distance between points G and N. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.7", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center Y and circumradius 3. Line Q is the angle bisector of angle HGA. Let M be the line through point F perpendicular to line Q. Point T is the reflection of point Y across line M. Point R is the reflection of point T across line Q. Find the length of segment RY. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.49", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGIJKL is a regular 11-gon with center H and circumradius 8. Let R be the angle bisector of angle CIA. Line V is the reflection of line R across point J. Let N be the midpoint of segment GL. Line X passes through point F parallel to line V. Let Y be the midpoint of segment GF. Let M be the midpoint of segment CY. Point P is the reflection of point M across line X. What is the distance between points P and N? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.91", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center L and circumradius 9. Line O is the perpendicular bisector of segment GL. Let M be the line through point F parallel to line O. Line Y is the reflection of line O across line M. Point S is the reflection of point A across line Y. Find the length of segment FS. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18.38", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center P and circumradius 3. Line Y is the angle bisector of angle BPE. Let Z be the line through point A parallel to line Y. Point L is the midpoint of segment EP. Point H is the intersection of segment BL and line Z. What is the distance between points H and B? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.5", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center X and circumradius 1. Point I is the reflection of point C across point G. Let M be the perpendicular bisector of segment XD. Point Q is the reflection of point F across line M. Let N be the midpoint of segment EI. Point S is the midpoint of segment NQ. Find the length of segment SI. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.45", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center O and circumradius 1. Let S be the circumcircle of polygon ABCDEFGHIJKL. Let Q be the reflection of circle S through point A. Line N is the perpendicular bisector of segment IH. Let W be the inversion of point F in circle Q. Let Z be the line through point W perpendicular to line N. Point X is the reflection of point L across line Z. What is the length of segment XJ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.89", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center V and circumradius 2. Let Z be the angle bisector of angle EDC. Line G is the angle bisector of angle ECV. Point M is the reflection of point B across line G. Point K is the reflection of point M across line Z. Point L is the reflection of point M across point A. Point U is the reflection of point L across point K. What is the length of segment MU? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.58", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center N and circumradius 8. Let S be the line through points G and I. Line Q passes through point L parallel to line S. Segment NB and line Q intersect at point M. Find the distance between points M and L. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.93", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center O and circumradius 5. Line L passes through points B and A. Point K is the reflection of point E across line L. Point M is the reflection of point K across point B. Point T is the reflection of point C across point M. Find the length of segment KT. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center P and circumradius 6. Let L be the perpendicular bisector of segment PB. Point R is the reflection of point A across line L. Point G is the midpoint of segment AR. What is the length of segment BG? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.45", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center K and circumradius 4. Let W be the rotation of point B by angle 2pi/3 around point H. Point J is obtained by rotating point W by angle pi/6 about point A. Line S is the perpendicular bisector of segment JG. Point M is the reflection of point B across line S. Point Z is the midpoint of segment HA. Find the length of segment MZ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.71", "dataset": "MATH"}
{"messages": [{"content": "ABCDFG is a regular hexagon with center E and circumradius 5. Point T is the midpoint of segment AB. Point O is the reflection of point T across point C. Let S be the midpoint of segment EC. Find the length of segment OS. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.66", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGHIJK is a regular 10-gon with center F and circumradius 5. Let S be the line through point D perpendicular to segment GB. Line O is the reflection of line S across point G. Let Y be the angle bisector of angle AEI. Line N is the reflection of line Y across line O. Point L is the reflection of point I across line N. What is the length of segment LF? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31.93", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center Z and circumradius 10. Line R is the angle bisector of angle HFD. Let L be the line through point A parallel to line R. Line X is the reflection of line L across line R. Let T be the midpoint of segment ZD. Point Y is the reflection of point T across line X. What is the distance between points B and Y? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.55", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center L and circumradius 12. Line X is the perpendicular bisector of segment BI. Line J passes through point B parallel to line X. Let S be the reflection of line J through point D. Point V is the reflection of point C across line S. Let R be the reflection of point V across point E. What is the length of segment IR? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33.87", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center I and circumradius 6. Let O be the perpendicular bisector of segment FE. Line P passes through point B parallel to line O. Let Y be the line through point E parallel to line P. Point V is the reflection of point F across line Y. Find the length of segment AV. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.87", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center P and circumradius 6. Let M be the midpoint of segment CF. Let N be the midpoint of segment FM. Let T be the midpoint of segment BP. Find the distance between points N and T. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.47", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFHIJKLM is a regular dodecagon with center G and circumradius 9. Point R is the reflection of point C across point D. Let X be the line through points K and R. Line Z passes through point L parallel to segment FG. Point V is the intersection of lines X and Z. Find the distance between points V and M. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.83", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEFGHI be a regular octagon with center B and circumradius 2. Line W is the angle bisector of angle HEA. Let O be the midpoint of segment CI. Line Q is the reflection of line W across point A. Point P is the reflection of point G across line Q. Find the length of segment OP. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.73", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center W and circumradius 8. Let L be the line through points W and C. Let K be the line through point F parallel to line L. Let M be the reflection of line K through point D. Point R is the reflection of point H across line M. Find the length of segment RW. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.77", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center G and circumradius 5. Let Y be the line through points A and E. Let L be the perpendicular bisector of segment ED. Let M be the perpendicular bisector of segment EC. Let V be the rotation of point B by angle pi/3 around point E. Line O is the reflection of line Y across point V. Let K be the reflection of line L across line M. Lines O and K intersect at point I. Let R be the reflection of point C across point I. What is the length of segment GR? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32.85", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center X and circumradius 4. Let L be the line through point G perpendicular to segment DE. Point K is the reflection of point A across line L. Point S is the midpoint of segment FK. What is the length of segment SB? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.01", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center H and circumradius 7. Let Q be the line through points C and B. Line R is the angle bisector of angle FBA. Line L is the reflection of line Q across line R. Point Z is the reflection of point E across line L. Find the length of segment DZ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center J and circumradius 1. Let N be the circumcircle of polygon ABCDE. Let V be the reflection of point J across point C. Let K be the inversion of point V in circle N. Let H be the midpoint of segment JK. What is the measure of angle DCH, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.94", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center Q and circumradius 3. Let Y be the angle bisector of angle FEA. Let Z be the reflection of line Y through point B. Point X is the reflection of point L across line Z. Point T is the midpoint of segment XA. What is the length of segment TD? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.76", "dataset": "MATH"}
{"messages": [{"content": "ABCEF is a regular 5-gon with center D and circumradius 4. Let W be the perpendicular bisector of segment BF. Line S passes through point D perpendicular to line W. Line R is the reflection of line W across point C. Let M be the line through point F perpendicular to line R. Let I be the reflection of line S across line M. Point V is the reflection of point F across line I. Point N is the reflection of point C across point F. What is the length of segment VN? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.47", "dataset": "MATH"}
{"messages": [{"content": "Let BCDEF be a regular 5-gon with center A and circumradius 1. Point T is the midpoint of segment FD. Let P be the reflection of point F across point C. Let S be the angle bisector of angle PAB. Point U is the reflection of point F across line S. Point V is the reflection of point U across point T. Find the distance between points V and A. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.88", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFHI be a regular octagon with center G and circumradius 1. Let J be the line through point H perpendicular to segment AE. Let Y be the reflection of line J through point D. Point T is the reflection of point H across line Y. What is the length of segment IT? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.55", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEFGHIJ be a regular 9-gon with center B and circumradius 11. Let L be the midpoint of segment IG. Point X is the reflection of point A across segment IG. Point K is the reflection of point L across segment HX. What is the distance between points K and I? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.84", "dataset": "MATH"}
{"messages": [{"content": "Let ABCEFG be a regular hexagon with center D and circumradius 6. Point R is the midpoint of segment CE. Let U be the circumcircle of polygon ABCEFG. Line J passes through points A and B. Let Z be the inversion of point R in circle U. Line K is the reflection of line J across point Z. Point M is the reflection of point A across line K. What is the distance between points M and Z? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27.5", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGIJK is a regular 10-gon with center H and circumradius 5. Point V is the midpoint of segment JE. Let W be the midpoint of segment FK. Point O is the midpoint of segment IW. What is the distance between points V and O? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.22", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHJKL is a regular 11-gon with center I and circumradius 8. Let Z be the perpendicular bisector of segment IE. Let S be the reflection of point J across point H. Point Y is the reflection of point C across segment BA. Let U be the circumcircle of polygon ABCDEFGHJKL. Let T be the reflection of circle U across line Z. Point O is the inversion of point S with respect to circle T. Find the distance between points Y and O. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.36", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIKLM be a regular dodecagon with center J and circumradius 5. Let O be the midpoint of segment JG. Let P be the line through point O perpendicular to segment KH. Point T is the reflection of point D across line P. Find the length of segment TD. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.67", "dataset": "MATH"}
{"messages": [{"content": "Let ACDE be a square with center B and circumradius 10. Let W be the reflection of point E across point A. Let G be the rotation of point W by angle 5pi/6 around point D. Let N be the reflection of point B across point C. Let J be the midpoint of segment NE. Let H be the reflection of point D across point C. Point P is the midpoint of segment HG. Line Z is the angle bisector of angle CPW. Point I is the reflection of point H across line Z. Find the distance between points J and I. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.99", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center X and circumradius 7. Point K is the reflection of point J across point H. Point U is obtained by rotating point E by angle 5pi/6 about point I. Let T be the reflection of point B across point D. Point M is the midpoint of segment ET. Let Z be the line through points T and H. Let R be the line through points K and X. Let L be the line through point U parallel to line Z. Point S is the intersection of lines R and L. Point W is the reflection of point S across segment AB. Find the distance between points M and W. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26.5", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center R and circumradius 6. Point Y is the reflection of point R across point D. Let V be the line through points E and A. Let P be the line through points D and C. Point X is the intersection of lines P and V. Point U is the midpoint of segment YX. What is the distance between points U and X? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.63", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center Z and circumradius 6. Point K is the midpoint of segment AC. Point P is the midpoint of segment ZC. Let X be the circumcircle of polygon ABCDE. Let N be the inversion of point P in circle X. Find the measure of angle NZK, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.26", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center J and circumradius 6. Let K be the rotation of point C by angle 5pi/12 around point J. Let P be the reflection of point D across point B. Let Q be the midpoint of segment CK. Let G be the midpoint of segment PK. Find the distance between points G and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.94", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHI be a regular 9-gon with center Q and circumradius 7. Point J is the reflection of point Q across point B. Let W be the midpoint of segment JF. Segments EH and FD intersect at point O. What is the distance between points O and W? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.8", "dataset": "MATH"}
{"messages": [{"content": "BCDE is a square with center A and circumradius 2. Point J is obtained by rotating point E by angle pi/12 about point B. Let L be the line through point J perpendicular to segment CB. Point T is the reflection of point D across line L. Point U is the reflection of point T across point E. Let I be the midpoint of segment BE. What is the length of segment IU? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.52", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEF be a regular 5-gon with center B and circumradius 12. Point O is the midpoint of segment AF. Let W be the midpoint of segment AE. Let X be the reflection of point W across point B. Point L is the reflection of point X across point D. Let I be the midpoint of segment WO. Find the length of segment LI. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27.38", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFGH be a regular 7-gon with center C and circumradius 12. Let I be the reflection of point B across point C. Let Q be the midpoint of segment IA. Let S be the line through points A and H. Point M is the reflection of point Q across line S. Find the distance between points M and A. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.81", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center V and circumradius 8. Let W be the midpoint of segment FH. Let L be the circumcircle of polygon ABCDEFGH. Circle X is the reflection of circle L across point W. Let P be the inversion of point E in circle X. Find the distance between points D and P. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.53", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center P and circumradius 11. Let U be the rotation of point P by angle pi around point F. Line T is the angle bisector of angle DBU. Let S be the line through points A and C. Let X be the reflection of line S through point U. Let V be the line through point D perpendicular to line S. Let L be the reflection of line V across line T. Lines L and X intersect at point H. What is the length of segment AH? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33.86", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center K and circumradius 11. Let T be the angle bisector of angle CAF. Point P is on line T. Let N be the line through points C and P. Point U is the reflection of point B across line N. Find the distance between points C and U. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.42", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFHIJKLM is a regular dodecagon with center G and circumradius 6. Let Z be the midpoint of segment FJ. Line P is the angle bisector of angle KZE. Let V be the line through point D parallel to line P. Line R passes through point B perpendicular to line V. Point U is the reflection of point L across line R. What is the distance between points C and U? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.59", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center W and circumradius 12. Line Q passes through points D and B. Line T is the reflection of line Q across point A. Let O be the line through points C and E. Point U is the intersection of lines O and T. Line H passes through point A perpendicular to line Q. Point Z is on line H. What is the measure of angle UAZ, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.2", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center S and circumradius 10. Line W passes through points C and F. Point R lies on line W. Let U be the angle bisector of angle ESR. Point Z is the reflection of point L across line U. What is the distance between points Z and S? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHI be a regular 9-gon with center O and circumradius 7. Point M is the reflection of point A across segment HF. Let Q be the reflection of point A across point I. Let Z be the reflection of point M across point H. What is the distance between points Z and Q? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.58", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center T and circumradius 11. Let W be the rotation of point E by angle pi/12 around point D. Line V is the perpendicular bisector of segment BW. Let Z be the angle bisector of angle CAT. Let X be the circumcircle of polygon ABCDEF. Point G is the reflection of point F across line V. Let S be the reflection of point B across point G. Line K is the reflection of line Z across point C. Let L be the reflection of circle X through point F. Line H is the reflection of line K across line V. Let R be the line through point W parallel to line H. Let M be the reflection of circle L across line R. Point J is the inversion of point S with respect to circle M. Find the distance between points S and J. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.8", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center Y and circumradius 7. Line O passes through points L and K. Point Q is obtained by rotating point F by angle pi/4 about point K. Point T is the midpoint of segment AQ. Line P passes through point T perpendicular to line O. Let X be the reflection of line P through point C. Point M is the reflection of point K across line X. Find the length of segment MA. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30.39", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center O and circumradius 6. Let F be the line through points A and D. Let E be the line through point C perpendicular to line F. Point T is the reflection of point O across line E. Line Q passes through point T perpendicular to line E. Let J be the midpoint of segment TB. Let W be the reflection of line Q through point A. Let G be the reflection of point J across point O. Point R is the reflection of point G across line W. What is the length of segment TR? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18.25", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center V and circumradius 4. Line R passes through point G perpendicular to segment KL. Point Z is the reflection of point E across line R. Line Y passes through points A and G. Let O be the line through points E and Z. Point Q is the intersection of lines O and Y. Find the distance between points Q and J. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.77", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center Z and circumradius 5. Let W be the circumcircle of polygon ABCDEFGHIJKL. Let M be the reflection of point K across point L. Let Y be the perpendicular bisector of segment ML. Let O be the reflection of circle W across line Y. Point V is the inversion of point L with respect to circle O. What is the length of segment VL? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.16", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center Z and circumradius 5. Line K is the angle bisector of angle EAC. Point Y is the midpoint of segment BD. Let Q be the midpoint of segment YD. Let O be the angle bisector of angle EQY. Point J is the reflection of point B across line K. Point W is the reflection of point J across point Z. Point F is the reflection of point Z across line O. Point N is the reflection of point F across point W. What is the distance between points D and N? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.26", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKM is a regular dodecagon with center L and circumradius 7. Point T is the midpoint of segment GK. Let Y be the midpoint of segment DT. Find the length of segment IY. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.46", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJK be a regular 11-gon with center S and circumradius 2. Let O be the perpendicular bisector of segment BJ. Let W be the circumcircle of polygon ABCDEFGHIJK. Let Z be the line through point F perpendicular to line O. Let T be the reflection of circle W across line Z. Let Q be the inversion of point G in circle T. Find the distance between points H and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.1", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center Q and circumradius 4. Let V be the circumcircle of polygon ABCDEFGHIJKL. Let M be the reflection of circle V through point B. Point S is the midpoint of segment BJ. Let P be the inversion of point E in circle M. Find the distance between points P and S. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.96", "dataset": "MATH"}
{"messages": [{"content": "Let BCDEFG be a regular hexagon with center A and circumradius 12. Point V is the reflection of point F across point G. Let Z be the circumcircle of polygon BCDEFG. Circle K is the reflection of circle Z across point F. Let S be the reflection of circle K through point A. Line I is the angle bisector of angle AVE. Let O be the reflection of circle S through point B. Point M is the reflection of point E across line I. Point H is the inversion of point F with respect to circle O. What is the distance between points H and M? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27.57", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEFGHI be a regular octagon with center B and circumradius 12. Line K is the perpendicular bisector of segment EC. Point U is the midpoint of segment HB. Line V is the reflection of line K across point I. Point Z is on line V. Let J be the midpoint of segment UZ. Find the measure of angle DCJ, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.96", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center W and circumradius 1. Point Y is obtained by rotating point D by angle 2pi/3 about point E. Point N is the reflection of point G across point C. Let O be the line through points H and A. Let T be the perpendicular bisector of segment CN. Point X is the reflection of point J across line O. Point M is the reflection of point Y across line T. Find the distance between points X and M. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.86", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center Y and circumradius 6. Point Z is obtained by rotating point E by angle 3pi/4 about point F. Let J be the circumcircle of polygon ABCDEFGH. Circle Q is the reflection of circle J across point C. Let O be the line through points E and Z. Let S be the reflection of circle Q across line O. Point W is the inversion of point C with respect to circle S. Find the distance between points Y and W. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.45", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center Y and circumradius 12. Let J be the line through points D and H. Let X be the reflection of point C across point H. Point M is the midpoint of segment CF. Point Z is the reflection of point M across line J. Find the length of segment XZ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38.13", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center Y and circumradius 1. Point O is obtained by rotating point C by angle 11pi/12 about point D. Point G is the midpoint of segment DC. Let U be the midpoint of segment OG. Point X is the midpoint of segment ED. Point H is the reflection of point Y across point X. Find the length of segment UH. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.82", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center L and circumradius 12. Point Q is the midpoint of segment CE. Point M is the reflection of point Q across point F. Point N is the reflection of point B across segment DC. Find the length of segment MN. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51.26", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKM be a regular dodecagon with center L and circumradius 12. Point T is obtained by rotating point D by angle pi/4 about point J. Let P be the midpoint of segment KG. Point Q is the midpoint of segment TL. Let X be the circumcircle of polygon ABCDEFGHIJKM. Point W is the inversion of point P with respect to circle X. What is the length of segment WQ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.53", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center U and circumradius 2. Point Y is the midpoint of segment KC. Line M is the perpendicular bisector of segment YG. Point O is the reflection of point F across line M. Let R be the midpoint of segment YG. Point V is the reflection of point O across segment YG. What is the length of segment RV? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.59", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center Y and circumradius 7. Point I is obtained by rotating point E by angle 2pi/3 about point G. Line N is the angle bisector of angle GBI. Let X be the reflection of line N through point H. Point Z is the reflection of point A across line X. Let M be the reflection of point Z across point Y. What is the length of segment DM? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.74", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center W and circumradius 7. Let Q be the perpendicular bisector of segment HJ. Let V be the line through point W perpendicular to line Q. Let U be the reflection of line V through point E. Point P is the reflection of point A across line U. What is the length of segment PB? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22.65", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center L and circumradius 11. Let I be the perpendicular bisector of segment EC. Let U be the midpoint of segment FD. Point X is the reflection of point U across line I. Find the length of segment AX. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.55", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center X and circumradius 12. Line N is the angle bisector of angle CHG. Line U passes through point B perpendicular to line N. Let W be the line through point D parallel to line U. Point Z is the reflection of point E across line W. Find the length of segment GZ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18.52", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center U and circumradius 3. Point O is the reflection of point A across point C. Let Y be the midpoint of segment BO. Point N is the midpoint of segment CY. Point T is the midpoint of segment CN. What is the length of segment TD? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.89", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center M and circumradius 12. Line Z passes through points D and E. Let I be the midpoint of segment DA. Point Q is the reflection of point B across line Z. Let N be the midpoint of segment AQ. What is the length of segment IN? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.41", "dataset": "MATH"}
{"messages": [{"content": "BCDEFG is a regular hexagon with center A and circumradius 1. Let P be the perpendicular bisector of segment DB. Point L is the midpoint of segment DA. Point T is the reflection of point L across line P. Point Y is the reflection of point A across point B. Find the length of segment TY. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.5", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGHI is a regular octagon with center F and circumradius 9. Let Y be the angle bisector of angle BID. Point Q is the reflection of point H across line Y. Line U is the reflection of line Y across point F. Let P be the reflection of point Q across point H. Line U and segment AG intersect at point T. Point R is the reflection of point T across point E. What is the length of segment RP? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19.08", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center U and circumradius 8. Let Q be the midpoint of segment UE. Let X be the midpoint of segment HB. Let O be the circumcircle of polygon ABCDEFGH. Point L is obtained by rotating point Q by angle 11pi/12 about point B. Line M passes through points B and L. Line Y is the angle bisector of angle DEX. Circle P is the reflection of circle O across line Y. Line M and segment CG intersect at point J. Point V is the inversion of point C with respect to circle P. Find the length of segment JV. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.21", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHJKLM is a regular dodecagon with center I and circumradius 11. Let R be the reflection of point M across point D. Line O is the angle bisector of angle IDB. Line Y passes through points R and K. Let X be the reflection of line O through point G. Line Q passes through point L perpendicular to line X. Line T passes through point B perpendicular to line X. Let S be the reflection of line O across line Y. Let P be the reflection of line Q across line T. Point V is the intersection of lines P and S. What is the distance between points A and V? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22.37", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center X and circumradius 11. Point K is the midpoint of segment XH. Let V be the circumcircle of polygon ABCDEFGHIJ. Let O be the angle bisector of angle XAH. Point L is the reflection of point E across line O. Let W be the reflection of circle V through point K. Let Z be the inversion of point L in circle W. Find the distance between points F and Z. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30.36", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center Z and circumradius 11. Point Y is the reflection of point A across point D. Point W is the reflection of point E across point Y. Point K is the midpoint of segment DB. Let H be the reflection of point K across point Z. Find the length of segment WH. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "64.84", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center M and circumradius 6. Point K is the midpoint of segment GC. Point N is obtained by rotating point G by angle 3pi/4 about point F. Line Y is the perpendicular bisector of segment GD. Let X be the reflection of line Y through point A. Line Z passes through points F and N. Lines X and Z intersect at point U. Find the length of segment KU. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.57", "dataset": "MATH"}
{"messages": [{"content": "ABCDF is a regular 5-gon with center E and circumradius 3. Let G be the circumcircle of polygon ABCDF. Line T is the angle bisector of angle DFE. Point H is the reflection of point F across point C. Let P be the reflection of circle G across line T. Let Y be the inversion of point C in circle P. Find the length of segment HY. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.88", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center U and circumradius 9. Point Y is obtained by rotating point C by angle pi/4 about point A. Let N be the midpoint of segment BG. Line X passes through points D and C. Let I be the line through point Y parallel to line X. Point Q is the reflection of point N across line I. What is the distance between points U and Q? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.47", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center L and circumradius 9. Let R be the reflection of point L across point B. Line V passes through points D and E. Let S be the circumcircle of polygon ABCDEFGHIJ. Let P be the reflection of circle S across line V. Point U is the inversion of point R with respect to circle P. Point O is the inversion of point U with respect to circle S. What is the length of segment EO? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.87", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center P and circumradius 10. Let Z be the circumcircle of polygon ABCDEFGH. Let S be the perpendicular bisector of segment CD. Let I be the reflection of circle Z through point G. Line N passes through point D parallel to line S. Line X is the reflection of line N across point H. Let Y be the inversion of point F in circle I. Point K is the reflection of point C across line X. What is the length of segment YK? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34.6", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEF be a regular 5-gon with center B and circumradius 4. Let W be the line through points E and C. Let V be the line through point B parallel to line W. Segment CF and line V intersect at point H. Find the distance between points D and H. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.52", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFHI be a regular octagon with center G and circumradius 2. Let R be the reflection of point A across point I. Line Q is the perpendicular bisector of segment DR. Line X is the reflection of line Q across point B. Point Z is the reflection of point A across line X. Point U is the reflection of point I across line X. What is the length of segment UZ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.53", "dataset": "MATH"}
{"messages": [{"content": "ABDEFG is a regular hexagon with center C and circumradius 1. Point L is the midpoint of segment AB. Point N is the midpoint of segment GL. Let W be the midpoint of segment DN. Find the length of segment BW. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.7", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center M and circumradius 7. Point S is the midpoint of segment BF. Point N is the reflection of point S across segment AE. Point K is the reflection of point C across segment AE. What is the distance between points K and N? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.12", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center M and circumradius 3. Let L be the circumcircle of polygon ABCDEFGH. Line N is the perpendicular bisector of segment GE. Let I be the line through point H parallel to line N. Circle Z is the reflection of circle L across line I. Point W is the inversion of point D with respect to circle Z. Let K be the midpoint of segment GD. Find the measure of angle KDW, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.39", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEFGHI be a regular octagon with center B and circumradius 9. Let O be the circumcircle of polygon ACDEFGHI. Point N is the midpoint of segment IH. Line Z passes through points A and N. Point M is the reflection of point H across line Z. Point Y is the inversion of point N with respect to circle O. Find the length of segment YM. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.31", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center O and circumradius 7. Point X is the reflection of point O across point B. Let P be the midpoint of segment XJ. Point W is the midpoint of segment CF. Let V be the angle bisector of angle BPW. Point N is the reflection of point B across line V. Find the length of segment NB. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.77", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center M and circumradius 5. Let S be the line through points C and M. Line P is the perpendicular bisector of segment AB. Line V is the reflection of line S across line P. Point Y is the midpoint of segment DA. Point H is on line S with |HD| = 11. Let Q be the midpoint of segment DH. Segment CY and line V intersect at point W. Let U be the reflection of point Q across point D. What is the distance between points W and U? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.02", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFHI be a regular octagon with center G and circumradius 8. Point L is the midpoint of segment GB. Let M be the circumcircle of polygon ABCDEFHI. Let P be the angle bisector of angle LHD. Point V is the reflection of point A across line P. Let Z be the inversion of point V in circle M. What is the distance between points Z and I? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.5", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center Z and circumradius 6. Let S be the reflection of point Z across point B. Point L is the midpoint of segment DA. Point U is the reflection of point E across point L. Let I be the midpoint of segment SE. What is the distance between points I and U? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.55", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKM is a regular dodecagon with center L and circumradius 5. Let X be the circumcircle of polygon ABCDEFGHIJKM. Let O be the reflection of circle X through point E. Line S passes through points M and D. Let Y be the inversion of point L in circle O. Point Z is the reflection of point Y across line S. Find the length of segment FZ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.14", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center T and circumradius 3. Point K is the midpoint of segment BT. Let R be the circumcircle of polygon ABCDEFGHIJ. Let U be the inversion of point K in circle R. Find the distance between points U and B. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center X and circumradius 6. Let L be the perpendicular bisector of segment DF. Line Q passes through point F parallel to line L. Point S is the reflection of point D across line Q. Let H be the line through points E and S. Lines Q and H intersect at point Z. Point O is the reflection of point Z across line L. Find the length of segment OZ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.39", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center R and circumradius 11. Let J be the vector from point A to point F. Point Q is obtained by rotating point B by angle pi/3 about point H. Point W is the translation of point R by vector J. Line T is the perpendicular bisector of segment BR. Point S is the reflection of point W across line T. Let P be the midpoint of segment SQ. Point V is the reflection of point D across point S. Find the length of segment VP. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46.7", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center N and circumradius 10. Let T be the reflection of point G across point C. Let L be the line through point G parallel to segment AH. Line V passes through point T perpendicular to line L. Point I is the reflection of point G across line V. Find the distance between points F and I. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "74.31", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDFG be a regular hexagon with center E and circumradius 2. Let T be the circumcircle of polygon ABCDFG. Let W be the angle bisector of angle DAE. Point K is the midpoint of segment BC. Let S be the reflection of circle T through point K. Point R is the inversion of point D with respect to circle S. Point U is the reflection of point R across line W. Find the length of segment CU. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.36", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center J and circumradius 7. Line U passes through points G and E. Line S passes through point F perpendicular to line U. Let Y be the line through point F perpendicular to line S. Point Q is the reflection of point B across line Y. What is the length of segment QA? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22.19", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center H and circumradius 2. Line T passes through points E and C. Line N passes through point A perpendicular to line T. Point G is the reflection of point E across line N. Let K be the reflection of point C across point D. Point R is the reflection of point B across segment AK. Find the length of segment RG. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.16", "dataset": "MATH"}
{"messages": [{"content": "ABCDFGHI is a regular octagon with center E and circumradius 12. Line N passes through points G and C. Point J is the reflection of point B across line N. Let R be the line through points C and E. Line W is the reflection of line R across line N. Let Z be the reflection of line W through point A. Line U passes through point A perpendicular to line Z. Point K is the reflection of point J across line U. What is the length of segment BK? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.97", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center W and circumradius 4. Let S be the midpoint of segment BD. Point J is the midpoint of segment FE. Line Z passes through points W and J. Let K be the line through point A parallel to line Z. Point V is the reflection of point S across line K. Find the distance between points V and B. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.21", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center U and circumradius 5. Point G is the midpoint of segment AU. Point Y is obtained by rotating point A by angle 5pi/12 about point U. Point O is the midpoint of segment BE. Line K passes through points C and D. Point I is the reflection of point O across line K. Let M be the midpoint of segment YG. Find the distance between points I and M. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.78", "dataset": "MATH"}
{"messages": [{"content": "ABDEFG is a regular hexagon with center C and circumradius 4. Let Y be the rotation of point F by angle 3pi/4 around point A. Let X be the angle bisector of angle FYG. Point L is the reflection of point F across line X. Point P is the reflection of point L across segment YF. What is the length of segment YP? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.8", "dataset": "MATH"}
{"messages": [{"content": "Let BCDEFGHIJKLM be a regular dodecagon with center A and circumradius 6. Let T be the line through points D and B. Point P is the reflection of point K across line T. Let Q be the midpoint of segment JP. Find the length of segment CQ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.52", "dataset": "MATH"}
{"messages": [{"content": "Let TG = 9. ABCDEF is a regular hexagon with center G and circumradius 10. Let H be the rotation of point G by angle pi around point D. Point W is the midpoint of segment AG. Point M is the midpoint of segment HW. Line V passes through points G and A. Point Q is the midpoint of segment MF. Point I is the reflection of point Q across line V. What is the length of segment IW? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.6", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center L and circumradius 12. Let T be the line through points B and F. Line J passes through point D parallel to line T. Let O be the reflection of line T through point A. Line Q is the reflection of line J across line T. Line R is the reflection of line O across line Q. Line X is the reflection of line O across line R. Line X and segment DH intersect at point P. What is the distance between points B and P? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.35", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center T and circumradius 7. Let P be the rotation of point A by angle 5pi/12 around point E. Point I is the midpoint of segment FT. Point J is the reflection of point A across segment PI. Point R is the reflection of point J across point P. What is the distance between points F and R? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29.78", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJK is a regular 11-gon with center T and circumradius 3. Let U be the rotation of point G by angle 3pi/4 around point B. Let O be the circumcircle of polygon ABCDEFGHIJK. Circle P is the reflection of circle O across point F. Line S is the perpendicular bisector of segment TU. Let Y be the reflection of circle P across line S. Let V be the line through points A and I. Circle N is the reflection of circle Y across line V. Let R be the inversion of point A in circle N. What is the length of segment RF? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.07", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center R and circumradius 4. Point X is the reflection of point F across point A. Let I be the line through points B and H. Point T is the reflection of point C across line I. Point Q is the reflection of point G across point X. What is the measure of angle TGQ, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.39", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center U and circumradius 1. Line Y passes through point F parallel to segment HU. Line K passes through points D and A. Let N be the reflection of line K through point E. Lines Y and N intersect at point I. Find the distance between points I and E. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.33", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEGHI be a regular octagon with center F and circumradius 11. Let K be the circumcircle of polygon ABCDEGHI. Line N is the angle bisector of angle GIA. Let T be the angle bisector of angle IBC. Point W is the reflection of point H across line T. Let V be the reflection of circle K across line N. Let L be the inversion of point B in circle V. Find the length of segment WL. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28.32", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center P and circumradius 10. Line M passes through points C and A. Line L is the perpendicular bisector of segment EF. Let J be the reflection of line L through point E. Lines M and J intersect at point O. Find the distance between points O and F. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25.17", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center Y and circumradius 10. Point U is obtained by rotating point Y by angle 3pi/4 about point F. Let P be the circumcircle of polygon ABCDEFGH. Let Q be the reflection of point C across point U. Point T is the inversion of point Q with respect to circle P. Let R be the midpoint of segment YT. Point I is the midpoint of segment BR. What is the distance between points F and I? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.41", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center Q and circumradius 4. Let M be the circumcircle of polygon ABCDEFGHIJKL. Point P is the reflection of point B across point I. Point V is the reflection of point I across point P. Point U is the inversion of point P with respect to circle M. What is the length of segment UV? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.97", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center K and circumradius 1. Point E is the reflection of point K across segment DC. Let X be the reflection of point E across point C. Let H be the midpoint of segment KD. Find the distance between points H and X. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.8", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center Y and circumradius 4. Point T is the midpoint of segment CY. Let L be the midpoint of segment TB. Point V is the reflection of point C across point D. Let E be the reflection of point L across point A. Point W is the midpoint of segment EV. What is the length of segment WB? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.1", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center H and circumradius 12. Line J passes through points C and B. Point I is the reflection of point H across line J. Let S be the reflection of point F across point I. Point W is the reflection of point S across line J. What is the distance between points W and C? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41.57", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center M and circumradius 2. Let W be the midpoint of segment BA. Let V be the midpoint of segment MW. Let H be the reflection of point A across point V. Point U is the midpoint of segment AV. What is the length of segment UH? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.37", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center S and circumradius 8. Let J be the angle bisector of angle DBE. Let I be the line through point D perpendicular to line J. Line K passes through point G parallel to line J. Point Z is the midpoint of segment SF. Line T passes through points Z and C. Let P be the reflection of line K across line T. Let R be the reflection of line I through point E. Lines P and R intersect at point L. Point W is the midpoint of segment GL. What is the length of segment SW? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.59", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center X and circumradius 5. Let P be the line through points C and E. Let G be the midpoint of segment BC. Point L is the reflection of point G across line P. Find the distance between points X and L. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.8", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center M and circumradius 3. Let G be the reflection of point M across point C. Let O be the midpoint of segment BG. Line T passes through points B and D. Point F is the reflection of point O across line T. Let W be the midpoint of segment AM. Find the length of segment FW. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.67", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center O and circumradius 1. Line R is the perpendicular bisector of segment BF. Let T be the line through point C parallel to line R. Point Q is the reflection of point F across line T. Let Y be the reflection of point Q across point C. Find the length of segment YQ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.7", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFGH be a regular 7-gon with center C and circumradius 1. Let J be the circumcircle of polygon ABDEFGH. Let M be the reflection of circle J through point E. Point R is the reflection of point C across point A. Let Y be the inversion of point C in circle M. What is the distance between points Y and R? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.41", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center X and circumradius 7. Line Z is the perpendicular bisector of segment HI. Let U be the line through point L parallel to line Z. Point Y is the reflection of point H across line U. Point T is the reflection of point X across point D. What is the distance between points T and Y? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30.24", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGH is a regular 7-gon with center F and circumradius 3. Let Z be the rotation of point E by angle pi/3 around point B. Let M be the reflection of point F across point C. Let X be the reflection of point A across point Z. Let P be the reflection of point X across point E. Point Y is the reflection of point P across segment MX. Find the distance between points Y and B. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.59", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center F and circumradius 4. Line L passes through points D and B. Line Q passes through point A perpendicular to line L. Line H is the reflection of line Q across point D. Let G be the reflection of point D across point A. Line O is the reflection of line H across point B. Let T be the reflection of line H across line O. Point U is the reflection of point G across line T. What is the distance between points U and A? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "76.11", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHI be a regular 9-gon with center V and circumradius 7. Let L be the angle bisector of angle BDH. Line R passes through point E parallel to line L. Line R and segment FA intersect at point M. Let T be the midpoint of segment FA. What is the distance between points M and T? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.11", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center O and circumradius 1. Point L is the midpoint of segment IG. Point T is obtained by rotating point A by angle pi/2 about point D. Let M be the circumcircle of polygon ABCDEFGHIJ. Point P is the reflection of point J across segment BA. Let N be the inversion of point T in circle M. Point Y is the midpoint of segment PL. What is the distance between points N and Y? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.1", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center X and circumradius 12. Let R be the line through points K and D. Point U is the reflection of point E across line R. Point S is the intersection of line R and segment BX. What is the length of segment US? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.42", "dataset": "MATH"}
{"messages": [{"content": "YE = 3. Let ABCDFGHIJKL be a regular 11-gon with center E and circumradius 7. Point X is the midpoint of segment FE. Let U be the angle bisector of angle CKJ. Let R be the reflection of line U through point X. Point S is the reflection of point X across line R. What is the distance between points S and D? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.28", "dataset": "MATH"}
{"messages": [{"content": "ABCEFGHI is a regular octagon with center D and circumradius 5. Line U is the perpendicular bisector of segment AI. Let M be the midpoint of segment HI. Let V be the reflection of line U through point E. Segment GB and line V intersect at point X. What is the distance between points X and M? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.29", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center L and circumradius 5. Line O is the angle bisector of angle LDB. Let M be the circumcircle of polygon ABCDEF. Let Z be the line through point B perpendicular to line O. Circle N is the reflection of circle M across line Z. Point Y is the inversion of point E with respect to circle N. Let K be the reflection of point L across point Y. Find the length of segment DK. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.98", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center Q and circumradius 7. Let X be the line through points F and A. Line T passes through point Q parallel to segment FD. Let P be the reflection of line X through point C. Let H be the midpoint of segment FD. Lines T and P intersect at point K. Find the distance between points K and H. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18.52", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center J and circumradius 8. Point P is obtained by rotating point J by angle pi/6 about point H. Let R be the line through points D and A. Line N passes through point G perpendicular to segment PA. Line Z is the reflection of line R across line N. Line S is the reflection of line Z across point F. Point X is the intersection of line S and segment PA. Find the distance between points E and X. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.43", "dataset": "MATH"}
{"messages": [{"content": "Let EP = 3. Let ABCDFG be a regular hexagon with center P and circumradius 6. Let L be the line through points F and D. Point X is the reflection of point B across line L. Point R is the midpoint of segment XG. Point O is the midpoint of segment DB. Find the measure of angle RDO, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.09", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIKL is a regular 11-gon with center J and circumradius 2. Let V be the rotation of point A by angle 5pi/12 around point E. Let Y be the midpoint of segment EF. Line R passes through points Y and I. Line T passes through point G perpendicular to line R. Line O passes through point J perpendicular to line T. Point M is the reflection of point F across line O. Point P is the midpoint of segment MV. What is the length of segment PL? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.2", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFGHIJKLM be a regular dodecagon with center C and circumradius 1. Point W is the reflection of point B across point I. Point N is the reflection of point A across point B. Point O is the reflection of point N across segment WG. What is the distance between points K and O? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.21", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center M and circumradius 6. Line I passes through points E and B. Let O be the circumcircle of polygon ABCDEFG. Let Q be the reflection of circle O across line I. Line R is the reflection of line I across point G. Circle N is the reflection of circle Q across line R. Let H be the inversion of point M in circle N. What is the distance between points F and H? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24.82", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center X and circumradius 6. Let N be the reflection of point C across point E. Let T be the angle bisector of angle NBA. Point J is the reflection of point D across point F. Point R is the reflection of point I across line T. What is the distance between points R and J? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.8", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJK is a regular 11-gon with center P and circumradius 12. Point N is the midpoint of segment IK. Point Z is the midpoint of segment AC. Let T be the midpoint of segment NB. Find the length of segment ZT. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.94", "dataset": "MATH"}
{"messages": [{"content": "ACDEF is a regular 5-gon with center B and circumradius 8. Let G be the circumcircle of polygon ACDEF. Let W be the midpoint of segment FB. Let U be the midpoint of segment DB. Let L be the inversion of point W in circle G. Point S is the midpoint of segment UB. What is the distance between points S and L? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.66", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center X and circumradius 6. Line Q passes through points B and D. Point S is the reflection of point X across line Q. Line N passes through points S and A. Point Y is the intersection of lines N and Q. Find the length of segment YE. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.96", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center I and circumradius 10. Line R is the angle bisector of angle FDC. Let P be the reflection of point G across point F. Line V is the angle bisector of angle BPG. Point L is the midpoint of segment BC. Point K is the reflection of point L across line R. Point N is the reflection of point K across line V. Find the length of segment NP. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.81", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center Z and circumradius 12. Let P be the midpoint of segment FE. Let W be the line through points E and D. Point K is the midpoint of segment CZ. Let X be the reflection of line W through point P. Point Q is the reflection of point D across line X. What is the distance between points K and Q? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.3", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center X and circumradius 1. Let I be the circumcircle of polygon ABCDEF. Let T be the reflection of point A across point C. Point O is the inversion of point T with respect to circle I. Point U is the midpoint of segment XB. Find the distance between points U and O. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.57", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFGHIJK be a regular 10-gon with center C and circumradius 6. Line U passes through points J and C. Let O be the line through points B and D. Point Y is the intersection of lines O and U. Point T is the midpoint of segment AY. What is the distance between points C and T? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.85", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center K and circumradius 7. Let Q be the midpoint of segment HF. Point P is the midpoint of segment KQ. Point Y is the reflection of point P across segment FK. Find the length of segment YQ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.53", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center S and circumradius 5. Point M is the midpoint of segment SB. Line I is the perpendicular bisector of segment MF. Point L is the reflection of point E across segment AF. Point X is the reflection of point L across line I. What is the distance between points S and X? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.38", "dataset": "MATH"}
{"messages": [{"content": "Let ABCEFGHIJKLM be a regular dodecagon with center D and circumradius 5. Let V be the midpoint of segment BF. Let X be the midpoint of segment DV. Line R is the perpendicular bisector of segment LJ. Point O is the reflection of point X across line R. Find the distance between points O and G. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.95", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center G and circumradius 9. Line N is the angle bisector of angle GDB. Point O is the reflection of point G across line N. Line F is the angle bisector of angle CAO. Line H passes through point D parallel to line F. Let L be the midpoint of segment BD. Point T is the reflection of point O across line H. Point S is the reflection of point L across point C. Point K is the reflection of point T across line N. What is the length of segment SK? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.59", "dataset": "MATH"}
{"messages": [{"content": "Let BCDEFG be a regular hexagon with center A and circumradius 4. Point X is the reflection of point E across point F. Let S be the midpoint of segment EA. Let N be the midpoint of segment XS. What is the length of segment ND? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHJKLM be a regular dodecagon with center I and circumradius 2. Let S be the reflection of point D across point L. Let Q be the midpoint of segment SL. Point R is the reflection of point H across point K. What is the length of segment RQ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.64", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGHIJ is a regular 9-gon with center F and circumradius 10. Let T be the reflection of point B across point I. Line L is the angle bisector of angle CGD. Let N be the reflection of point F across point T. Point Z is the reflection of point N across line L. Find the distance between points Z and G. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48.84", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center K and circumradius 9. Let H be the reflection of point E across point C. Let L be the line through point H parallel to segment AF. Point R is the reflection of point D across line L. Point T is the midpoint of segment AF. Point Y is the reflection of point T across segment RD. Find the measure of angle KAY, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.05", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center Y and circumradius 6. Line O is the angle bisector of angle KCE. Let N be the line through point Y parallel to line O. Let U be the reflection of line N across line O. Let Z be the angle bisector of angle BFJ. Let W be the reflection of line Z across line U. Let P be the reflection of line W through point A. Point T is the reflection of point G across line P. Let M be the midpoint of segment HT. What is the length of segment MB? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.8", "dataset": "MATH"}
{"messages": [{"content": "Let BCDEFGHIJKLM be a regular dodecagon with center A and circumradius 3. Point U is the midpoint of segment LE. Point Q is the reflection of point B across segment LE. Let T be the reflection of point G across point Q. What is the length of segment UT? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.91", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEG be a regular hexagon with center F and circumradius 8. Line S passes through points B and E. Line S and segment GD intersect at point Y. Point W is the midpoint of segment AY. Find the length of segment FW. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.46", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center K and circumradius 8. Line O passes through points A and D. Point M is the reflection of point B across point D. Let Z be the line through points E and D. Let J be the reflection of line Z across line O. Point Y is the reflection of point M across line J. Find the distance between points Y and E. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center T and circumradius 5. Point S is obtained by rotating point C by angle 3pi/4 about point B. Point F is the midpoint of segment CE. Let M be the midpoint of segment DB. Point G is the reflection of point F across segment DS. Find the distance between points G and M. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.07", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center J and circumradius 2. Point M is the reflection of point A across point E. Point G is the intersection of segments MJ and AD. Point V is the reflection of point A across segment MJ. Find the length of segment GV. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.25", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center Z and circumradius 12. Point I is obtained by rotating point A by angle 3pi/4 about point B. Let V be the line through points Z and B. Line P passes through point I parallel to line V. Let H be the reflection of line P across line V. Point T is the intersection of line H and segment ZA. What is the distance between points T and D? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.94", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center K and circumradius 1. Let L be the circumcircle of polygon ABCDEFGHIJ. Let V be the angle bisector of angle FAJ. Point P is the reflection of point C across line V. Point X is the inversion of point P with respect to circle L. Find the distance between points X and J. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.49", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center L and circumradius 4. Let Q be the midpoint of segment AB. Point M is the reflection of point D across point Q. Point V is the reflection of point Q across segment EM. What is the distance between points D and V? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.61", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center K and circumradius 11. Let X be the midpoint of segment DG. Point Z is the reflection of point X across point K. Point U is the midpoint of segment FZ. What is the distance between points B and U? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.58", "dataset": "MATH"}
{"messages": [{"content": "Let ABCEFGHI be a regular octagon with center D and circumradius 11. Let X be the reflection of point I across point H. Point L is the midpoint of segment HX. Let K be the reflection of point A across point H. Find the length of segment KL. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.78", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center R and circumradius 8. Let Z be the circumcircle of polygon ABCDEFGHI. Let X be the reflection of circle Z through point G. Let Q be the midpoint of segment DE. Point M is the midpoint of segment CQ. Point Y is the inversion of point M with respect to circle X. Find the distance between points E and Y. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.67", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center Z and circumradius 6. Let P be the reflection of point Z across point A. Let S be the rotation of point P by angle pi/4 around point C. Point O is the midpoint of segment EP. Let M be the perpendicular bisector of segment ZS. Line K passes through point C perpendicular to line M. Point J is the reflection of point O across line K. Find the length of segment JG. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19.71", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center Q and circumradius 3. Let T be the reflection of point D across point C. Let J be the midpoint of segment QT. Let X be the reflection of point A across point J. What is the distance between points X and B? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.2", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center X and circumradius 7. Let Q be the line through points F and D. Point K is the reflection of point C across segment FB. Point H is the reflection of point C across line Q. What is the measure of angle KCH, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.09", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center W and circumradius 1. Let T be the perpendicular bisector of segment BW. Let K be the midpoint of segment FD. Line X passes through point G perpendicular to segment FD. Let P be the reflection of line X across line T. Line I is the perpendicular bisector of segment CG. Let O be the reflection of line I across line X. Lines O and P intersect at point Z. Point U is the reflection of point K across line T. What is the distance between points U and Z? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.56", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center I and circumradius 6. Let W be the reflection of point A across point B. Point J is the midpoint of segment FB. Point Y is the reflection of point W across point F. Point V is the midpoint of segment JY. Point K is the midpoint of segment AE. What is the length of segment KV? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.35", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center M and circumradius 8. Point T is the reflection of point D across point G. Line W is the angle bisector of angle TFG. Let U be the reflection of point A across point B. Point Y is the reflection of point U across line W. Find the distance between points Y and G. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22.63", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center Q and circumradius 3. Point S is the midpoint of segment HQ. Line J is the perpendicular bisector of segment SG. Point Y is the reflection of point Q across line J. What is the length of segment HY? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.63", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center Y and circumradius 11. Let V be the midpoint of segment CD. Let Q be the midpoint of segment FB. Point U is the reflection of point B across segment VD. What is the measure of angle UQV, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.52", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center M and circumradius 11. Let L be the line through point C perpendicular to segment AC. Point T is the reflection of point F across line L. Let S be the reflection of point T across point A. Point Z is the reflection of point S across segment AC. What is the distance between points Z and F? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31.11", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFHIJK be a regular 10-gon with center G and circumradius 10. Point L is the reflection of point B across point F. Let Y be the reflection of point B across point L. Let V be the midpoint of segment YI. Find the distance between points B and V. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "46.08", "dataset": "MATH"}
{"messages": [{"content": "ABDEFGHI is a regular octagon with center C and circumradius 6. Let S be the rotation of point F by angle pi/12 around point B. Let K be the perpendicular bisector of segment DF. Line L passes through point F parallel to line K. Let U be the perpendicular bisector of segment HS. Let J be the reflection of line U across line L. Line J and segment HS intersect at point O. What is the length of segment AO? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.35", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEG be a regular hexagon with center F and circumradius 2. Point H is the reflection of point A across point C. Point Q is the midpoint of segment GH. Let O be the reflection of point Q across point C. Point I is the midpoint of segment OB. What is the length of segment ID? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.12", "dataset": "MATH"}
{"messages": [{"content": "Let BCDEF be a regular 5-gon with center A and circumradius 11. Let M be the reflection of point D across point C. Let X be the circumcircle of polygon BCDEF. Point L is the reflection of point F across point M. Point K is the inversion of point L with respect to circle X. Find the length of segment KD. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.26", "dataset": "MATH"}
{"messages": [{"content": "BN = 10. Let ACDEFG be a regular hexagon with center N and circumradius 7. Let S be the reflection of point E across point C. Line L is the perpendicular bisector of segment DS. Let O be the line through points C and G. Point I is the reflection of point G across line L. Point J is the reflection of point I across line O. What is the length of segment JA? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.09", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center O and circumradius 7. Point Y is the intersection of segments FK and OH. Point R is the reflection of point Y across point K. Point Z is the reflection of point E across point B. What is the length of segment ZR? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.0", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEFGHIJK be a regular 10-gon with center B and circumradius 11. Let O be the line through points G and D. Line L passes through point A perpendicular to line O. Point M is the reflection of point I across point F. Point U is the reflection of point J across line L. Point V is the midpoint of segment UM. What is the distance between points K and V? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.65", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center R and circumradius 7. Point M is the midpoint of segment CR. Let T be the circumcircle of polygon ABCDEFGH. Let J be the inversion of point M in circle T. What is the distance between points J and A? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.65", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center U and circumradius 7. Point N is obtained by rotating point U by angle pi/12 about point A. Line M is the angle bisector of angle DAE. Line L passes through points N and E. Line F is the reflection of line L across line M. Point S is the reflection of point N across line F. Find the length of segment SU. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.96", "dataset": "MATH"}
{"messages": [{"content": "Let LZ = 3. Let ABCDEFGHIJKM be a regular dodecagon with center Z and circumradius 4. Let R be the line through point D parallel to segment HZ. Point S is on line R. Point P is the reflection of point S across segment HZ. Find the distance between points P and S. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.93", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center Q and circumradius 6. Let W be the line through points A and D. Line P is the reflection of line W across point B. Point S is the reflection of point C across line P. Let T be the midpoint of segment SD. Find the distance between points T and C. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center S and circumradius 5. Let I be the circumcircle of polygon ABCDEF. Let H be the line through points A and C. Let M be the midpoint of segment ES. Let L be the reflection of circle I through point D. Let Q be the inversion of point A in circle L. Point R is the reflection of point Q across line H. Find the length of segment MR. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.72", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIK be a regular 10-gon with center J and circumradius 4. Line O is the perpendicular bisector of segment DK. Line T passes through point H parallel to line O. Point P is the midpoint of segment KF. Point V is the reflection of point P across line T. What is the length of segment DV? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.09", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFGHIJKLM be a regular dodecagon with center C and circumradius 6. Let S be the circumcircle of polygon ABDEFGHIJKLM. Let Z be the line through points K and D. Let V be the midpoint of segment LG. Line Y is the reflection of line Z across point C. Circle R is the reflection of circle S across line Y. Point U is the reflection of point A across line Y. Let T be the midpoint of segment DU. Let O be the inversion of point T in circle R. What is the distance between points V and O? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.56", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center L and circumradius 11. Let Z be the circumcircle of polygon ABCDE. Let K be the perpendicular bisector of segment BL. Point R is the reflection of point C across line K. Point P is the inversion of point R with respect to circle Z. What is the length of segment EP? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.11", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGIJKLM be a regular dodecagon with center H and circumradius 5. Let Z be the reflection of point G across point J. Point Y is the midpoint of segment MC. Point W is the midpoint of segment HY. What is the distance between points W and Z? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.28", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center M and circumradius 1. Point Q is the reflection of point E across segment BA. Let X be the reflection of point G across point C. Line V is the angle bisector of angle HEQ. Point O is the reflection of point H across line V. Let P be the midpoint of segment XM. What is the distance between points O and P? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.39", "dataset": "MATH"}
{"messages": [{"content": "Let ABDE be a square with center C and circumradius 5. Line V passes through points E and D. Let T be the circumcircle of polygon ABDE. Let M be the reflection of circle T across line V. Point J is the inversion of point A with respect to circle M. What is the measure of angle AJE, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.79", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center Z and circumradius 10. Let O be the reflection of point G across point J. Point U is the midpoint of segment JB. Point Y is the midpoint of segment HO. Find the length of segment UY. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.65", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center N and circumradius 11. Let M be the line through point E parallel to segment AD. Let V be the reflection of point G across point F. Point W is the reflection of point V across line M. What is the distance between points N and W? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.75", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGIJK be a regular 10-gon with center H and circumradius 10. Let U be the midpoint of segment EB. Let V be the perpendicular bisector of segment IE. Point P is the reflection of point D across point U. Point W is the reflection of point P across line V. What is the measure of angle JDW, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.31", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGIJKLM is a regular dodecagon with center H and circumradius 10. Line Y is the perpendicular bisector of segment KF. Let W be the angle bisector of angle AHB. Let V be the reflection of line Y across line W. Line U passes through points A and D. Let N be the reflection of line U across line V. Point Z is the reflection of point G across line N. Find the length of segment EZ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32.35", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center Z and circumradius 7. Let U be the line through points F and L. Let Y be the perpendicular bisector of segment EK. Let X be the reflection of line Y through point J. Line V is the reflection of line X across line U. Line M passes through point Z parallel to segment IJ. Let O be the reflection of line M across line U. Lines O and V intersect at point R. What is the distance between points R and E? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.98", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center G and circumradius 9. Let U be the perpendicular bisector of segment EG. Point W lies on line U at distance 7 from point C. Line F passes through point B perpendicular to segment GE. Point T is the reflection of point W across segment GE. Point V is the reflection of point T across line F. What is the distance between points T and V? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.56", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center R and circumradius 5. Line N is the angle bisector of angle ABC. Let F be the reflection of line N through point C. Let W be the line through points C and A. Point G is the intersection of lines F and W. Let X be the midpoint of segment GD. Find the distance between points A and X. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.31", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center X and circumradius 1. Let Q be the reflection of point E across point A. Line N passes through points Q and H. Line S passes through point H perpendicular to line N. Point L is the reflection of point E across line S. What is the distance between points D and L? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.43", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center Z and circumradius 2. Point X is the midpoint of segment AF. Line Y is the perpendicular bisector of segment CZ. Line N is the reflection of line Y across point X. Point U is the reflection of point E across line N. Find the length of segment UF. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.38", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center N and circumradius 9. Let Y be the line through points G and L. Point X is the intersection of line Y and segment IA. Point O is the reflection of point L across segment IA. What is the length of segment OX? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.66", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center T and circumradius 6. Let R be the midpoint of segment CA. Point Z is the midpoint of segment ER. Point J is the midpoint of segment BZ. What is the distance between points J and T? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.25", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center P and circumradius 5. Point W is the midpoint of segment GF. Line S is the perpendicular bisector of segment DW. Let Z be the line through point P perpendicular to line S. Point Y is the reflection of point F across line Z. Point X is the reflection of point G across line Z. Let Q be the reflection of point X across point A. Find the distance between points Y and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.27", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFHIJKL is a regular 11-gon with center G and circumradius 2. Let S be the circumcircle of polygon ABCDEFHIJKL. Let O be the angle bisector of angle KDL. Point T is the reflection of point A across line O. Let R be the reflection of circle S through point J. Let U be the inversion of point F in circle R. What is the length of segment UT? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.17", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center N and circumradius 9. Let Z be the line through points N and A. Line X is the angle bisector of angle DEA. Let L be the reflection of line X across line Z. Point P is the reflection of point B across line L. What is the length of segment PG? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.8", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center S and circumradius 4. Let R be the rotation of point F by angle pi/4 around point S. Point Q is the reflection of point B across point F. Let O be the line through points R and Q. Let N be the angle bisector of angle RSC. Let H be the reflection of line N across line O. Point P is the reflection of point E across line H. What is the distance between points P and F? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.14", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center U and circumradius 11. Point K is the midpoint of segment EB. Segments KA and CJ intersect at point O. Find the distance between points K and O. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.74", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center T and circumradius 8. Line M is the perpendicular bisector of segment AT. Let N be the reflection of point D across point F. Point V is obtained by rotating point T by angle 5pi/6 about point F. Line S passes through point N perpendicular to line M. Point L is on line S. Let W be the angle bisector of angle LVF. Point P is the reflection of point T across line W. Find the length of segment PV. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.45", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center X and circumradius 1. Let R be the circumcircle of polygon ABCDEFGHI. Let M be the midpoint of segment HI. Point T is the inversion of point M with respect to circle R. What is the distance between points C and T? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.94", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center Y and circumradius 9. Line S is the angle bisector of angle YDC. Let U be the midpoint of segment AY. Let O be the reflection of line S through point F. Point Q is the reflection of point U across line O. Find the length of segment QG. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32.09", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center Q and circumradius 9. Point Y is obtained by rotating point J by angle pi/6 about point I. Let O be the angle bisector of angle GIY. Line M passes through point B perpendicular to line O. Let V be the reflection of line M through point L. Let N be the reflection of line V through point K. Point U is the reflection of point B across line N. What is the distance between points U and D? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.44", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGI be a regular octagon with center H and circumradius 1. Line L passes through points I and H. Point V is the reflection of point I across point E. Let X be the perpendicular bisector of segment GE. Line W passes through point V perpendicular to line L. Line U is the reflection of line X across line W. Point R is the reflection of point A across line U. What is the length of segment AR? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.07", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center O and circumradius 9. Let L be the circumcircle of polygon ABCDEFGH. Point Z is the reflection of point O across point F. Let N be the inversion of point Z in circle L. Point M is the midpoint of segment OE. Point W is the midpoint of segment ME. Let K be the perpendicular bisector of segment OE. Point V is the reflection of point W across line K. What is the distance between points N and V? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.32", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJK be a regular 11-gon with center M and circumradius 1. Point T is the midpoint of segment KB. Point W is the midpoint of segment DE. Line Q is the angle bisector of angle WTJ. Point P is the reflection of point I across line Q. What is the length of segment AP? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.55", "dataset": "MATH"}
{"messages": [{"content": "ABCDFGHIJKLM is a regular dodecagon with center E and circumradius 1. Point T is the midpoint of segment GI. Let W be the angle bisector of angle GEC. Line P is the angle bisector of angle BAG. Point Y is the reflection of point B across line P. Point Q is the reflection of point T across line W. Find the length of segment YQ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.39", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center I and circumradius 12. Point J is obtained by rotating point D by angle pi/6 about point B. Let R be the line through points J and A. Line Z is the reflection of line R across point B. Point V is the reflection of point B across line R. Point Q is the reflection of point V across line Z. Point K is the midpoint of segment VQ. Find the measure of angle KBC, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.79", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center M and circumradius 8. Line O is the perpendicular bisector of segment ME. Point H is the midpoint of segment BM. Point G is the reflection of point H across line O. Find the distance between points G and M. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDEG is a regular hexagon with center F and circumradius 1. Let T be the reflection of point E across point C. Let K be the reflection of point C across point B. Line M is the angle bisector of angle CEF. Point Z is the midpoint of segment AB. Point Y is the reflection of point Z across line M. Let U be the reflection of point Y across point T. What is the length of segment UK? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.79", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center M and circumradius 6. Let K be the rotation of point E by angle pi/12 around point C. Let X be the vector from point E to point B. Let Y be point C translated by vector X. Point I is the reflection of point K across point M. Let R be the line through points Y and I. Point T is the reflection of point B across line R. Point J is the reflection of point C across segment GD. Find the length of segment JT. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.41", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center U and circumradius 12. Point Z is the midpoint of segment KF. Let P be the reflection of point I across point Z. Let R be the line through point P parallel to segment AG. Point X is the reflection of point J across line R. Let Y be the reflection of point Z across point G. What is the distance between points X and Y? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30.84", "dataset": "MATH"}
{"messages": [{"content": "BCDEFGHIJKLM is a regular dodecagon with center A and circumradius 2. Let X be the perpendicular bisector of segment FA. Line Z is the reflection of line X across point F. Point U is the reflection of point G across line Z. Find the distance between points U and A. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.38", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center O and circumradius 12. Point Y is the midpoint of segment ED. Let X be the angle bisector of angle HEC. Let N be the line through point Y parallel to line X. Point Q is the reflection of point O across line N. Find the length of segment JQ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18.77", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center X and circumradius 9. Let J be the reflection of point X across point C. Point G is the midpoint of segment DE. Let L be the midpoint of segment GE. What is the distance between points L and J? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21.7", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center R and circumradius 2. Let N be the perpendicular bisector of segment DA. Line O passes through point C perpendicular to line N. Point H is the reflection of point R across line O. Line P is the angle bisector of angle RHD. Point Z is the reflection of point R across line P. Find the distance between points Z and C. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.31", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center U and circumradius 5. Let S be the midpoint of segment IU. Let Z be the line through points C and J. Point Q is the midpoint of segment HD. Let V be the line through points Q and F. Let M be the reflection of line Z through point F. Lines V and M intersect at point O. What is the length of segment OS? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.61", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center W and circumradius 4. Line G passes through point C perpendicular to segment AB. Let V be the circumcircle of polygon ABCDE. Circle K is the reflection of circle V across line G. Let T be the inversion of point B in circle K. Point N is the midpoint of segment TA. Find the distance between points E and N. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.39", "dataset": "MATH"}
{"messages": [{"content": "Let VA = 8. BCDEFG is a regular hexagon with center A and circumradius 3. Line P is the angle bisector of angle AEB. Line Q is the reflection of line P across point C. Line J is the reflection of line Q across point G. Point H is the reflection of point F across line J. Find the distance between points H and F. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.59", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center Y and circumradius 8. Let X be the midpoint of segment FA. Line I passes through point X parallel to segment GF. Point O lies on line I. Find the measure of angle OXA, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.69", "dataset": "MATH"}
{"messages": [{"content": "ABCEFGHIJKLM is a regular dodecagon with center D and circumradius 2. Let X be the circumcircle of polygon ABCEFGHIJKLM. Circle R is the reflection of circle X across point M. Circle T is the reflection of circle R across point F. Point W is the inversion of point G with respect to circle T. What is the length of segment WG? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.13", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGHIJK is a regular 10-gon with center F and circumradius 9. Let P be the perpendicular bisector of segment IE. Let L be the reflection of line P through point A. Point X is the reflection of point G across line L. Let M be the reflection of point G across point X. Let Z be the reflection of point M across point B. Find the distance between points Z and G. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47.81", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center W and circumradius 3. Line M is the angle bisector of angle GDW. Let O be the line through points W and D. Line P is the reflection of line O across line M. Point T is the reflection of point C across line P. What is the distance between points B and T? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.63", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJK be a regular 11-gon with center R and circumradius 2. Let X be the midpoint of segment AD. Let S be the line through points I and D. Line T passes through point X parallel to line S. Point O is the reflection of point A across line T. Let N be the reflection of point O across point R. Find the length of segment BN. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.77", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center V and circumradius 3. Let O be the circumcircle of polygon ABCDEFGHIJ. Let W be the reflection of circle O through point G. Circle P is the reflection of circle W across point V. Let L be the inversion of point D in circle P. Point R is the midpoint of segment CB. Point X is the midpoint of segment LA. Let Y be the midpoint of segment RX. Find the length of segment BY. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.23", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center K and circumradius 9. Point T is the reflection of point K across segment AC. Line N passes through point T perpendicular to segment AC. Let I be the midpoint of segment KA. Point Y is the reflection of point I across segment AC. Point H is the reflection of point I across line N. Find the distance between points H and Y. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center H and circumradius 6. Let O be the angle bisector of angle BEC. Let Z be the rotation of point C by angle pi/4 around point F. Let G be the angle bisector of angle DZC. Line R is the reflection of line G across line O. Line L is the reflection of line O across line R. Point S is the intersection of lines R and L. Point K is the reflection of point Z across point S. Point Y is the reflection of point K across line R. Find the distance between points A and Y. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.82", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center M and circumradius 9. Let X be the midpoint of segment FM. Point T is the midpoint of segment AG. Let W be the midpoint of segment DT. Find the distance between points X and W. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.67", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center X and circumradius 1. Line M is the angle bisector of angle CBE. Let Y be the line through point A parallel to line M. Let Z be the reflection of line Y through point D. Point H is the reflection of point C across line Z. What is the distance between points E and H? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.65", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center Z and circumradius 10. Line X passes through points A and Z. Line X and segment DB intersect at point U. Point Q is the reflection of point C across segment DB. Let M be the reflection of point E across point B. Find the measure of angle UQM, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.26", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center S and circumradius 9. Point J is the midpoint of segment BE. Let O be the midpoint of segment JE. Point V is the midpoint of segment BO. Point Z is the reflection of point V across segment DB. What is the distance between points O and Z? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.45", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center I and circumradius 7. Let J be the perpendicular bisector of segment IA. Point R is the midpoint of segment BD. Line U passes through point I perpendicular to line J. Point M is the reflection of point R across line U. Find the distance between points E and M. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.5", "dataset": "MATH"}
{"messages": [{"content": "BCDEFGHIJK is a regular 10-gon with center A and circumradius 11. Let Z be the midpoint of segment KE. Let W be the line through points C and J. Let M be the line through point Z parallel to line W. Line S passes through point K parallel to line M. Point O is the reflection of point Z across segment GC. Point T is the reflection of point O across line S. What is the length of segment KT? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHJKL be a regular 11-gon with center I and circumradius 1. Line S passes through points L and I. Point M is the reflection of point I across point J. Line O passes through point M perpendicular to line S. Point W is the reflection of point G across line O. What is the length of segment WK? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.49", "dataset": "MATH"}
{"messages": [{"content": "ABCDF is a regular 5-gon with center E and circumradius 11. Line H is the perpendicular bisector of segment ED. Point S is the midpoint of segment EB. Line Z passes through point D perpendicular to line H. Point Q is the reflection of point S across line Z. Find the length of segment EQ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.5", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center W and circumradius 5. Point U is the midpoint of segment GE. Point X is the reflection of point D across segment HU. Point Z is the reflection of point A across segment HU. What is the distance between points Z and X? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.24", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center N and circumradius 7. Line O passes through points N and C. Let S be the midpoint of segment AB. Let L be the midpoint of segment GA. Point I is the reflection of point S across line O. What is the length of segment IL? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.2", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHJKLM is a regular dodecagon with center I and circumradius 11. Point T is the reflection of point K across point E. Point Q is the reflection of point A across segment BK. Let S be the perpendicular bisector of segment MH. Let N be the reflection of line S through point T. Line R passes through points A and Q. Lines R and N intersect at point U. Find the length of segment UK. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37.24", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDFGHIJKL be a regular 11-gon with center E and circumradius 8. Let U be the circumcircle of polygon ABCDFGHIJKL. Circle W is the reflection of circle U across point H. Let N be the reflection of circle W through point D. Let M be the reflection of circle N through point G. Let T be the inversion of point B in circle M. What is the length of segment TF? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30.35", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJK be a regular 11-gon with center W and circumradius 5. Line S passes through points J and C. Line T passes through point F parallel to line S. Line O passes through points A and K. Point Y is the intersection of lines O and T. Find the distance between points Y and W. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33.71", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center O and circumradius 12. Let I be the midpoint of segment OE. Let Q be the line through points B and I. Let H be the line through point F perpendicular to line Q. Let U be the reflection of point E across point C. Let N be the line through points U and A. Let L be the reflection of line H across line N. Lines N and L intersect at point V. What is the measure of angle VDB, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.05", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center O and circumradius 7. Let U be the reflection of point G across point F. Line Y passes through point U perpendicular to segment OH. Let V be the circumcircle of polygon ABCDEFGH. Circle X is the reflection of circle V across line Y. Line L passes through points H and B. Circle Q is the reflection of circle X across line L. Point M is the inversion of point D with respect to circle Q. Point K is the midpoint of segment OH. Find the distance between points M and K. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.32", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center Y and circumradius 11. Line Q passes through point G parallel to segment YC. Line V passes through points C and B. Lines V and Q intersect at point N. Point P is the midpoint of segment FN. Find the length of segment PN. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.46", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center Z and circumradius 10. Let N be the circumcircle of polygon ABCDEF. Let X be the midpoint of segment BZ. Let H be the inversion of point X in circle N. Find the length of segment HD. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26.46", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJLM be a regular dodecagon with center K and circumradius 10. Point R is the reflection of point H across segment GL. Point O is the intersection of segments RI and GL. Point Q is the midpoint of segment FI. Find the distance between points O and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.11", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center W and circumradius 12. Point U is the reflection of point A across point E. Let X be the midpoint of segment KW. Point O is obtained by rotating point X by angle 7pi/12 about point F. Line S passes through point C parallel to segment KO. Line S and segment BU intersect at point V. What is the distance between points V and D? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.13", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center X and circumradius 1. Let R be the reflection of point A across point E. Let Z be the midpoint of segment XA. Point U is the reflection of point B across point X. Point N is the midpoint of segment ZR. Let Q be the midpoint of segment UN. Find the length of segment UQ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.18", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center K and circumradius 2. Let I be the angle bisector of angle HCE. Line U passes through point D parallel to line I. Let R be the reflection of line U through point E. Point Z is the reflection of point C across line R. What is the distance between points Z and K? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.57", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center V and circumradius 5. Let U be the perpendicular bisector of segment HF. Line S passes through point A parallel to line U. Point Y is the reflection of point H across line S. Let Q be the reflection of point B across point Y. Point X is the reflection of point E across line S. Point T is the midpoint of segment BX. Find the length of segment QT. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.38", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center N and circumradius 3. Let V be the line through points B and E. Let M be the angle bisector of angle DEG. Line I is the reflection of line M across line V. Line T passes through points D and G. Point Z is the intersection of lines T and I. Find the distance between points F and Z. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.08", "dataset": "MATH"}
{"messages": [{"content": "Let BCDEFG be a regular hexagon with center A and circumradius 8. Let X be the reflection of point G across point C. Line I passes through points X and F. Point K is the reflection of point G across point F. Point J is the reflection of point F across point C. Line S passes through point K perpendicular to segment JK. Point L is the reflection of point E across line S. Line I and segment JK intersect at point O. What is the length of segment OL? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26.23", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center W and circumradius 12. Let J be the midpoint of segment WE. Point U is the reflection of point D across point W. Point O is the midpoint of segment UJ. Let K be the perpendicular bisector of segment CO. Point R is the reflection of point B across line K. What is the length of segment JR? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19.09", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGHIJK is a regular 10-gon with center F and circumradius 12. Point M is the midpoint of segment AH. Let N be the midpoint of segment JF. Point Y is the reflection of point M across segment GN. What is the length of segment BY? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.99", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center G and circumradius 1. Let E be the circumcircle of polygon ABCD. Line U is the angle bisector of angle GCB. Point P is the reflection of point A across line U. Let N be the line through point P parallel to line U. Circle I is the reflection of circle E across line N. Point T is the reflection of point G across point C. Let O be the inversion of point T in circle I. Find the distance between points P and O. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.53", "dataset": "MATH"}
{"messages": [{"content": "ABCEFGHIJKL is a regular 11-gon with center D and circumradius 3. Point T is the reflection of point K across segment JG. Let N be the perpendicular bisector of segment TB. Let Z be the line through point C parallel to segment KF. Point R is the reflection of point J across line Z. Point Y is the reflection of point J across line N. Find the distance between points R and Y. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.77", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDFGHI be a regular octagon with center E and circumradius 3. Line Q passes through points G and A. Line Z passes through point B parallel to segment IA. Point M is the intersection of lines Z and Q. What is the measure of angle ADM, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.39", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center Y and circumradius 9. Let R be the line through points A and E. Line T is the reflection of line R across point D. Point G is the reflection of point F across line T. Point K is the midpoint of segment YA. Point X is the reflection of point C across point K. Point S is the reflection of point E across segment GA. Find the length of segment SX. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.69", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJLM be a regular dodecagon with center K and circumradius 7. Let O be the reflection of point B across point L. Point V is the midpoint of segment KJ. Point W is the reflection of point O across segment DG. Find the distance between points W and V. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25.35", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center X and circumradius 7. Let R be the circumcircle of polygon ABCDEFGH. Line K passes through points B and H. Point I is the reflection of point A across line K. Point S is the inversion of point I with respect to circle R. Find the distance between points S and C. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18.29", "dataset": "MATH"}
{"messages": [{"content": "BCDEFG is a regular hexagon with center A and circumradius 7. Point N is the midpoint of segment GB. Let S be the reflection of point N across point F. Line X passes through points G and E. Point R is the reflection of point S across line X. What is the length of segment RE? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.5", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center I and circumradius 4. Let L be the midpoint of segment GD. Let W be the midpoint of segment LI. Line P passes through points W and B. Line P and segment GE intersect at point M. Find the distance between points B and M. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.84", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center U and circumradius 4. Let M be the midpoint of segment DB. Line Q is the angle bisector of angle JIG. Point N is the reflection of point M across line Q. Point O is the midpoint of segment NJ. Find the distance between points U and O. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.31", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJK is a regular 11-gon with center X and circumradius 5. Line N passes through point X perpendicular to segment XK. Point U is the reflection of point A across line N. Line O passes through point U perpendicular to segment XK. Let W be the reflection of point G across point F. Point Q is the reflection of point W across line O. What is the distance between points J and Q? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.29", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center N and circumradius 10. Point S is the reflection of point K across point E. Let R be the angle bisector of angle AJK. Point X is the reflection of point S across line R. Point Q is the reflection of point X across point H. Find the length of segment QB. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56.03", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center L and circumradius 12. Line S is the perpendicular bisector of segment LG. Line K is the reflection of line S across point H. Point V is the reflection of point F across line K. What is the distance between points V and C? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26.83", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center I and circumradius 3. Let N be the perpendicular bisector of segment IB. Point M is the reflection of point C across line N. Point L is the midpoint of segment IC. Point U is the reflection of point M across point L. What is the length of segment EU? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.18", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center T and circumradius 6. Let Y be the reflection of point D across point A. Line Z is the angle bisector of angle HFY. Point O is the midpoint of segment CK. Point P is the reflection of point O across line Z. What is the length of segment OP? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.92", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center M and circumradius 5. Line W is the angle bisector of angle GAM. Let Z be the rotation of point I by angle pi/12 around point D. Point V lies on line W at distance 4 from point F. Line L is the angle bisector of angle JEZ. Let P be the reflection of line L across line W. Line R is the perpendicular bisector of segment GB. Let T be the line through point V perpendicular to line R. Line S passes through point C parallel to line T. Lines S and P intersect at point Y. Find the distance between points Y and G. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.14", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center J and circumradius 1. Let V be the midpoint of segment CD. Let M be the midpoint of segment AC. Find the length of segment MV. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.95", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center J and circumradius 3. Let M be the rotation of point A by angle pi/12 around point F. Let P be the line through points A and M. Let T be the circumcircle of polygon ABCDEF. Circle U is the reflection of circle T across line P. Point I is the inversion of point C with respect to circle U. Point Y is the reflection of point I across point M. Find the length of segment JY. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.65", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center N and circumradius 8. Let V be the circumcircle of polygon ABCDEFGHIJKL. Line Y passes through points I and A. Point Z is the midpoint of segment NC. Let W be the perpendicular bisector of segment DC. Let U be the reflection of line W across line Y. Let R be the inversion of point Z in circle V. Point P is the reflection of point D across line U. What is the measure of angle RJP, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.88", "dataset": "MATH"}
{"messages": [{"content": "ABCDF is a regular 5-gon with center E and circumradius 4. Let N be the midpoint of segment BC. Let W be the perpendicular bisector of segment EF. Point M is the reflection of point A across line W. Point I is the reflection of point A across point M. What is the length of segment NI? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.43", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center W and circumradius 6. Let H be the line through points B and A. Point X is the reflection of point A across point E. Line T is the reflection of line H across point W. Line M is the reflection of line H across line T. Point Y is the reflection of point X across point A. Point J is the reflection of point Y across line M. Point P is the reflection of point B across point J. Find the distance between points P and W. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "99.66", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJK is a regular 11-gon with center V and circumradius 5. Let W be the perpendicular bisector of segment BE. Line P passes through point C perpendicular to line W. Let U be the reflection of point F across point A. Point M is the reflection of point K across line P. What is the distance between points U and M? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.04", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGIJ is a regular 9-gon with center H and circumradius 8. Let R be the perpendicular bisector of segment AI. Line P is the reflection of line R across point B. Let Q be the line through point A perpendicular to line P. Point O is the reflection of point E across line Q. What is the distance between points O and I? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.86", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center S and circumradius 3. Point P is the midpoint of segment SB. Let F be the reflection of point C across point S. Let W be the angle bisector of angle APE. Point K is the reflection of point F across line W. Find the length of segment PK. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.75", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHI be a regular 9-gon with center X and circumradius 9. Point W is the reflection of point E across point C. Let L be the angle bisector of angle WFX. Line S passes through point B parallel to line L. Let T be the reflection of line S across line L. Line Y is the reflection of line T across point C. Point O is the reflection of point B across line Y. What is the distance between points O and I? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31.07", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFGHI be a regular octagon with center C and circumradius 6. Point Y is the midpoint of segment IG. Point O is the reflection of point Y across segment BD. Point R is the midpoint of segment OY. What is the distance between points A and R? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.09", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center J and circumradius 3. Line V passes through point J parallel to segment CE. Point H is the midpoint of segment AE. Line T passes through points D and H. Let O be the circumcircle of polygon ABCDE. Let Q be the line through points D and C. Circle X is the reflection of circle O across line T. Point M is the reflection of point H across segment CE. Let K be the line through point E parallel to line Q. Lines K and V intersect at point S. Circle W is the reflection of circle X across point S. Point I is the reflection of point C across point M. Let R be the inversion of point A in circle W. What is the distance between points I and R? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.22", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center V and circumradius 11. Let Q be the circumcircle of polygon ABCDEFGHIJKL. Let P be the midpoint of segment GF. Let X be the midpoint of segment GI. Point R is the inversion of point P with respect to circle Q. What is the distance between points R and X? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.19", "dataset": "MATH"}
{"messages": [{"content": "BCDEFG is a regular hexagon with center A and circumradius 11. Let R be the line through points A and G. Line Q passes through points C and B. Line K is the reflection of line R across line Q. Point P is the reflection of point D across line K. What is the measure of angle PBE, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.09", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFHIJKL is a regular 11-gon with center G and circumradius 1. Let N be the midpoint of segment KL. Let X be the midpoint of segment EC. Let P be the line through point K perpendicular to segment NX. Let Z be the midpoint of segment NX. Let W be the reflection of point L across point C. Point O is the reflection of point D across line P. Let R be the reflection of point W across point O. Find the length of segment RZ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center M and circumradius 3. Line Y is the angle bisector of angle BEM. Let R be the line through point M parallel to line Y. Line O passes through points A and F. Lines R and O intersect at point Q. What is the distance between points Q and C? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.06", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJK be a regular 11-gon with center V and circumradius 1. Point L is the reflection of point F across point J. Line X passes through points A and K. Let N be the line through point I parallel to line X. Point M is the reflection of point L across line N. Find the distance between points H and M. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.72", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center M and circumradius 9. Let V be the line through points F and E. Let P be the angle bisector of angle AFG. Let Z be the perpendicular bisector of segment CI. Point N is the reflection of point I across line V. Let Q be the reflection of point C across point F. Let T be the reflection of line P across line Z. Point Y is the reflection of point N across line T. What is the length of segment QY? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.59", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center S and circumradius 10. Point T is the midpoint of segment BF. Point W is the reflection of point S across point C. Let X be the midpoint of segment DT. Find the length of segment XW. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.84", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGHI is a regular octagon with center F and circumradius 2. Let L be the reflection of point G across point I. Let V be the line through points L and H. Point K is the reflection of point D across line V. Point U is the reflection of point G across line V. Let T be the midpoint of segment AU. Point P is the reflection of point K across point H. What is the distance between points T and P? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.14", "dataset": "MATH"}
{"messages": [{"content": "ABDEFGHIJKLM is a regular dodecagon with center C and circumradius 8. Let R be the circumcircle of polygon ABDEFGHIJKLM. Let P be the perpendicular bisector of segment AC. Circle X is the reflection of circle R across line P. Let W be the line through point I perpendicular to line P. Point O is the inversion of point G with respect to circle X. Line N passes through point O parallel to line W. Segment BK and line N intersect at point V. What is the length of segment FV? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.94", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEF be a regular 5-gon with center C and circumradius 2. Point L is the reflection of point A across point E. Let H be the midpoint of segment LB. Point X is the midpoint of segment HD. What is the distance between points X and L? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.32", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center Q and circumradius 10. Point H is the reflection of point Q across point E. Let Z be the angle bisector of angle ECA. Point G is the reflection of point H across line Z. Point I is the reflection of point C across point G. Find the length of segment DI. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "55.68", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center K and circumradius 2. Point R is the reflection of point E across point A. Point W is the reflection of point K across point A. Line J is the perpendicular bisector of segment WR. Point I is the reflection of point C across line J. Find the length of segment WI. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center N and circumradius 8. Let O be the line through points B and G. Let M be the midpoint of segment AD. Let L be the midpoint of segment DM. Point X is the reflection of point N across line O. Point P is the midpoint of segment NX. Find the distance between points P and L. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.79", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center M and circumradius 12. Line U is the angle bisector of angle DCB. Let O be the reflection of line U through point B. Point F is the reflection of point A across point D. Let Z be the reflection of point F across point A. Let R be the line through point A perpendicular to line O. Point W is the reflection of point B across line R. Point G is the midpoint of segment DZ. Find the distance between points G and W. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.49", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center O and circumradius 5. Line V passes through points F and O. Line R passes through points G and J. Let T be the reflection of line R across line V. Point M is the reflection of point I across line T. What is the measure of angle FDM, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.36", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center K and circumradius 4. Let S be the perpendicular bisector of segment KA. Line J passes through point F perpendicular to line S. Point T is the reflection of point K across line J. Let Q be the midpoint of segment TE. What is the distance between points K and Q? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.29", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center O and circumradius 9. Let K be the angle bisector of angle CEB. Let Q be the line through point C perpendicular to line K. Point G is the reflection of point D across line Q. Point U is the midpoint of segment AE. Point V is the midpoint of segment EG. What is the distance between points V and U? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.21", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center M and circumradius 10. Line Z passes through points E and M. Let V be the rotation of point C by angle 11pi/12 around point D. Line P passes through point H parallel to line Z. Line S is the reflection of line Z across line P. Point R is the reflection of point F across line S. Point O is the midpoint of segment VR. What is the distance between points V and O? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18.84", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center S and circumradius 11. Let V be the midpoint of segment EB. Point O is the midpoint of segment EV. Point X is the reflection of point F across segment SO. What is the length of segment XB? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.26", "dataset": "MATH"}
{"messages": [{"content": "Let ACDE be a square with center B and circumradius 4. Let G be the reflection of point B across point E. Line U is the perpendicular bisector of segment EB. Line Z is the reflection of line U across point G. Point X is the reflection of point D across line Z. Point T is the reflection of point X across segment AC. What is the length of segment XT? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50.91", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGIJKLM is a regular dodecagon with center H and circumradius 6. Let Y be the circumcircle of polygon ABCDEFGIJKLM. Line S passes through points K and A. Let N be the reflection of circle Y through point M. Let V be the line through points G and M. Lines S and V intersect at point W. Let X be the inversion of point J in circle N. What is the distance between points X and W? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.38", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center P and circumradius 10. Line Q is the perpendicular bisector of segment GI. Line M passes through point F parallel to line Q. Point T is the reflection of point G across point F. Point V is the reflection of point T across line M. What is the measure of angle VCP, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.63", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center Q and circumradius 2. Let M be the rotation of point Q by angle 3pi/4 around point B. Let P be the angle bisector of angle GMD. Let V be the line through points F and E. Let K be the reflection of line P through point H. Line U is the reflection of line P across line K. Lines U and V intersect at point X. What is the length of segment DX? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.3", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center I and circumradius 8. Line U passes through points G and C. Let W be the line through point A perpendicular to line U. Point Y is the reflection of point B across line W. Point H is the reflection of point Y across line U. Let O be the midpoint of segment GC. What is the distance between points O and H? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.74", "dataset": "MATH"}
{"messages": [{"content": "Let ABCE be a square with center D and circumradius 7. Point K is the reflection of point E across point B. Let U be the circumcircle of polygon ABCE. Let S be the inversion of point K in circle U. Point V is the midpoint of segment KD. Find the distance between points S and V. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.17", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center U and circumradius 9. Line K passes through points C and A. Line Z passes through point E parallel to line K. Line N passes through point U parallel to line K. Let M be the reflection of line N across line Z. Point I is the reflection of point F across line M. Find the distance between points I and A. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDFGHIJKLM be a regular dodecagon with center E and circumradius 6. Let Z be the line through points C and F. Let N be the reflection of line Z through point A. Point S is the intersection of segment EK and line N. What is the length of segment AS? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.94", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center Q and circumradius 8. Let N be the reflection of point Q across point H. Let M be the reflection of point I across point E. Point U is the midpoint of segment NH. What is the distance between points U and M? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30.74", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHI be a regular 9-gon with center K and circumradius 8. Line W passes through points I and D. Point J is the intersection of segment AG and line W. Let V be the midpoint of segment AG. What is the measure of angle IJV, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.75", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center U and circumradius 3. Line O is the angle bisector of angle FHA. Line X passes through point K parallel to line O. Line X and segment LB intersect at point M. Find the length of segment UM. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.85", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFH is a regular 7-gon with center G and circumradius 3. Let U be the circumcircle of polygon ABCDEFH. Let Y be the reflection of circle U through point A. Let J be the inversion of point H in circle Y. Let I be the reflection of point J across point H. Find the length of segment EI. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.22", "dataset": "MATH"}
{"messages": [{"content": "Let ET = 1. ABCDFG is a regular hexagon with center T and circumradius 3. Let Q be the perpendicular bisector of segment BT. Point Z is the intersection of segments DG and FB. Point W is the reflection of point Z across line Q. What is the length of segment WC? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.97", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center L and circumradius 9. Let Z be the circumcircle of polygon ABCDEF. Let Y be the line through points B and L. Circle K is the reflection of circle Z across point F. Let U be the reflection of circle K across line Y. Point Q is the inversion of point B with respect to circle U. Let N be the inversion of point Q in circle K. What is the length of segment QN? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26.28", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center X and circumradius 5. Line U is the angle bisector of angle FIA. Line Z passes through point X parallel to line U. Let O be the reflection of line Z across line U. Line O and segment EA intersect at point V. What is the measure of angle HAV, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.05", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center U and circumradius 5. Point K is obtained by rotating point B by angle pi/3 about point A. Line S passes through points E and G. Let I be the perpendicular bisector of segment KH. Line Q is the reflection of line S across point A. Let Y be the reflection of line Q across line I. Point T is the reflection of point A across line Y. Point J is the reflection of point T across point U. Point V is the midpoint of segment AD. What is the length of segment JV? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.14", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJK be a regular 11-gon with center Z and circumradius 2. Let Y be the line through points I and A. Line N is the reflection of line Y across point H. Point L is the reflection of point A across line N. Let P be the angle bisector of angle FEJ. Let R be the midpoint of segment LK. Point U is the reflection of point R across line P. What is the distance between points U and L? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.21", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center O and circumradius 3. Point L is the midpoint of segment DC. Let W be the circumcircle of polygon ABCD. Circle E is the reflection of circle W across point D. Let J be the inversion of point O in circle E. Point I is the reflection of point B across point L. Find the distance between points I and J. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.35", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIKL be a regular 11-gon with center J and circumradius 10. Let R be the midpoint of segment LG. Point O is the midpoint of segment RL. Let P be the midpoint of segment GO. Find the distance between points P and J. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.74", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center U and circumradius 3. Let J be the midpoint of segment CA. Let W be the midpoint of segment AU. Line X is the angle bisector of angle JCW. Point M is the reflection of point D across line X. Point Z is the midpoint of segment FM. Find the length of segment AZ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.56", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center J and circumradius 1. Let L be the circumcircle of polygon ABCDEF. Line Y is the angle bisector of angle AFC. Let G be the midpoint of segment BD. Let M be the reflection of circle L through point F. Point T is the midpoint of segment CG. Circle X is the reflection of circle M across line Y. Let I be the inversion of point T in circle X. Find the distance between points B and I. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.26", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEFG be a regular hexagon with center B and circumradius 8. Let K be the perpendicular bisector of segment FE. Let Z be the line through point D parallel to line K. Let R be the reflection of line Z across line K. Point Q is the reflection of point D across line R. What is the distance between points Q and F? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21.17", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center M and circumradius 7. Line T is the angle bisector of angle DGF. Point J is the reflection of point D across line T. Line X is the perpendicular bisector of segment JF. Point N is the reflection of point D across point B. Line L passes through point H perpendicular to line X. Point Z is the reflection of point N across line L. Find the distance between points G and Z. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.93", "dataset": "MATH"}
{"messages": [{"content": "ABCDFGHI is a regular octagon with center E and circumradius 11. Let V be the circumcircle of polygon ABCDFGHI. Line P passes through points I and C. Circle J is the reflection of circle V across line P. Point U is the inversion of point B with respect to circle J. What is the distance between points U and C? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.33", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHJKLM is a regular dodecagon with center I and circumradius 4. Let O be the line through point K parallel to segment BE. Let R be the midpoint of segment BE. Point N is the reflection of point E across line O. Find the distance between points N and R. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.68", "dataset": "MATH"}
{"messages": [{"content": "Let ABCEFGHI be a regular octagon with center D and circumradius 10. Let W be the circumcircle of polygon ABCEFGHI. Let Z be the angle bisector of angle FED. Circle M is the reflection of circle W across point I. Point R is the reflection of point D across point C. Let Y be the inversion of point R in circle M. Let V be the reflection of line Z through point A. Point T is the reflection of point Y across line V. What is the distance between points G and T? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48.81", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center O and circumradius 10. Let S be the reflection of point G across point A. Let N be the angle bisector of angle OFB. Let J be the line through points A and B. Let T be the line through point E perpendicular to line N. Let K be the reflection of line J across line T. Lines J and K intersect at point I. What is the length of segment SI? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "44.34", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFG be a regular hexagon with center C and circumradius 11. Let Y be the midpoint of segment EF. Point R is the midpoint of segment BY. Let H be the line through points R and A. Point V is the reflection of point D across line H. What is the distance between points V and A? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19.05", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center W and circumradius 3. Point T is the reflection of point A across point B. Let Q be the line through points E and W. Point X is the reflection of point T across line Q. Let U be the midpoint of segment WD. Find the distance between points U and X. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.29", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center H and circumradius 11. Let J be the line through points H and D. Let O be the line through point H perpendicular to line J. Point U is the midpoint of segment EA. Line Z is the reflection of line O across point B. Let M be the line through point D parallel to line Z. Point T is the reflection of point U across line M. What is the length of segment TH? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.6", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center X and circumradius 12. Point Z is the reflection of point X across segment DA. Let U be the line through points B and C. Point O is the reflection of point Z across line U. Find the distance between points Z and O. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50.91", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center X and circumradius 11. Let Z be the circumcircle of polygon ABCDEFGH. Let R be the reflection of circle Z through point H. Line I passes through points X and F. Circle O is the reflection of circle R across line I. Let P be the inversion of point E in circle O. Find the distance between points P and F. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.14", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center X and circumradius 7. Let U be the perpendicular bisector of segment HE. Let O be the line through points J and D. Line T is the reflection of line O across line U. Let W be the line through point D perpendicular to line T. Point Z is the reflection of point J across line W. What is the distance between points Z and D? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.31", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center U and circumradius 12. Point S is the midpoint of segment BE. Let P be the circumcircle of polygon ABCDEFG. Let T be the reflection of circle P through point F. Let J be the angle bisector of angle UGS. Line K is the reflection of line J across point D. Let W be the reflection of point U across point F. Let R be the reflection of point W across point C. Circle H is the reflection of circle T across line K. Point M is the inversion of point E with respect to circle H. What is the distance between points M and R? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "57.02", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center L and circumradius 6. Point V is the midpoint of segment AC. Let P be the midpoint of segment VD. Point J is the reflection of point F across segment AP. Find the length of segment BJ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.87", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center T and circumradius 8. Line R passes through points A and C. Line X passes through point D parallel to line R. Let O be the perpendicular bisector of segment FA. Let U be the reflection of line O across line X. Lines X and U intersect at point W. Find the measure of angle EWG, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.79", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center S and circumradius 11. Let P be the angle bisector of angle CFD. Line Y is the reflection of line P across point H. Line K passes through point A perpendicular to line Y. Line T passes through point D perpendicular to line K. Let W be the reflection of line T through point H. Point N is the reflection of point J across line W. What is the length of segment JN? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.85", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHJK be a regular 10-gon with center I and circumradius 6. Point X is the midpoint of segment CG. Point P is the reflection of point X across segment BD. Point M is the midpoint of segment BD. Point U is the midpoint of segment DM. What is the distance between points U and P? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.28", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center M and circumradius 1. Let O be the circumcircle of polygon ABCDEFG. Circle I is the reflection of circle O across point E. Point Q is the inversion of point G with respect to circle I. Let V be the reflection of point Q across point F. What is the length of segment VE? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.96", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center V and circumradius 9. Z is placed 2 units away from K. Line R is the angle bisector of angle ZKC. Point P is the midpoint of segment CK. Line X passes through point C perpendicular to line R. Point M is the midpoint of segment DP. Point S is the reflection of point M across line X. Find the distance between points C and S. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.99", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center I and circumradius 3. Let S be the reflection of point B across point D. Let W be the reflection of point F across point S. Line P is the angle bisector of angle ACG. Point O is the reflection of point W across line P. Find the length of segment OG. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.68", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center Z and circumradius 10. Point X is the reflection of point A across point E. Point N is the midpoint of segment ZD. Point H is the reflection of point X across segment NZ. Find the measure of angle NAH, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.52", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKM be a regular dodecagon with center L and circumradius 1. Point U is the reflection of point M across point E. Line T is the angle bisector of angle KMJ. Point Y is the reflection of point K across line T. Point O is the reflection of point F across point U. What is the distance between points O and Y? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.75", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center G and circumradius 5. Line L is the perpendicular bisector of segment CG. Line Z passes through point E perpendicular to segment DC. Let N be the midpoint of segment DC. Line Q is the reflection of line L across line Z. Point R is the reflection of point N across line Q. Find the distance between points C and R. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.61", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center H and circumradius 4. Let R be the midpoint of segment FA. Line O is the perpendicular bisector of segment RH. Point Y is the reflection of point C across point D. Point Q is the reflection of point F across line O. Find the distance between points Y and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.29", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center Z and circumradius 7. Let I be the midpoint of segment ZF. Point Q is the midpoint of segment CI. Point H is the reflection of point D across point F. What is the length of segment HQ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.18", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center L and circumradius 5. Line O is the angle bisector of angle GJD. Point S is the reflection of point H across line O. Let W be the circumcircle of polygon ABCDEFGHIJ. Point K is the inversion of point S with respect to circle W. Point U is the reflection of point D across point B. Find the length of segment KU. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.66", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center I and circumradius 3. Let N be the reflection of point E across point D. Line J passes through points I and A. Point G is the reflection of point N across line J. Line S is the angle bisector of angle BNG. Let R be the reflection of line J across line S. Line R and segment EG intersect at point P. Find the distance between points P and N. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.81", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center O and circumradius 6. Line X is the angle bisector of angle ECF. Let U be the circumcircle of polygon ABCDEF. Let V be the reflection of circle U across line X. Let P be the inversion of point A in circle V. Let R be the midpoint of segment DP. Find the length of segment ER. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJK be a regular 11-gon with center V and circumradius 10. Line N passes through points A and H. Point L is the reflection of point B across line N. Point Q is the midpoint of segment HL. Find the length of segment VQ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.95", "dataset": "MATH"}
{"messages": [{"content": "Let ABCEFGHIJKLM be a regular dodecagon with center D and circumradius 4. Point V is the reflection of point B across point H. Let W be the reflection of point H across point G. Point N is the reflection of point V across segment LA. Find the distance between points N and W. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22.48", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center U and circumradius 6. Point P is the reflection of point B across point A. Point X is obtained by rotating point A by angle pi/6 about point G. Let W be the midpoint of segment XF. Point L is the midpoint of segment UW. Point V is the reflection of point L across segment GU. What is the distance between points V and P? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.15", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center U and circumradius 1. Let H be the midpoint of segment UB. Let M be the reflection of point C across point A. Let K be the angle bisector of angle UMC. Point R is on line K with |RU| = 8. Line J passes through point H perpendicular to line K. Point W is obtained by rotating point M by angle 5pi/6 about point C. Let L be the circumcircle of polygon ABCD. Point F is the reflection of point W across point D. Let Y be the line through point F perpendicular to line J. Let O be the reflection of line Y across line K. Let S be the reflection of circle L across line O. Let E be the inversion of point R in circle S. Find the distance between points E and H. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.44", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDF be a regular 5-gon with center E and circumradius 11. Point J is the midpoint of segment EB. Let W be the perpendicular bisector of segment CA. Let P be the line through point A perpendicular to line W. Point I is the reflection of point J across line P. What is the distance between points I and B? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.7", "dataset": "MATH"}
{"messages": [{"content": "ABDEFGH is a regular 7-gon with center C and circumradius 9. Let S be the perpendicular bisector of segment GC. Let M be the line through point A parallel to line S. Let Y be the reflection of line S across line M. Point O is the reflection of point A across line Y. Find the length of segment OH. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.69", "dataset": "MATH"}
{"messages": [{"content": "Let ABCEFGHIJK be a regular 10-gon with center D and circumradius 8. Let T be the line through point G parallel to segment DK. Point V is the reflection of point B across line T. Let N be the midpoint of segment DK. Let O be the reflection of point N across point J. Find the length of segment OV. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.99", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGHI is a regular octagon with center F and circumradius 4. Let J be the midpoint of segment HI. Let K be the circumcircle of polygon ABCDEGHI. Point W is the inversion of point J with respect to circle K. Point V is the reflection of point W across point G. Find the distance between points V and E. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.7", "dataset": "MATH"}
{"messages": [{"content": "Let ABDE be a square with center C and circumradius 2. Let Z be the reflection of point B across point D. Let V be the angle bisector of angle ZBE. Line I passes through point E perpendicular to line V. Point N is the reflection of point Z across line I. What is the measure of angle ZNB, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.36", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEFG be a regular hexagon with center B and circumradius 9. Let J be the perpendicular bisector of segment CG. Let N be the line through point F parallel to line J. Point R is the midpoint of segment BA. Point H is the reflection of point R across line N. What is the measure of angle HCB, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.52", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEG be a regular hexagon with center F and circumradius 8. Point T is the midpoint of segment DG. Let N be the angle bisector of angle CGD. Point P is the reflection of point F across line N. Point H is the reflection of point G across point P. What is the distance between points H and T? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.07", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center L and circumradius 4. Let X be the perpendicular bisector of segment EB. Let U be the line through point C parallel to line X. Line V passes through point A perpendicular to line U. Point N is the reflection of point L across line V. What is the measure of angle NAD, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.09", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center K and circumradius 11. Point Q is obtained by rotating point F by angle pi/4 about point B. Let W be the angle bisector of angle BGK. Line O passes through point C parallel to line W. Line O and segment QD intersect at point J. Find the distance between points J and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.3", "dataset": "MATH"}
{"messages": [{"content": "ZN = 12. ABCD is a square with center N and circumradius 4. Let I be the reflection of point B across point D. Let L be the reflection of point A across point I. Let T be the circumcircle of polygon ABCD. Point O is the inversion of point L with respect to circle T. What is the length of segment LO? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.67", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center I and circumradius 3. Point T is the midpoint of segment HC. Segments BE and GD intersect at point Q. What is the length of segment TQ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.57", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHI be a regular 9-gon with center U and circumradius 10. Let V be the perpendicular bisector of segment BC. Let P be the midpoint of segment FE. Point R is the reflection of point P across line V. Find the distance between points B and R. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.86", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center R and circumradius 9. Line F passes through points E and R. Line T is the reflection of line F across point C. Point L is the reflection of point C across line T. Point U is the reflection of point L across segment BA. Find the distance between points U and A. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26.42", "dataset": "MATH"}
{"messages": [{"content": "ABDEFGHI is a regular octagon with center C and circumradius 12. Point Z is obtained by rotating point H by angle 11pi/12 about point G. Let S be the circumcircle of polygon ABDEFGHI. Let W be the perpendicular bisector of segment CB. Line K is the angle bisector of angle BEG. Let J be the reflection of line W across line K. Let R be the inversion of point Z in circle S. Point M is the reflection of point R across line J. What is the distance between points G and M? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.98", "dataset": "MATH"}
{"messages": [{"content": "ABCDFGHIJKLM is a regular dodecagon with center E and circumradius 10. Line Z is the perpendicular bisector of segment GI. Let X be the perpendicular bisector of segment FL. Let O be the line through point I parallel to line Z. Line P is the reflection of line X across line O. Let U be the reflection of line O through point J. Point S is the intersection of lines U and P. Find the length of segment JS. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.73", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center P and circumradius 6. Let X be the midpoint of segment EC. Line Q is the angle bisector of angle FXC. Point V is the reflection of point E across line Q. Let S be the midpoint of segment VP. Find the distance between points S and D. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.05", "dataset": "MATH"}
{"messages": [{"content": "ABCDEG is a regular hexagon with center F and circumradius 7. Line M passes through points G and A. Let Z be the circumcircle of polygon ABCDEG. Circle O is the reflection of circle Z across line M. Let S be the inversion of point D in circle O. What is the distance between points C and S? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.09", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center O and circumradius 8. Line M passes through points O and D. Line K is the reflection of line M across point C. Let Z be the line through point C perpendicular to line K. Point U is on line Z. What is the measure of angle UEB, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.45", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGIJK is a regular 10-gon with center H and circumradius 9. Let P be the vector from point I to point D. Point Z is the translation of point F by vector P. Point L is obtained by rotating point H by angle 2pi/3 about point Z. Point Y is the translation of point A by vector P. Line M passes through points A and Y. Line X is the perpendicular bisector of segment CZ. Line R is the perpendicular bisector of segment LC. Let U be the reflection of line R across line X. Point Q is the intersection of lines U and M. Find the distance between points Q and D. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.08", "dataset": "MATH"}
{"messages": [{"content": "BCDEF is a regular 5-gon with center A and circumradius 12. Let Z be the rotation of point C by angle 3pi/4 around point F. Line S is the perpendicular bisector of segment ED. Let R be the line through point Z perpendicular to segment BC. Let G be the reflection of line R across line S. Point W is the reflection of point F across point C. Point N is the reflection of point W across line G. What is the distance between points N and W? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29.79", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center N and circumradius 12. Let Z be the line through points E and D. Let K be the midpoint of segment NA. Let Y be the reflection of point E across point A. Let G be the reflection of line Z through point K. Line V passes through points C and Y. Let I be the line through point A perpendicular to line V. Let F be the line through point N parallel to line I. Lines F and G intersect at point T. Find the distance between points K and T. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.9", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDFGHIJ be a regular 9-gon with center E and circumradius 6. Let P be the angle bisector of angle GAC. Let R be the circumcircle of polygon ABCDFGHIJ. Circle K is the reflection of circle R across line P. Let Z be the inversion of point C in circle K. Let N be the inversion of point Z in circle R. What is the length of segment NH? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.48", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFG be a regular hexagon with center C and circumradius 8. Point V is the reflection of point E across point A. Point R is the midpoint of segment CD. Let M be the line through points R and V. Point O is the reflection of point B across line M. What is the distance between points V and O? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21.17", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center W and circumradius 1. Let V be the line through points E and L. Line R passes through point G perpendicular to line V. Let Q be the midpoint of segment GF. Line Y is the reflection of line R across point J. Point N is the reflection of point J across line Y. Find the length of segment NQ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.38", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center W and circumradius 7. Point M is the midpoint of segment JG. Point P is the reflection of point E across point H. Point R is the reflection of point B across segment DP. What is the length of segment MR? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.83", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center X and circumradius 7. Line W is the angle bisector of angle JGI. Line N is the reflection of line W across point C. Let Y be the reflection of point K across point I. Point S is the reflection of point A across line N. Point M is the reflection of point I across point S. Find the distance between points Y and M. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "81.67", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJK is a regular 11-gon with center U and circumradius 11. Line Q is the angle bisector of angle IGU. Point R is the reflection of point U across line Q. Let O be the circumcircle of polygon ABCDEFGHIJK. Circle S is the reflection of circle O across point H. Let Z be the inversion of point A in circle S. Find the length of segment RZ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.75", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center P and circumradius 10. Let J be the circumcircle of polygon ABCDEFGH. Let K be the line through points C and D. Let R be the angle bisector of angle CHF. Let Z be the line through points F and H. Lines K and Z intersect at point M. Circle O is the reflection of circle J across line R. Let Q be the inversion of point F in circle O. What is the length of segment MQ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27.06", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center J and circumradius 3. Point Z is the reflection of point G across point E. Line I is the perpendicular bisector of segment JG. Line S passes through point Z parallel to line I. Point T is the reflection of point D across line S. What is the distance between points T and J? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.42", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJK is a regular 11-gon with center S and circumradius 4. Line P passes through points A and D. Let Z be the line through point F parallel to line P. Point L is the reflection of point G across line Z. Let W be the midpoint of segment KS. Find the length of segment WL. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.15", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center X and circumradius 7. Let Z be the angle bisector of angle BAE. Let L be the rotation of point E by angle 3pi/4 around point B. Line T is the reflection of line Z across point L. Point W is the reflection of point B across line T. Point O is the reflection of point D across point X. Point K is the reflection of point B across point W. What is the distance between points O and K? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "104.55", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEG be a regular hexagon with center F and circumradius 2. Let P be the circumcircle of polygon ABCDEG. Let V be the reflection of circle P through point C. Point O is the inversion of point A with respect to circle V. Let I be the midpoint of segment DO. Find the distance between points I and G. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.21", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFHIJKLM is a regular dodecagon with center G and circumradius 9. Let R be the reflection of point K across point J. Let T be the angle bisector of angle AJR. Point P is the reflection of point B across line T. Let W be the line through points D and H. Line S passes through point F parallel to line W. Let U be the reflection of point K across point P. Point X is the intersection of line S and segment UD. What is the length of segment RX? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.79", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center I and circumradius 7. Point M is obtained by rotating point E by angle 5pi/12 about point B. Line W is the angle bisector of angle ICM. Point P is the reflection of point M across line W. Point K is the midpoint of segment EB. Point F is the midpoint of segment MP. What is the length of segment KF? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.16", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center S and circumradius 11. Line M passes through points J and C. Let L be the line through point I parallel to line M. Let W be the reflection of line L across line M. Point U is the reflection of point J across line W. What is the length of segment IU? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19.51", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center Q and circumradius 1. Point P is obtained by rotating point A by angle pi about point E. Point I is the reflection of point E across segment BP. Line X passes through point H perpendicular to segment BP. Point V is the reflection of point I across line X. What is the length of segment BV? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.33", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center Y and circumradius 12. Let J be the circumcircle of polygon ABCDEFGHI. Let S be the reflection of point E across point F. Let V be the reflection of circle J through point S. Point R is the inversion of point I with respect to circle V. Let N be the inversion of point H in circle V. What is the distance between points N and R? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.18", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDF be a regular 5-gon with center E and circumradius 1. Line N passes through points D and F. Let R be the line through point A parallel to line N. Let Q be the perpendicular bisector of segment CB. Let H be the reflection of line R across line Q. Let J be the circumcircle of polygon ABCDF. Let T be the reflection of line N across line H. Let S be the reflection of point F across point E. Let V be the reflection of circle J through point S. Point L is the reflection of point S across line T. Point M is the inversion of point L with respect to circle V. Find the length of segment MA. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.44", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center W and circumradius 5. Let Y be the angle bisector of angle CED. Let P be the line through points B and E. Point S is the reflection of point D across line P. Line I is the reflection of line Y across point C. Point R is the reflection of point B across line I. Let M be the midpoint of segment ES. Find the length of segment MR. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.51", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHI be a regular 9-gon with center M and circumradius 9. Point Q is the reflection of point F across point M. Let L be the perpendicular bisector of segment QI. Let S be the reflection of point Q across point I. Let N be the midpoint of segment AF. Point Z is the reflection of point N across line L. What is the length of segment SZ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.64", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFHI be a regular octagon with center G and circumradius 6. Let M be the reflection of point A across point E. Let Y be the line through point M perpendicular to segment AI. Point T is the reflection of point C across line Y. Point L is the reflection of point F across segment DH. Find the distance between points L and T. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18.93", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center V and circumradius 9. Let K be the line through points E and H. Let P be the line through point D parallel to segment BF. Lines K and P intersect at point R. What is the length of segment RV? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.69", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHI be a regular 9-gon with center X and circumradius 1. Let O be the circumcircle of polygon ABCDEFGHI. Point Q is the reflection of point D across point H. Point P is the reflection of point I across point Q. Let M be the reflection of circle O through point Q. Point K is the inversion of point F with respect to circle M. What is the length of segment KP? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.85", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center U and circumradius 7. Point K is the midpoint of segment FE. Point Q is the reflection of point K across point B. Point Z is the midpoint of segment JQ. Let M be the reflection of point Z across point E. What is the length of segment BM? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28.82", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center T and circumradius 3. Point Y is the midpoint of segment AB. Let R be the circumcircle of polygon ABCD. Let N be the inversion of point Y in circle R. Let Q be the midpoint of segment AN. What is the distance between points Q and A? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.5", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center E and circumradius 10. Let V be the circumcircle of polygon ABCD. Point K is obtained by rotating point E by angle 3pi/4 about point D. Line L passes through points E and K. Point I is the reflection of point C across line L. Point H is the midpoint of segment AK. Let W be the midpoint of segment IH. Let Z be the inversion of point W in circle V. What is the distance between points Z and H? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.15", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center S and circumradius 2. Let V be the perpendicular bisector of segment SE. Point Z is the reflection of point G across line V. Let X be the line through points B and F. Line O passes through points H and C. Point W is the intersection of lines X and O. Line M is the reflection of line V across point W. Point P is the reflection of point F across line M. What is the length of segment ZP? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.78", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center T and circumradius 4. Point U is obtained by rotating point C by angle 7pi/12 about point G. Point P is obtained by rotating point H by angle 11pi/12 about point T. Let V be the line through points U and B. Point X is the reflection of point H across line V. Let S be the reflection of point A across point X. Point W is the reflection of point C across point S. Point Y is the midpoint of segment WP. What is the distance between points Y and C? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center Z and circumradius 3. Line V passes through points Z and G. Line M passes through point L parallel to line V. Let W be the reflection of line M across line V. Point X is the reflection of point A across line W. Let T be the midpoint of segment GX. What is the length of segment TL? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.97", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center Z and circumradius 3. Let Q be the perpendicular bisector of segment GL. Let R be the line through point H parallel to line Q. Point S is the reflection of point D across line R. Point W is the reflection of point S across line Q. Find the length of segment WK. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.2", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center R and circumradius 3. Line N is the perpendicular bisector of segment BR. Let U be the midpoint of segment AF. Let Y be the reflection of point E across point C. Point V is the reflection of point Y across line N. Point P is the reflection of point V across segment UA. Find the length of segment PA. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center Y and circumradius 10. Point V is the reflection of point E across point B. Let N be the perpendicular bisector of segment EC. Point P is the reflection of point E across point A. Let M be the line through point V perpendicular to line N. Let K be the line through points D and P. Let J be the reflection of line M through point C. Point I is the intersection of lines K and J. Find the distance between points I and B. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32.45", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center J and circumradius 1. Let P be the circumcircle of polygon ABCD. Let S be the midpoint of segment BJ. Point M is the midpoint of segment SA. Let T be the inversion of point M in circle P. What is the length of segment MT? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.23", "dataset": "MATH"}
{"messages": [{"content": "Let BCDEFGHIJK be a regular 10-gon with center A and circumradius 5. Let N be the line through points D and B. Line M is the reflection of line N across point K. Point V is the reflection of point F across line M. Line P passes through points V and E. Point L is the intersection of lines P and N. Find the length of segment GL. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.29", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center T and circumradius 11. Point M is the reflection of point B across point A. Let N be the reflection of point D across point A. Let Y be the circumcircle of polygon ABCDEF. Point X is the inversion of point N with respect to circle Y. What is the measure of angle XAM, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.09", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEG be a regular hexagon with center F and circumradius 9. Let N be the vector from point F to point B. Point Z is the translation of point C by vector N. Line V is the perpendicular bisector of segment FE. Line Q is the reflection of line V across point Z. Point U is the reflection of point B across line Q. Point S is the midpoint of segment CU. Find the length of segment SG. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35.72", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIKLM be a regular dodecagon with center J and circumradius 2. Let Y be the angle bisector of angle ABD. Point Z is the midpoint of segment JI. Line P is the reflection of line Y across point M. Point V is the reflection of point J across line P. What is the length of segment ZV? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.55", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEG be a regular hexagon with center F and circumradius 2. Point S is the reflection of point E across point D. Point W is the reflection of point D across segment CS. Point L is the midpoint of segment AW. What is the distance between points B and L? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.0", "dataset": "MATH"}
{"messages": [{"content": "ACDEFGHIJKL is a regular 11-gon with center B and circumradius 3. Let Q be the vector from point G to point C. Line R is the angle bisector of angle ICK. Let Y be point I translated by vector Q. Let S be the perpendicular bisector of segment CD. Line O passes through point A perpendicular to line S. Line W is the reflection of line R across line O. Point U is the reflection of point J across line W. Find the distance between points U and Y. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.56", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center O and circumradius 12. Let N be the circumcircle of polygon ABCDEF. Line Q is the angle bisector of angle BCD. Let Y be the midpoint of segment AC. Let K be the line through point F perpendicular to line Q. Let Z be the inversion of point Y in circle N. Point S is the reflection of point Z across line K. What is the distance between points Z and S? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center K and circumradius 1. Line O is the angle bisector of angle EBA. Let I be the perpendicular bisector of segment DE. Let Q be the reflection of line I across line O. Point J is the reflection of point F across line Q. Find the distance between points J and A. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHJKLM is a regular dodecagon with center I and circumradius 10. Line S passes through points E and L. Point T is the reflection of point K across point E. Point P is the reflection of point T across line S. Point Y is the midpoint of segment IP. What is the length of segment KY? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.13", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center Q and circumradius 11. Let X be the circumcircle of polygon ABCDEFGH. Point V is the midpoint of segment HC. Let T be the inversion of point V in circle X. Let Z be the midpoint of segment TH. What is the distance between points A and Z? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.37", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center L and circumradius 11. Line R is the angle bisector of angle GHB. Point P is the reflection of point G across line R. Let V be the midpoint of segment PD. Let Y be the midpoint of segment PH. What is the length of segment VY? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.0", "dataset": "MATH"}
{"messages": [{"content": "ACDEFGHIJKLM is a regular dodecagon with center B and circumradius 12. Line T passes through point D perpendicular to segment AJ. Let W be the line through point E perpendicular to line T. Point P is the reflection of point F across line W. Find the distance between points F and P. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.22", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFGHIJKLM be a regular dodecagon with center C and circumradius 9. Let W be the circumcircle of polygon ABDEFGHIJKLM. Circle N is the reflection of circle W across point H. Point O is the inversion of point I with respect to circle N. Line X passes through points K and D. Point Y is the reflection of point O across line X. Point T is the reflection of point B across line X. Find the distance between points T and Y. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.54", "dataset": "MATH"}
{"messages": [{"content": "ABCDFGHI is a regular octagon with center E and circumradius 4. Let X be the perpendicular bisector of segment DF. Line T passes through point F parallel to line X. Let R be the line through point H perpendicular to line T. Line O is the perpendicular bisector of segment EG. Line S is the reflection of line O across line R. Point K is the reflection of point C across line S. Point N is the midpoint of segment KE. Find the distance between points I and N. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.67", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center Q and circumradius 4. Line I is the angle bisector of angle GEA. Line O is the reflection of line I across point B. Point Y is the reflection of point G across point E. Let Z be the line through point E perpendicular to line O. Point J is the reflection of point F across line Z. Find the length of segment JY. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.3", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKM is a regular dodecagon with center L and circumradius 8. Let Y be the rotation of point C by angle 5pi/12 around point D. Let O be the perpendicular bisector of segment IF. Line V passes through point G parallel to line O. Point S is the intersection of line V and segment EJ. What is the length of segment YS? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.12", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center W and circumradius 4. Let R be the circumcircle of polygon ABCDEFGH. Circle Z is the reflection of circle R across point F. Let Q be the reflection of circle Z through point W. Point L is the inversion of point H with respect to circle Q. What is the length of segment GL? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.45", "dataset": "MATH"}
{"messages": [{"content": "ABDEFG is a regular hexagon with center C and circumradius 5. Let W be the line through points F and C. Let N be the reflection of point E across point B. Let Z be the midpoint of segment AN. Let S be the line through point Z perpendicular to segment BD. Line V is the reflection of line S across line W. Let M be the perpendicular bisector of segment BD. Line O is the reflection of line M across line V. Lines O and W intersect at point K. Let L be the circumcircle of polygon ABDEFG. Point Y is the midpoint of segment DK. Let I be the inversion of point Y in circle L. What is the length of segment IC? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.59", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center Z and circumradius 6. Line N is the perpendicular bisector of segment BC. Line R passes through point Z perpendicular to line N. Line T passes through point C parallel to line R. Point W is the reflection of point B across point C. Point X is the midpoint of segment WJ. Point S is the reflection of point A across line T. Find the length of segment XS. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.94", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFH be a regular 7-gon with center G and circumradius 4. Point L is the midpoint of segment GH. Point Z is the midpoint of segment BC. Point T is the reflection of point C across segment EZ. Let I be the midpoint of segment LT. Find the length of segment FI. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.66", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center Y and circumradius 4. Point O is the reflection of point D across segment BK. Let T be the perpendicular bisector of segment JK. Line Z passes through point L parallel to line T. Point U is the intersection of line Z and segment BK. Let Q be the midpoint of segment OU. What is the length of segment UQ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.48", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center Z and circumradius 8. Let V be the perpendicular bisector of segment FD. Line O passes through point D parallel to line V. Point J is the reflection of point Z across line O. Let R be the midpoint of segment CB. Point I is the reflection of point R across point C. What is the distance between points J and I? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center T and circumradius 3. Point H is the midpoint of segment DC. Let Q be the circumcircle of polygon ABCDEF. Circle W is the reflection of circle Q across point B. Let X be the inversion of point H in circle W. Find the distance between points X and H. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.16", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center U and circumradius 6. Let P be the midpoint of segment AU. Let H be the reflection of point P across point A. Let I be the midpoint of segment DP. What is the distance between points H and I? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.5", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center X and circumradius 12. Line Z passes through points X and B. Line T passes through point C perpendicular to line Z. Let I be the midpoint of segment FD. Point W is the reflection of point E across line T. Find the distance between points W and I. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31.55", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center V and circumradius 3. Let N be the reflection of point A across point F. Let U be the perpendicular bisector of segment AF. Line Q is the reflection of line U across point C. Let K be the line through point N perpendicular to line Q. Let H be the reflection of line K through point D. Lines H and Q intersect at point L. Point T is the reflection of point E across point L. Find the measure of angle TVD, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.05", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center K and circumradius 3. Let W be the circumcircle of polygon ABCDEF. Let R be the reflection of circle W through point F. Point O is the inversion of point B with respect to circle R. Line U is the angle bisector of angle CBO. Let L be the reflection of circle W across line U. Point Y is the inversion of point A with respect to circle L. What is the length of segment EY? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.17", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJK is a regular 11-gon with center Q and circumradius 9. Let R be the angle bisector of angle HCA. Point W is the midpoint of segment DA. Line S passes through point F perpendicular to line R. Point N is the reflection of point W across line S. Point O is the reflection of point A across point N. What is the distance between points O and F? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.47", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGHIJKLM is a regular dodecagon with center F and circumradius 1. Point Y is the midpoint of segment HL. Let Z be the line through point Y perpendicular to segment EM. Point U is the reflection of point B across line Z. Let O be the perpendicular bisector of segment MB. Line R passes through point A parallel to segment EM. Line Q is the reflection of line O across line R. Point S is the reflection of point U across line Q. What is the length of segment SG? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.19", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center G and circumradius 4. Line S is the perpendicular bisector of segment AG. Let O be the circumcircle of polygon ABCDEF. Let X be the reflection of circle O across line S. Point L is the inversion of point D with respect to circle X. Find the distance between points L and A. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHJ be a regular 9-gon with center I and circumradius 2. Point S is obtained by rotating point C by angle pi about point H. Let L be the circumcircle of polygon ABCDEFGHJ. Let V be the midpoint of segment HA. Line Y passes through points J and S. Point X is the reflection of point V across line Y. Point R is the inversion of point X with respect to circle L. What is the length of segment ER? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.71", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGI be a regular octagon with center H and circumradius 12. Line Z is the angle bisector of angle ECD. Line N is the reflection of line Z across point I. Point Y is obtained by rotating point F by angle pi about point C. Let X be the midpoint of segment IA. Let W be the reflection of line N across line Z. Line K is the reflection of line Z across line W. Let T be the reflection of point Y across point I. Point Q is the reflection of point T across line K. What is the distance between points X and Q? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "250.4", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGHI is a regular octagon with center F and circumradius 8. Let X be the reflection of point E across point I. Let O be the midpoint of segment IH. Point P is obtained by rotating point F by angle 11pi/12 about point I. Line W passes through points H and G. Let U be the midpoint of segment IX. Line K passes through point P parallel to line W. Point J is the reflection of point U across line K. Find the distance between points O and J. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.19", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center M and circumradius 1. Let V be the midpoint of segment IE. Line L passes through point J parallel to segment IE. Line T is the perpendicular bisector of segment DC. Line N is the reflection of line L across point E. Let W be the reflection of line T through point D. Let Z be the line through point M parallel to line N. Lines W and Z intersect at point Y. Point P is the midpoint of segment DC. Point X is the reflection of point P across point V. What is the length of segment YX? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.17", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center Z and circumradius 2. Segments AC and BI intersect at point Q. Point U is the reflection of point D across segment BI. Find the distance between points Q and U. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.95", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEFGHIJKLM be a regular dodecagon with center B and circumradius 5. Point S is the midpoint of segment AD. Point N is the midpoint of segment LH. What is the distance between points N and S? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.61", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEGHIJKLM be a regular dodecagon with center F and circumradius 9. Let R be the angle bisector of angle FAH. Let X be the reflection of point F across point L. Line V is the perpendicular bisector of segment XM. Let Z be the reflection of line V across line R. Point P is the reflection of point L across line Z. What is the distance between points P and D? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21.06", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center Z and circumradius 6. Point V is the reflection of point A across point I. Let T be the perpendicular bisector of segment EF. Let Y be the circumcircle of polygon ABCDEFGHIJKL. Let Q be the inversion of point V in circle Y. Line M passes through point J perpendicular to line T. Point O is the reflection of point G across line M. Find the length of segment QO. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.04", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center L and circumradius 10. Let O be the rotation of point B by angle pi around point E. Let Y be the rotation of point O by angle 3pi/4 around point C. Line J passes through point Y parallel to segment LA. Point K is the reflection of point O across segment LA. Let P be the line through point Y perpendicular to line J. Point Q is the reflection of point K across line P. Find the distance between points K and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.49", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGIJ is a regular 9-gon with center H and circumradius 11. Let X be the line through points E and C. Let L be the reflection of line X through point I. Point O is the reflection of point G across line L. Point R is the midpoint of segment HA. Find the length of segment OR. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51.95", "dataset": "MATH"}
{"messages": [{"content": "ABCEF is a regular 5-gon with center D and circumradius 3. Point P is the reflection of point F across segment AB. Let Y be the reflection of point F across point D. Point W is the reflection of point Y across segment AB. What is the measure of angle WPA, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.94", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center I and circumradius 6. Line W passes through points D and B. Point Z is the reflection of point G across line W. Point T is the midpoint of segment FA. Point L is the reflection of point Z across point A. Find the length of segment TL. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.3", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center V and circumradius 3. Let Z be the line through points H and G. Point P is the midpoint of segment AV. Line M passes through point F perpendicular to line Z. Point S is the reflection of point I across line M. Let U be the line through points S and F. Point K is the reflection of point G across line U. What is the length of segment KP? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.65", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center O and circumradius 6. Let T be the angle bisector of angle CBO. Let K be the circumcircle of polygon ABCDEF. Line I passes through point E perpendicular to line T. Let U be the reflection of circle K through point A. Let P be the inversion of point O in circle U. Let J be the reflection of point P across point B. Point Z is the reflection of point J across line I. Let H be the inversion of point Z in circle K. Find the distance between points H and A. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.14", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center R and circumradius 1. Point Q is the reflection of point D across segment CB. Point I is the reflection of point G across segment CB. Point T is the reflection of point I across point H. Find the length of segment TQ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.54", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center V and circumradius 8. Point H is the midpoint of segment BV. Let I be the angle bisector of angle CAV. Point Q is the midpoint of segment HV. Line N is the reflection of line I across point V. Line J passes through points Q and E. Point O is the intersection of lines N and J. Find the distance between points O and H. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.27", "dataset": "MATH"}
{"messages": [{"content": "ABDEFGHI is a regular octagon with center C and circumradius 2. Line U is the angle bisector of angle FGE. Line S is the perpendicular bisector of segment ID. Let X be the reflection of line U across line S. Point Q is the reflection of point A across line X. What is the measure of angle QFB, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.36", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center Y and circumradius 4. Let W be the midpoint of segment CY. Point J is the midpoint of segment HW. Point N is the midpoint of segment JA. Let P be the midpoint of segment JN. Let I be the reflection of point F across point P. What is the distance between points I and C? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.18", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center S and circumradius 9. Let M be the angle bisector of angle KSC. Let V be the line through point S perpendicular to line M. Let Q be the reflection of point J across point F. Let Z be the reflection of line V through point Q. Point N is the reflection of point G across line Z. Point R is the midpoint of segment HN. Find the length of segment RL. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38.43", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center K and circumradius 2. Point Z is the reflection of point B across point D. Let Q be the midpoint of segment BC. Point U is the reflection of point H across point C. Point T is the reflection of point U across segment CZ. Find the length of segment QT. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.73", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIKLM be a regular dodecagon with center J and circumradius 5. Let Y be the angle bisector of angle DMB. Line S passes through point L perpendicular to line Y. Point T is the reflection of point J across point I. Point O is the reflection of point D across line S. Point V is the reflection of point O across point H. Find the length of segment TV. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19.36", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center Y and circumradius 6. Let Q be the midpoint of segment BA. Let N be the reflection of point G across point I. Line L is the angle bisector of angle GYA. Let U be the reflection of line L through point Q. Let S be the reflection of line U through point I. Line K passes through point N perpendicular to line S. Point R is the reflection of point Y across line K. Find the length of segment QR. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21.08", "dataset": "MATH"}
{"messages": [{"content": "ABCEFGHIJKLM is a regular dodecagon with center D and circumradius 5. Line W passes through point A perpendicular to segment GJ. Line R is the reflection of line W across point K. Point S is the reflection of point A across line R. What is the length of segment SH? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19.32", "dataset": "MATH"}
{"messages": [{"content": "Let ABCEF be a regular 5-gon with center D and circumradius 7. Let G be the reflection of point E across point F. Let L be the circumcircle of polygon ABCEF. Circle O is the reflection of circle L across point A. Point T is the midpoint of segment EC. Let K be the inversion of point G in circle O. What is the distance between points T and K? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18.27", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center V and circumradius 8. Line I is the angle bisector of angle GAB. Point R is the reflection of point E across point F. Let S be the reflection of point R across point G. Point J is the reflection of point S across line I. What is the distance between points J and B? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.5", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJK is a regular 11-gon with center L and circumradius 2. Line W is the angle bisector of angle KDE. Line M passes through points J and G. Let Q be the reflection of line W across line M. Lines Q and M intersect at point P. Find the distance between points B and P. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.46", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center P and circumradius 5. Point Y is obtained by rotating point F by angle pi/2 about point L. Point R is the reflection of point Y across point A. Line V is the perpendicular bisector of segment FL. Let N be the line through point P perpendicular to line V. Point S is the reflection of point R across line N. Find the length of segment SJ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.73", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center X and circumradius 11. Point Q is the midpoint of segment CF. Let H be the midpoint of segment XG. Point T is the midpoint of segment DQ. Let Z be the angle bisector of angle FAT. Line J passes through point H perpendicular to line Z. Point S is the reflection of point B across line J. What is the length of segment AS? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.83", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center R and circumradius 3. Let V be the midpoint of segment BC. Point N is the reflection of point V across segment GE. Point J is the midpoint of segment CN. Find the distance between points R and J. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.83", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJK is a regular 11-gon with center M and circumradius 5. Line S is the perpendicular bisector of segment MG. Line Z passes through point D perpendicular to segment CE. Line Y passes through point F perpendicular to segment CE. Line X passes through point I parallel to line Z. Line L passes through point M parallel to line S. Line U is the reflection of line X across line Y. Lines U and L intersect at point Q. Find the distance between points C and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "50.35", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center W and circumradius 7. Let Z be the perpendicular bisector of segment FB. Let V be the line through point D parallel to segment EA. Let K be the reflection of line Z through point F. Lines V and K intersect at point P. Let L be the midpoint of segment AP. Point J is the reflection of point L across line Z. What is the distance between points H and J? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.69", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center I and circumradius 1. Point V is the reflection of point E across point I. Let L be the line through points V and A. Point N is the reflection of point I across line L. Let F be the midpoint of segment NI. What is the length of segment EF? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.43", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center M and circumradius 2. Let S be the rotation of point E by angle 7pi/12 around point B. Let G be the perpendicular bisector of segment AC. Point K is the midpoint of segment AM. Line L is the reflection of line G across point C. Point T is the reflection of point B across point S. Point Z is the reflection of point C across line L. Point Q is the midpoint of segment TZ. What is the length of segment KQ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.07", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center S and circumradius 3. Let I be the reflection of point E across point D. Line Z is the perpendicular bisector of segment BD. Point H is the reflection of point I across line Z. Let G be the reflection of point H across point A. What is the length of segment GA? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center X and circumradius 9. Let W be the line through points B and E. Line M is the angle bisector of angle BHD. Line Q passes through point A parallel to line W. Point V is the intersection of line Q and segment DG. Let O be the line through point F perpendicular to line M. Point S is the reflection of point V across line O. What is the measure of angle SFE, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.75", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center M and circumradius 8. Point F is the midpoint of segment EC. Point U is the reflection of point F across point B. Let L be the midpoint of segment UA. Find the length of segment LC. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.41", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center X and circumradius 9. Point Y is the midpoint of segment FH. Let T be the angle bisector of angle IKJ. Line P passes through point A perpendicular to line T. Point S is the reflection of point I across line P. What is the length of segment SY? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32.11", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHJKLM is a regular dodecagon with center I and circumradius 9. Point W is the midpoint of segment AI. Let P be the reflection of point E across point G. Point O is the reflection of point B across point W. What is the distance between points O and P? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.07", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center Z and circumradius 8. Let Q be the reflection of point G across point A. Let V be the midpoint of segment DQ. Let U be the reflection of point Z across point Q. Find the length of segment VU. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28.35", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center S and circumradius 3. Let V be the perpendicular bisector of segment FC. Line T is the reflection of line V across point L. Point M is the reflection of point L across line T. Line N passes through point M parallel to line T. Point R is the reflection of point S across line N. Find the length of segment RD. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.3", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center Z and circumradius 7. Let X be the rotation of point I by angle pi/4 around point D. Let W be the line through point Z parallel to segment DB. Let S be the line through point D perpendicular to line W. Point N is the reflection of point X across line S. What is the distance between points N and A? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21.68", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center O and circumradius 5. Point U is the reflection of point O across point J. Line P passes through point D perpendicular to segment HE. Line Q passes through point U parallel to segment HE. Line N passes through point G parallel to line P. Point Y is the intersection of lines N and Q. Let X be the midpoint of segment DY. What is the distance between points B and X? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.39", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center Y and circumradius 1. Line P passes through point G perpendicular to segment LH. Let O be the reflection of line P through point L. Let Z be the reflection of line O across line P. Point W is the reflection of point A across line Z. What is the distance between points W and Y? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.46", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center X and circumradius 1. Let T be the angle bisector of angle XDC. Let L be the line through point A perpendicular to line T. Line G is the reflection of line L across point E. Let W be the line through point E perpendicular to line G. Point F is the reflection of point C across line W. What is the length of segment BF? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.69", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center X and circumradius 12. Let N be the line through points C and X. Let P be the line through point B parallel to line N. Let S be the reflection of point I across point C. Point Q is the intersection of line P and segment JG. Find the length of segment SQ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41.32", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center Y and circumradius 8. Line T is the angle bisector of angle YAJ. Point N is the reflection of point L across line T. Let M be the reflection of point G across point H. Point Q is the reflection of point M across line T. What is the distance between points Q and N? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.4", "dataset": "MATH"}
{"messages": [{"content": "ABCDF is a regular 5-gon with center E and circumradius 2. Point I is the reflection of point F across point B. Let J be the rotation of point E by angle pi/2 around point C. Line S is the angle bisector of angle BDJ. Let W be the midpoint of segment FA. Point T is the reflection of point W across line S. Let R be the reflection of point I across point W. What is the length of segment RT? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.31", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center Y and circumradius 10. Let M be the rotation of point D by angle 3pi/4 around point A. Let N be the circumcircle of polygon ABCDEF. Let S be the line through point M perpendicular to segment DA. Let I be the reflection of circle N across line S. Point K is the inversion of point F with respect to circle I. What is the length of segment KE? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "51.72", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDFG be a regular hexagon with center E and circumradius 4. Let Z be the rotation of point B by angle pi/4 around point E. Point I is the midpoint of segment FC. Let T be the line through points E and Z. Point X is on line T. Line L passes through points I and X. Point R is the reflection of point Z across line L. Find the distance between points R and I. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.98", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center P and circumradius 12. Point J is the midpoint of segment HF. Let L be the midpoint of segment IJ. Point W is the reflection of point L across segment GB. What is the length of segment WP? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.24", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEFGHI be a regular octagon with center B and circumradius 5. Line T is the angle bisector of angle CEF. Let P be the midpoint of segment HI. Let M be the angle bisector of angle IBE. Point X is the reflection of point P across line M. Let N be the midpoint of segment PG. Let J be the line through point N parallel to line T. Point Z is the reflection of point X across line J. Point Q is the midpoint of segment ZH. What is the length of segment QH? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.01", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center K and circumradius 6. Let Q be the line through point K parallel to segment EF. Line T passes through points B and D. Lines T and Q intersect at point M. Point U is the reflection of point D across segment EF. Find the length of segment UM. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.54", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHI be a regular 9-gon with center Z and circumradius 2. Let U be the perpendicular bisector of segment ZC. Let S be the line through point A parallel to line U. Segment DH and line S intersect at point M. Point T is the reflection of point Z across segment DH. What is the distance between points M and T? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.69", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGHIJKLM is a regular dodecagon with center F and circumradius 3. Let N be the reflection of point F across point G. Let O be the line through points N and E. Let Q be the line through point H parallel to line O. Point R is the reflection of point C across line Q. Let S be the midpoint of segment BA. Find the length of segment SR. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.69", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center V and circumradius 3. Let K be the reflection of point C across point G. Point N is the reflection of point F across point K. Let L be the midpoint of segment HA. Line S passes through points L and N. Point I is the reflection of point E across line S. What is the distance between points I and F? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.45", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center P and circumradius 7. Let X be the circumcircle of polygon ABCDEFGHIJKL. Let V be the reflection of circle X through point G. Let T be the reflection of point C across point J. Let U be the reflection of circle V through point C. Point M is the inversion of point T with respect to circle U. What is the distance between points J and M? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27.21", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDFGHIJK be a regular 10-gon with center E and circumradius 10. Line L passes through points K and G. Let Z be the line through point F parallel to line L. Point W is the reflection of point B across line Z. Let S be the midpoint of segment ID. What is the length of segment WS? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.87", "dataset": "MATH"}
{"messages": [{"content": "ACDEFGHIJKLM is a regular dodecagon with center B and circumradius 11. Let Z be the circumcircle of polygon ACDEFGHIJKLM. Point N is the midpoint of segment AJ. Point P is the inversion of point N with respect to circle Z. What is the distance between points L and P? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center Y and circumradius 6. Let Z be the rotation of point B by angle pi around point C. Let T be the perpendicular bisector of segment BA. Point V is the reflection of point Z across line T. Point N is the reflection of point V across segment BE. Point J is the midpoint of segment CN. What is the distance between points J and D? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.92", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGHIJKL is a regular 11-gon with center F and circumradius 12. Let V be the circumcircle of polygon ABCDEGHIJKL. Let W be the midpoint of segment AE. Let P be the reflection of circle V through point A. Point Z is the reflection of point F across segment AE. Point S is the inversion of point W with respect to circle P. Find the distance between points Z and S. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.63", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center L and circumradius 3. Let Y be the line through point B perpendicular to segment GH. Let P be the line through point E parallel to line Y. Point X is the reflection of point G across line P. Point J is the reflection of point D across segment GH. Find the length of segment XJ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.42", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center Y and circumradius 3. Line G is the angle bisector of angle ACY. Let Z be the reflection of point E across point B. Let V be the line through point Z parallel to line G. Line H passes through point D parallel to line V. Point X is the reflection of point B across line H. Find the distance between points X and C. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.2", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center Y and circumradius 5. Let U be the circumcircle of polygon ABCDEFGH. Let Z be the reflection of circle U through point A. Let O be the perpendicular bisector of segment HE. Point V is the midpoint of segment HA. Point S is the inversion of point V with respect to circle Z. Point K is on line O. Point J lies on line O at distance 1 from point K. Line T passes through point S parallel to line O. Let Q be the reflection of line T through point J. Let P be the midpoint of segment AB. Point M is the midpoint of segment EB. Let L be the line through point M parallel to line Q. Point I is the reflection of point P across line L. Find the length of segment IE. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.35", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center Z and circumradius 4. Let J be the angle bisector of angle HEZ. Point S is the reflection of point D across line J. Let W be the circumcircle of polygon ABCDEFGH. Let U be the inversion of point S in circle W. What is the length of segment HU? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.43", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center L and circumradius 5. Let H be the circumcircle of polygon ABCDEF. Point O is obtained by rotating point B by angle pi/12 about point F. Line R is the perpendicular bisector of segment OB. Point U is the reflection of point L across line R. Let V be the line through points A and O. Let Z be the reflection of circle H across line V. Let Q be the inversion of point D in circle Z. Find the distance between points U and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.72", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center Q and circumradius 1. Let P be the perpendicular bisector of segment AB. Let L be the line through point E parallel to line P. Let U be the reflection of line L across line P. Let S be the line through point D perpendicular to segment BQ. Point Y is the reflection of point C across point E. Let J be the reflection of line P across line S. Let M be the line through point F parallel to line J. Line R passes through point Y parallel to line M. Lines R and U intersect at point I. Point O is the reflection of point I across line P. What is the distance between points A and O? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.46", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDFGHIJK be a regular 10-gon with center E and circumradius 9. Let L be the angle bisector of angle DFJ. Line X is the reflection of line L across point B. Point V is the reflection of point J across line X. Let W be the reflection of point V across point A. Find the length of segment WG. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30.89", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center Y and circumradius 5. Line K passes through points A and J. Point N is obtained by rotating point E by angle pi/12 about point H. Line Z is the angle bisector of angle BFN. Line M is the reflection of line K across line Z. Line P passes through point N parallel to line K. Point L is the intersection of lines P and M. Find the distance between points L and D. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36.95", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center V and circumradius 1. Let M be the line through point J perpendicular to segment IC. Line Z is the reflection of line M across point L. Let Y be the line through point G parallel to line Z. Line Y and segment IC intersect at point R. What is the distance between points R and B? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.45", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKM be a regular dodecagon with center L and circumradius 9. Line P is the perpendicular bisector of segment JE. Let U be the perpendicular bisector of segment LB. Let T be the reflection of line U across line P. Line W is the reflection of line P across line T. Point X is the intersection of lines W and T. What is the length of segment EX? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.15", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center I and circumradius 6. Line V is the perpendicular bisector of segment EF. Let O be the reflection of line V through point H. Line M passes through point C perpendicular to line O. Point Q is the reflection of point E across line M. Find the length of segment QB. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.59", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDFGHIJKLM be a regular dodecagon with center E and circumradius 1. Let Z be the reflection of point F across point D. Let V be the perpendicular bisector of segment LZ. Let P be the line through point G perpendicular to line V. Point R is the reflection of point M across line P. Point N is the midpoint of segment AB. Find the distance between points R and N. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.61", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center N and circumradius 12. Line O is the angle bisector of angle CFE. Let T be the line through point A perpendicular to line O. Let M be the circumcircle of polygon ABCDEF. Let R be the reflection of circle M across line T. Let V be the inversion of point C in circle R. Find the length of segment AV. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.86", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center G and circumradius 2. Line Q is the perpendicular bisector of segment DC. Let F be the line through point D perpendicular to line Q. Point H is the reflection of point A across line F. Point E is the reflection of point H across point G. Let L be the midpoint of segment CD. What is the length of segment EL? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.83", "dataset": "MATH"}
{"messages": [{"content": "ABDEFGHI is a regular octagon with center C and circumradius 4. Line X passes through points G and E. Let Y be the reflection of line X through point B. Let Z be the reflection of line Y through point H. Point T is the reflection of point B across line Z. Find the distance between points T and I. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.32", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHJKLM be a regular dodecagon with center I and circumradius 12. Let X be the line through points F and H. Line V passes through point E parallel to line X. Point Y is the intersection of line V and segment AG. What is the length of segment DY? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.42", "dataset": "MATH"}
{"messages": [{"content": "ABCE is a square with center D and circumradius 12. Let F be the midpoint of segment AE. Let H be the reflection of point D across point F. Line R passes through point A perpendicular to segment CH. Point L is the reflection of point F across line R. Find the measure of angle LFB, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.36", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJK be a regular 11-gon with center Q and circumradius 4. Line O passes through points K and J. Point U is the reflection of point F across line O. Let P be the midpoint of segment DK. Find the length of segment UP. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.46", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center P and circumradius 7. Let R be the circumcircle of polygon ABCDEFGHIJKL. Let Q be the midpoint of segment IK. Point M is the inversion of point Q with respect to circle R. Let T be the midpoint of segment MH. Find the distance between points I and T. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.54", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center M and circumradius 1. Let W be the midpoint of segment AM. Line O is the angle bisector of angle DEB. Line Q is the reflection of line O across point A. Point Y is the reflection of point B across line Q. Let T be the midpoint of segment YE. Let X be the reflection of point W across point M. What is the distance between points T and X? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.84", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEG be a regular hexagon with center F and circumradius 6. Let K be the angle bisector of angle GFD. Point Z is the reflection of point F across point G. Point M is the midpoint of segment ZE. Point P is the reflection of point Z across line K. What is the length of segment PM? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.59", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDFG be a regular hexagon with center E and circumradius 3. Point S is the reflection of point D across point B. Let P be the circumcircle of polygon ABCDFG. Let T be the reflection of circle P through point S. Let Y be the reflection of circle T through point E. Point J is the inversion of point A with respect to circle Y. What is the length of segment JF? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.58", "dataset": "MATH"}
{"messages": [{"content": "ACDEFGHIJKL is a regular 11-gon with center B and circumradius 7. Line V is the angle bisector of angle HBK. Line W passes through point C parallel to segment AK. Line M is the reflection of line V across line W. Point P is the intersection of segment AK and line M. Find the length of segment PB. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.91", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center M and circumradius 4. Let Y be the angle bisector of angle BMG. Line Z passes through points F and H. Line K is the reflection of line Y across line Z. Point N is the reflection of point M across line K. What is the length of segment NE? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.54", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center X and circumradius 9. Let W be the circumcircle of polygon ABCDE. Point V is the midpoint of segment BC. Let Z be the line through points C and V. Let Y be the reflection of circle W across line Z. Let H be the inversion of point E in circle Y. Find the measure of angle HXD, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.89", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFHIJK be a regular 10-gon with center G and circumradius 5. Let W be the circumcircle of polygon ABCDEFHIJK. Point X is the midpoint of segment GI. Circle M is the reflection of circle W across point C. Let R be the inversion of point E in circle M. Let U be the midpoint of segment JX. What is the distance between points U and R? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.36", "dataset": "MATH"}
{"messages": [{"content": "QV = 11. Let ABCDEFG be a regular 7-gon with center V and circumradius 1. Let U be the midpoint of segment FA. Point S is the reflection of point G across point A. Point R is the reflection of point S across segment DA. Find the distance between points R and U. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.72", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center Q and circumradius 11. Line H is the angle bisector of angle EAD. Point R is the reflection of point C across line H. Line N is the angle bisector of angle REA. Line S is the reflection of line N across line H. Line M passes through point Q parallel to line S. Point P is the reflection of point D across line M. Let W be the reflection of point P across point D. Find the length of segment FW. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31.84", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center X and circumradius 12. Point G is the midpoint of segment CD. Let M be the line through point C perpendicular to segment EC. Point P is the reflection of point G across line M. Point Y is the reflection of point E across segment PG. What is the length of segment YG? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.87", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center N and circumradius 1. Let T be the midpoint of segment JL. Let M be the midpoint of segment DN. Line U is the angle bisector of angle MHE. Point Q is the reflection of point T across line U. Point O is the reflection of point I across point H. What is the distance between points Q and O? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.81", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFHI be a regular octagon with center G and circumradius 1. Let T be the midpoint of segment BE. Line Z is the perpendicular bisector of segment GD. Point W is the reflection of point B across line Z. Let Q be the reflection of point T across point W. Point O is the midpoint of segment CB. Find the distance between points O and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.63", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHI be a regular 9-gon with center V and circumradius 8. Let Q be the angle bisector of angle EVH. Let K be the reflection of point V across point B. Point T is the reflection of point C across point K. Let L be the reflection of line Q through point D. Point J is the reflection of point T across line L. What is the distance between points J and V? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "44.87", "dataset": "MATH"}
{"messages": [{"content": "Let ABDE be a square with center C and circumradius 2. Let H be the line through points C and D. Let N be the circumcircle of polygon ABDE. Let F be the angle bisector of angle DEB. Line S is the reflection of line F across line H. Circle L is the reflection of circle N across line S. Point K is the inversion of point A with respect to circle L. What is the distance between points K and D? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.77", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center H and circumradius 12. Let L be the reflection of point C across point H. Let Y be the circumcircle of polygon ABCDE. Line R is the perpendicular bisector of segment CH. Let J be the reflection of line R through point C. Circle I is the reflection of circle Y across line J. Circle K is the reflection of circle I across line R. Let S be the inversion of point D in circle K. Point M is the midpoint of segment LS. Find the length of segment MB. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.06", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center X and circumradius 5. Point F is obtained by rotating point X by angle pi about point C. Let R be the perpendicular bisector of segment BD. Let Q be the line through points F and A. Line N is the reflection of line Q across line R. Point U is the reflection of point X across line N. Find the length of segment UB. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.79", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center P and circumradius 4. Point V is the reflection of point E across point H. Line I is the perpendicular bisector of segment CB. Let Z be the line through point D parallel to line I. Point Y is the reflection of point V across line Z. What is the distance between points G and Y? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.13", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center V and circumradius 3. Line Z passes through points V and G. Let S be the line through points H and A. Line X passes through point F parallel to line Z. Lines S and X intersect at point J. Let O be the midpoint of segment BH. Find the length of segment OJ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.07", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center V and circumradius 8. Let H be the angle bisector of angle EBD. Let J be the line through point A perpendicular to line H. Point N is the reflection of point D across line J. Let O be the reflection of point D across point N. What is the length of segment CO? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26.99", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center N and circumradius 1. Let X be the midpoint of segment HD. Let Z be the midpoint of segment XG. Let U be the angle bisector of angle NDZ. Line S is the reflection of line U across point Z. Point M is the midpoint of segment DX. Let R be the line through point M parallel to line S. Point V is the reflection of point G across line R. Find the distance between points E and V. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.32", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDF be a regular 5-gon with center E and circumradius 1. Line T passes through points A and D. Point O is obtained by rotating point C by angle 3pi/4 about point D. Let Y be the perpendicular bisector of segment OA. Line G passes through point B parallel to line Y. Point S is the intersection of lines T and G. Find the distance between points S and F. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.05", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center N and circumradius 10. Point J is the midpoint of segment CD. Point W is the reflection of point J across segment CB. Let Z be the line through points A and W. Point Q is the reflection of point C across line Z. Find the distance between points Q and D. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19.61", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center O and circumradius 11. Point R is the reflection of point G across point E. Let Q be the line through points C and R. Point W is the reflection of point A across line Q. Point S is the midpoint of segment WH. What is the length of segment SC? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.16", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center H and circumradius 8. Point R is the midpoint of segment CH. Line G is the angle bisector of angle BCR. Point N is the reflection of point R across line G. Let Q be the midpoint of segment HR. Point J is the midpoint of segment RC. Let K be the midpoint of segment JA. Let L be the reflection of point K across point N. Find the distance between points Q and L. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.39", "dataset": "MATH"}
{"messages": [{"content": "ABCE is a square with center D and circumradius 2. Line N is the perpendicular bisector of segment ED. Let Q be the reflection of line N through point E. Point X is the reflection of point E across point A. Let O be the perpendicular bisector of segment BC. Let G be the reflection of line Q across line O. Segment BX and line G intersect at point S. What is the distance between points S and A? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.24", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJK be a regular 11-gon with center R and circumradius 3. Let W be the reflection of point K across point G. Line Y passes through points C and A. Let S be the reflection of point A across point W. Line L passes through point I perpendicular to line Y. Let Z be the circumcircle of polygon ABCDEFGHIJK. Circle V is the reflection of circle Z across line L. Let P be the inversion of point S in circle V. What is the distance between points R and P? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.27", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center U and circumradius 9. Let N be the reflection of point U across point C. Let J be the reflection of point U across point A. Let W be the midpoint of segment NB. What is the distance between points W and J? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.38", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center S and circumradius 5. Let R be the angle bisector of angle AEF. Point X is the reflection of point C across line R. Let U be the circumcircle of polygon ABCDEF. Let O be the inversion of point X in circle U. Find the length of segment AO. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.12", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center Y and circumradius 5. Point K is the midpoint of segment AY. Line O is the angle bisector of angle FGD. Let Q be the midpoint of segment BE. Let R be the reflection of line O through point D. Point T is the reflection of point K across line R. Let V be the midpoint of segment DQ. Point M is the reflection of point T across point V. Find the length of segment MY. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.68", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center Q and circumradius 8. Let F be the perpendicular bisector of segment CB. Let L be the line through point B perpendicular to line F. Let O be the midpoint of segment BQ. Point Z is the reflection of point O across line L. Find the distance between points B and Z. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center K and circumradius 6. Let U be the circumcircle of polygon ABCDEFGHIJ. Let S be the reflection of point A across point H. Circle N is the reflection of circle U across point I. Let T be the inversion of point S in circle N. Find the length of segment KT. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.46", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDFGHI be a regular octagon with center E and circumradius 5. Let T be the midpoint of segment GF. Point Q is the midpoint of segment TE. Point P is the midpoint of segment TF. Find the measure of angle QPF, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.96", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center U and circumradius 7. Let P be the perpendicular bisector of segment EA. Let O be the line through points C and A. Line K is the reflection of line P across line O. Point S is the midpoint of segment AC. Point L is the reflection of point A across line K. Let R be the reflection of point S across point L. Let T be the reflection of point R across point E. Find the distance between points L and T. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39.35", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center I and circumradius 9. Let R be the circumcircle of polygon ABCDEF. Line Z is the angle bisector of angle BFD. Point X is obtained by rotating point B by angle pi/4 about point E. Let Y be the reflection of circle R through point B. Point W is the inversion of point E with respect to circle Y. Circle V is the reflection of circle Y across point E. Point G is the inversion of point W with respect to circle V. Point J is the reflection of point G across line Z. Find the length of segment XJ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "44.61", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center M and circumradius 12. Let T be the midpoint of segment AC. Let Z be the midpoint of segment BT. Let P be the reflection of point Z across point F. What is the distance between points B and P? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center O and circumradius 9. Let U be the reflection of point B across point A. Line Z is the perpendicular bisector of segment UA. Point S is the reflection of point D across line Z. Point V is the midpoint of segment UO. Point Q is the reflection of point V across point C. What is the length of segment QS? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47.41", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center N and circumradius 2. Point Q is obtained by rotating point C by angle 7pi/12 about point E. Point X is the midpoint of segment BF. Let L be the midpoint of segment XG. Line O is the angle bisector of angle QNF. Point R is the reflection of point L across line O. What is the length of segment BR? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.98", "dataset": "MATH"}
{"messages": [{"content": "ABDEFGHIJKL is a regular 11-gon with center C and circumradius 3. Let Z be the circumcircle of polygon ABDEFGHIJKL. Line N is the angle bisector of angle GBL. Line Y is the perpendicular bisector of segment HF. Let V be the reflection of line N across line Y. Let X be the reflection of circle Z across line N. Let U be the midpoint of segment GF. Let Q be the line through points I and J. Let O be the inversion of point U in circle X. Lines V and Q intersect at point R. Find the distance between points O and R. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.22", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center T and circumradius 10. Line N passes through point F parallel to segment GI. Line U passes through point C parallel to line N. Point Q is the reflection of point B across line U. Find the length of segment QI. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGHIJK is a regular 10-gon with center F and circumradius 5. Point R is the midpoint of segment AH. Let T be the circumcircle of polygon ABCDEGHIJK. Circle O is the reflection of circle T across point I. Point Q is the midpoint of segment CR. Point V is the inversion of point Q with respect to circle O. Point P is the midpoint of segment VK. What is the length of segment PG? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.84", "dataset": "MATH"}
{"messages": [{"content": "ACDEFG is a regular hexagon with center B and circumradius 10. Line L is the perpendicular bisector of segment DB. Point W is the reflection of point D across segment GA. Let O be the angle bisector of angle GBF. Let K be the line through point W parallel to line O. Line N is the reflection of line L across line K. Let M be the line through points A and B. Point S is the midpoint of segment CG. Line Q is the reflection of line M across point F. Lines Q and N intersect at point V. What is the length of segment VS? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "86.75", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center S and circumradius 9. Let Y be the reflection of point F across point B. Let X be the perpendicular bisector of segment EY. Line J passes through point E parallel to line X. Point H is the reflection of point A across line J. What is the distance between points H and A? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22.47", "dataset": "MATH"}
{"messages": [{"content": "ABCEFGHIJKLM is a regular dodecagon with center D and circumradius 9. Let Z be the angle bisector of angle DMC. Point X is the midpoint of segment FM. Point Y is the reflection of point G across line Z. Point T is the reflection of point H across point X. Find the distance between points Y and T. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.26", "dataset": "MATH"}
{"messages": [{"content": "BCDEFGHIJKLM is a regular dodecagon with center A and circumradius 8. Point P is the reflection of point I across point B. Point N is obtained by rotating point L by angle 3pi/4 about point K. Line Z is the angle bisector of angle ACP. Let S be the line through point N parallel to line Z. Point Q is the reflection of point K across line S. Let T be the angle bisector of angle BQE. Point R is the reflection of point H across line T. Find the distance between points E and R. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.47", "dataset": "MATH"}
{"messages": [{"content": "ABDEFGHIJKLM is a regular dodecagon with center C and circumradius 6. Point N is obtained by rotating point F by angle 3pi/4 about point K. Let X be the circumcircle of polygon ABDEFGHIJKLM. Let V be the angle bisector of angle NDI. Let T be the reflection of circle X across line V. Let Y be the inversion of point L in circle T. What is the distance between points D and Y? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.29", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center V and circumradius 9. Let W be the perpendicular bisector of segment VA. Let N be the reflection of line W through point A. Point R is the reflection of point E across line N. What is the distance between points A and R? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.81", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center N and circumradius 9. Line T passes through points A and B. Let G be the reflection of line T through point E. Line Z passes through point A perpendicular to line G. Point L is the reflection of point D across line Z. What is the length of segment AL? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.12", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center J and circumradius 2. Let K be the circumcircle of polygon ABCDEF. Let T be the reflection of point E across point C. Let Z be the line through points D and T. Let Q be the reflection of circle K through point C. Let P be the reflection of circle Q across line Z. Point O is the inversion of point C with respect to circle P. What is the distance between points T and O? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.58", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFHIJKLM be a regular dodecagon with center G and circumradius 1. Let W be the midpoint of segment IA. Point Q is the midpoint of segment LM. Let N be the midpoint of segment EQ. Find the length of segment NW. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.34", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center U and circumradius 12. Point X is the reflection of point D across point H. Line W passes through points J and X. Let S be the perpendicular bisector of segment GH. Let R be the reflection of line W across line S. Point Z is the reflection of point U across line R. Find the length of segment DZ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.78", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center K and circumradius 3. Let H be the perpendicular bisector of segment KA. Point Q is the reflection of point D across line H. Let U be the reflection of point A across point E. Let R be the midpoint of segment EU. Find the distance between points R and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.5", "dataset": "MATH"}
{"messages": [{"content": "Let BCDEFGHIJKLM be a regular dodecagon with center A and circumradius 9. Point Y is obtained by rotating point B by angle pi/6 about point J. Line O is the angle bisector of angle JIK. Point U is the reflection of point H across line O. Let N be the angle bisector of angle GAU. Point V is the reflection of point D across line N. Find the distance between points V and Y. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.17", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center W and circumradius 4. Let Y be the angle bisector of angle WCD. Point S is the reflection of point F across line Y. Line T passes through points D and F. Point N is the reflection of point S across segment BC. Point P is the reflection of point B across line T. What is the length of segment PN? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.44", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center Q and circumradius 11. Line Z is the perpendicular bisector of segment DA. Let M be the line through point A parallel to line Z. Let O be the line through point D parallel to line M. Point R is the reflection of point B across line O. Point X is the midpoint of segment ER. Find the length of segment XE. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.16", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJK is a regular 11-gon with center Q and circumradius 8. Let N be the midpoint of segment HD. Let Y be the perpendicular bisector of segment NK. Point Z is the reflection of point A across line Y. Point O is the reflection of point Z across point D. What is the length of segment OZ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.2", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center V and circumradius 9. Point P is the reflection of point A across point E. Point N is the reflection of point C across point E. Line U passes through points A and N. Point S is the reflection of point P across line U. Point K is the midpoint of segment BD. Find the length of segment SK. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36.26", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center J and circumradius 4. Let Q be the circumcircle of polygon ABCD. Line V is the angle bisector of angle DCJ. Point U is obtained by rotating point D by angle 3pi/4 about point J. Let N be the rotation of point C by angle 11pi/12 around point A. Point G is the reflection of point U across line V. Point O is the inversion of point G with respect to circle Q. Let R be the reflection of circle Q across line V. Let X be the inversion of point O in circle R. What is the measure of angle BNX, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.18", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center W and circumradius 8. Let O be the perpendicular bisector of segment BW. Let V be the line through point E perpendicular to line O. Let L be the reflection of point F across point E. Let G be the reflection of point F across point A. Line X is the angle bisector of angle LFG. Let I be the reflection of line V across line X. Point Y is the reflection of point G across line I. What is the length of segment YA? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDF be a regular 5-gon with center E and circumradius 4. Line P passes through points F and E. Line S is the reflection of line P across point B. Line X passes through points B and C. Point Q is the intersection of lines X and S. Find the distance between points A and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.56", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center O and circumradius 9. Line M is the angle bisector of angle DBG. Point V is the reflection of point A across line M. Point W is the midpoint of segment CH. What is the length of segment VW? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.64", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center N and circumradius 4. Let U be the rotation of point N by angle pi/2 around point D. Line L is the perpendicular bisector of segment UN. Line J passes through point E perpendicular to line L. Let V be the perpendicular bisector of segment CD. Line P is the reflection of line J across line V. Point T is the reflection of point N across line P. Find the measure of angle ABT, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.62", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center M and circumradius 4. Let X be the line through point M parallel to segment EB. Point U is the midpoint of segment EB. Point I is the midpoint of segment UE. Point G is the reflection of point I across line X. What is the length of segment GB? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.22", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEGHI be a regular octagon with center F and circumradius 4. Let X be the reflection of point E across point B. Let Q be the circumcircle of polygon ABCDEGHI. Let V be the angle bisector of angle CFX. Line R passes through point X perpendicular to line V. Let P be the midpoint of segment FG. Point U is the reflection of point H across line R. Point J is the inversion of point P with respect to circle Q. Find the distance between points J and U. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30.88", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center Y and circumradius 2. Point X is the reflection of point B across segment FA. Let P be the angle bisector of angle YCI. Line T is the reflection of line P across point F. Let R be the reflection of point D across point H. Line Q passes through point K parallel to line T. Point U is the reflection of point J across line Q. What is the measure of angle XUR, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.83", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center R and circumradius 11. Let I be the angle bisector of angle GEA. Point Q is the reflection of point G across line I. Let W be the line through point E perpendicular to line I. Point K is the reflection of point Q across line W. What is the length of segment KD? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.72", "dataset": "MATH"}
{"messages": [{"content": "ABCEFG is a regular hexagon with center D and circumradius 12. Line U passes through points G and A. Let L be the line through point C parallel to segment EA. Point M is the intersection of lines L and U. Point Q is the reflection of point D across point F. Find the distance between points Q and M. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43.27", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center K and circumradius 2. Let R be the line through points B and K. Line N is the reflection of line R across point D. Let T be the circumcircle of polygon ABCDEF. Let S be the reflection of line N through point K. Let J be the reflection of circle T across line S. Let O be the inversion of point B in circle J. Point U is the inversion of point O with respect to circle T. What is the length of segment UF? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.5", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center Y and circumradius 5. Point S is obtained by rotating point G by angle 5pi/12 about point D. Point T is the intersection of segments FS and JC. Let U be the midpoint of segment TB. Let W be the circumcircle of polygon ABCDEFGHIJ. Let V be the inversion of point U in circle W. What is the length of segment GV? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.4", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJLM be a regular dodecagon with center K and circumradius 11. Let Y be the circumcircle of polygon ABCDEFGHIJLM. Point V is the reflection of point B across segment MG. Let O be the midpoint of segment KA. Point P is the inversion of point V with respect to circle Y. What is the distance between points P and O? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.28", "dataset": "MATH"}
{"messages": [{"content": "Let ABCEFGHIJKLM be a regular dodecagon with center D and circumradius 9. Let Q be the line through points J and F. Point Y is the reflection of point I across line Q. Let W be the midpoint of segment LY. Let R be the midpoint of segment MW. What is the length of segment ER? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.08", "dataset": "MATH"}
{"messages": [{"content": "ABCEFGHIJKL is a regular 11-gon with center D and circumradius 11. Let S be the circumcircle of polygon ABCEFGHIJKL. Let U be the reflection of circle S through point L. Let W be the perpendicular bisector of segment EF. Circle O is the reflection of circle U across line W. Point Z is the inversion of point I with respect to circle O. Find the distance between points C and Z. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25.66", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center M and circumradius 11. Let J be the reflection of point F across point D. Point V is the midpoint of segment CE. Let P be the reflection of point J across point V. What is the distance between points F and P? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center L and circumradius 9. Line T is the perpendicular bisector of segment DL. Let N be the perpendicular bisector of segment CB. Let S be the reflection of line N across line T. Point X is the reflection of point F across line S. What is the measure of angle XDL, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.62", "dataset": "MATH"}
{"messages": [{"content": "ABCEFGHIJKLM is a regular dodecagon with center D and circumradius 4. Let T be the reflection of point F across point K. Point N is the midpoint of segment TM. Point Y is the reflection of point N across segment BL. Find the distance between points Y and A. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.3", "dataset": "MATH"}
{"messages": [{"content": "Let CK = 5. ABDEFGHIJLMN is a regular dodecagon with center K and circumradius 10. Let X be the line through points J and I. Line S is the reflection of line X across point M. Let Q be the reflection of line S through point K. Point V is the intersection of segment JG and line Q. What is the length of segment VE? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.57", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGIJ is a regular 9-gon with center H and circumradius 9. Line S is the angle bisector of angle DJC. Point X is the reflection of point D across line S. Let V be the midpoint of segment GB. Point Z is the midpoint of segment XV. Find the length of segment ZJ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.23", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center L and circumradius 10. Point T is the reflection of point D across point B. Line Y passes through point T parallel to segment AD. Point R is the reflection of point E across line Y. What is the length of segment CR? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34.64", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center N and circumradius 3. Line Q passes through points E and H. Line K passes through point G parallel to segment AH. Point S is the intersection of lines K and Q. Find the length of segment SA. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.81", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center T and circumradius 9. Let Z be the midpoint of segment AC. Segments HC and BF intersect at point Y. Find the length of segment ZY. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.58", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center S and circumradius 11. Let M be the midpoint of segment EF. Point L is the reflection of point M across point C. Point R is the reflection of point B across segment EF. Find the length of segment LR. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "59.49", "dataset": "MATH"}
{"messages": [{"content": "ABCDEG is a regular hexagon with center F and circumradius 3. Let M be the circumcircle of polygon ABCDEG. Line R is the angle bisector of angle DCG. Point K is on line R. Let J be the inversion of point K in circle M. What is the distance between points D and J? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center Q and circumradius 1. Let S be the circumcircle of polygon ABCDEFGH. Line W passes through points D and B. Circle O is the reflection of circle S across line W. Let L be the inversion of point F in circle O. What is the length of segment LB? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.89", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center X and circumradius 12. Let Z be the line through point C parallel to segment AK. Let P be the midpoint of segment AK. Let O be the midpoint of segment LD. Point V is the reflection of point P across line Z. Point Y is the reflection of point V across point O. What is the distance between points Y and F? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30.23", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center V and circumradius 10. Let W be the line through points B and C. Point Q is the reflection of point G across line W. Point X is the reflection of point F across segment BD. What is the distance between points Q and X? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21.41", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center M and circumradius 9. Let P be the circumcircle of polygon ABCDEF. Let V be the midpoint of segment ED. Let S be the reflection of circle P through point B. Line I is the perpendicular bisector of segment BV. Circle X is the reflection of circle S across point V. Line R passes through point M parallel to line I. Segment BV and line R intersect at point T. Point J is the inversion of point T with respect to circle X. What is the distance between points J and F? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28.85", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center T and circumradius 10. Line U is the perpendicular bisector of segment GF. Let Q be the reflection of point D across point G. Let S be the midpoint of segment FC. Point L is the reflection of point S across line U. What is the distance between points Q and L? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25.85", "dataset": "MATH"}
{"messages": [{"content": "Let BCDEFGH be a regular 7-gon with center A and circumradius 7. Let I be the line through points D and E. Line O passes through point G parallel to line I. Point Z is the reflection of point C across line O. Let P be the rotation of point C by angle 2pi/3 around point G. Line N is the reflection of line I across point Z. Let Y be the angle bisector of angle FPD. Let T be the reflection of line N across line Y. Point R is the intersection of lines I and T. Find the length of segment FR. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1015.68", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGH is a regular 7-gon with center F and circumradius 6. Line Y passes through points D and G. Let R be the line through points F and C. Point O is the intersection of lines R and Y. Let P be the reflection of point E across point A. Find the distance between points P and O. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.4", "dataset": "MATH"}
{"messages": [{"content": "ABCDEG is a regular hexagon with center F and circumradius 5. Let W be the line through points E and C. Point N is the reflection of point E across point C. Line R passes through point B perpendicular to line W. Let H be the line through point F perpendicular to line R. Point U is the reflection of point N across line H. Find the length of segment UB. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.66", "dataset": "MATH"}
{"messages": [{"content": "Let ABCE be a square with center D and circumradius 3. Line X is the perpendicular bisector of segment DC. Let Q be the line through point D parallel to line X. Let P be the reflection of line Q across line X. Let T be the reflection of line X across line P. Point K is the reflection of point A across line T. What is the distance between points K and E? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.37", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center U and circumradius 6. Let Y be the rotation of point E by angle 5pi/6 around point F. Point T is the midpoint of segment YU. Let J be the midpoint of segment BT. Point M is the midpoint of segment JT. Point O is the midpoint of segment FE. Find the distance between points O and M. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.13", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center N and circumradius 8. Point K is the reflection of point E across point A. Point M is the reflection of point E across point D. Line V passes through points F and D. Point T is the reflection of point K across line V. What is the length of segment TM? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36.66", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKM be a regular dodecagon with center L and circumradius 8. Let S be the perpendicular bisector of segment IH. Let Y be the reflection of line S through point D. Line R is the reflection of line Y across point F. Point T is the reflection of point L across point G. Point X is the reflection of point G across line R. What is the distance between points T and X? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.37", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHI be a regular 9-gon with center Z and circumradius 9. Let U be the midpoint of segment ZG. Let Q be the angle bisector of angle ACI. Let O be the line through point A parallel to line Q. Let M be the reflection of line O through point U. Point N is the reflection of point Z across line M. Find the distance between points F and N. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22.37", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center L and circumradius 5. Point G is the midpoint of segment AD. Let T be the reflection of point C across point E. Let Y be the reflection of point G across point T. Find the length of segment YC. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31.84", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center Q and circumradius 9. Let P be the line through points D and H. Line Y passes through point G perpendicular to line P. Point Z lies on line Y. Line I passes through points D and A. Point U is the reflection of point Z across line I. What is the measure of angle FAU, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.96", "dataset": "MATH"}
{"messages": [{"content": "ABDEF is a regular 5-gon with center C and circumradius 4. Let J be the reflection of point F across point A. Line M is the perpendicular bisector of segment BF. Line Y passes through point J parallel to line M. Let G be the line through point D perpendicular to line Y. Point N is the reflection of point F across line G. Find the length of segment JN. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.35", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFHIJ be a regular 9-gon with center G and circumradius 7. Let K be the circumcircle of polygon ABCDEFHIJ. Point N is the midpoint of segment CI. Point R is the midpoint of segment EH. Let Z be the line through point F perpendicular to segment CI. Let S be the reflection of circle K across line Z. Point X is the inversion of point I with respect to circle S. Point U is the midpoint of segment NX. What is the distance between points R and U? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.16", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center O and circumradius 2. Line N is the angle bisector of angle CGB. Let L be the line through point C parallel to line N. Let J be the line through point H parallel to line L. Point M is the reflection of point A across line J. What is the distance between points M and A? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.6", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center M and circumradius 5. Let K be the line through points M and G. Let S be the line through point A parallel to line K. Point T is the reflection of point F across point E. Point P is the reflection of point T across line S. Point H is the reflection of point P across segment TG. Find the distance between points C and H. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24.05", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center Y and circumradius 3. Point Q is obtained by rotating point F by angle pi/6 about point C. Line Z is the angle bisector of angle BAF. Line I is the reflection of line Z across point B. Point U is the reflection of point A across point F. Point V is the reflection of point Q across line I. What is the distance between points V and U? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.41", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center V and circumradius 3. Let W be the rotation of point J by angle 7pi/12 around point I. Let M be the midpoint of segment DW. Point Y is the reflection of point W across point J. Line O passes through point G parallel to segment JV. Line N passes through point M perpendicular to segment JV. Point P is the reflection of point C across line N. Point X is the reflection of point P across segment JV. Let R be the reflection of line N through point Y. Point T is the intersection of lines R and O. Find the distance between points X and T. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.7", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center S and circumradius 12. Point Z is the midpoint of segment DA. Point G is the reflection of point B across segment CE. Let P be the midpoint of segment CG. Point R is the reflection of point A across point P. What is the length of segment RZ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41.33", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center N and circumradius 4. Line H passes through points D and B. Let J be the line through points A and B. Line R passes through point F parallel to line H. Lines J and R intersect at point Z. Let V be the circumcircle of polygon ABCDEF. Let I be the inversion of point Z in circle V. What is the distance between points I and Z? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.27", "dataset": "MATH"}
{"messages": [{"content": "BCDEFG is a regular hexagon with center A and circumradius 3. Let V be the reflection of point A across point E. Line X is the angle bisector of angle VGD. Let I be the reflection of line X through point D. Point M is the reflection of point E across line I. What is the distance between points G and M? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.57", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center I and circumradius 3. Point U is on segment FB. Point Z is the reflection of point A across segment DC. What is the measure of angle UBZ, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.96", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center N and circumradius 4. Line M passes through point D perpendicular to segment CB. Point U is the reflection of point N across point D. Line E is the reflection of line M across point N. Point J is the reflection of point N across line E. What is the distance between points J and U? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.65", "dataset": "MATH"}
{"messages": [{"content": "Let ABDE be a square with center C and circumradius 7. Line W is the perpendicular bisector of segment AC. Point R is the midpoint of segment DB. Point S is the reflection of point R across line W. Let I be the reflection of point E across point B. Find the distance between points S and I. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.41", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center Y and circumradius 4. Point W is the midpoint of segment DG. Let L be the rotation of point C by angle 5pi/6 around point B. Line R is the perpendicular bisector of segment BW. Let Z be the midpoint of segment LC. Point V is the reflection of point E across line R. What is the distance between points Z and V? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.35", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJK is a regular 11-gon with center P and circumradius 10. Let X be the circumcircle of polygon ABCDEFGHIJK. Point N is the reflection of point C across point F. Let V be the inversion of point N in circle X. Point O is the midpoint of segment VP. What is the length of segment IO? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.61", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center J and circumradius 8. Let P be the reflection of point B across point D. Point Q is the midpoint of segment PE. Point N is the reflection of point B across segment QC. What is the measure of angle NDE, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.62", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center I and circumradius 4. Let O be the circumcircle of polygon ABCDE. Let U be the reflection of circle O through point D. Point T is the midpoint of segment BC. Let J be the inversion of point I in circle U. What is the length of segment TJ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.65", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center Z and circumradius 7. Point U is the midpoint of segment BZ. Let W be the reflection of point U across point C. Point K is the midpoint of segment WZ. Find the length of segment KA. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.11", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center T and circumradius 6. Let K be the rotation of point F by angle 3pi/4 around point G. Let Z be the angle bisector of angle DKB. Line Y is the reflection of line Z across point J. Line V is the reflection of line Y across point H. Point Q is the reflection of point F across line V. What is the length of segment DQ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.89", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center S and circumradius 8. Let F be the angle bisector of angle SBD. Line P passes through points C and E. Line Z is the reflection of line P across line F. Lines P and Z intersect at point J. What is the distance between points J and E? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.75", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center W and circumradius 4. Point O is the midpoint of segment AB. Point P is the reflection of point H across point F. Line J is the perpendicular bisector of segment FO. Point V is the reflection of point C across line J. What is the distance between points V and P? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.11", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center H and circumradius 5. Point J is obtained by rotating point B by angle pi/12 about point D. Point F is obtained by rotating point D by angle 3pi/4 about point J. Let R be the circumcircle of polygon ABCD. Line M is the perpendicular bisector of segment BD. Point S is the reflection of point F across line M. Point V is the inversion of point S with respect to circle R. Point G is the reflection of point J across point S. Find the distance between points V and G. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30.32", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIKL be a regular 11-gon with center J and circumradius 6. Let X be the line through points A and C. Let M be the line through point I parallel to line X. Segment LH and line M intersect at point Q. Find the distance between points Q and H. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.01", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center O and circumradius 3. Point Y is obtained by rotating point C by angle pi/12 about point D. Let L be the reflection of point B across point A. Line R passes through point H parallel to segment FE. Point N is on segment FE. Line P is the reflection of line R across point N. Point X is the reflection of point L across line P. Point U is the reflection of point Y across point X. Find the length of segment UF. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31.93", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center M and circumradius 11. Line O passes through points L and H. Point Q is the intersection of line O and segment KE. Point X is the midpoint of segment GJ. What is the distance between points X and Q? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.68", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center M and circumradius 2. Point P is the midpoint of segment EA. Point Z is the reflection of point B across point E. Let H be the midpoint of segment ZP. Point Q is the midpoint of segment AH. What is the length of segment BQ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.67", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center Z and circumradius 11. Let M be the midpoint of segment IF. Point N is the midpoint of segment EM. Point X is the reflection of point F across point J. What is the length of segment XN? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37.67", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center N and circumradius 6. Point X is the reflection of point B across point C. Let O be the angle bisector of angle DNE. Let Z be the line through point X perpendicular to line O. Let U be the line through point N parallel to line Z. Point H is the reflection of point C across line U. Let J be the reflection of point H across point A. What is the distance between points C and J? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.12", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHJK is a regular 10-gon with center I and circumradius 5. Let P be the circumcircle of polygon ABCDEFGHJK. Let W be the reflection of circle P through point G. Let V be the inversion of point A in circle W. Let R be the angle bisector of angle FVE. Point N is the reflection of point J across line R. What is the length of segment HN? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.26", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center L and circumradius 4. Let K be the line through points B and F. Let N be the perpendicular bisector of segment FA. Line U is the reflection of line N across point E. Line J is the reflection of line K across point E. Line V passes through points D and C. Point R is the reflection of point C across line U. Point W is the intersection of lines V and J. What is the distance between points R and W? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.45", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center V and circumradius 12. Z is placed 10 units away from E. Let S be the perpendicular bisector of segment BV. Let X be the midpoint of segment BV. Let P be the reflection of point Z across point X. Point G is the reflection of point P across segment DE. Point W is the reflection of point G across line S. Find the distance between points W and P. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31.42", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center U and circumradius 10. Line N passes through points A and B. Let Z be the reflection of line N through point C. Let H be the reflection of line Z through point U. Line Y passes through point C perpendicular to line H. Point X is the reflection of point A across line Y. What is the distance between points X and E? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36.18", "dataset": "MATH"}
{"messages": [{"content": "ACDEFGHIJK is a regular 10-gon with center B and circumradius 9. Line O is the angle bisector of angle JGB. Line V is the reflection of line O across point E. Point M is the reflection of point F across line V. Let S be the midpoint of segment HM. Point Q is the reflection of point G across point S. What is the distance between points Q and K? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33.21", "dataset": "MATH"}
{"messages": [{"content": "ABCEFGHIJK is a regular 10-gon with center D and circumradius 11. Line M passes through points A and I. Let L be the line through point B perpendicular to line M. Line N passes through point G parallel to line L. Point X is the reflection of point H across line N. Find the measure of angle HGX, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.63", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center Q and circumradius 2. Let Y be the line through points C and Q. Let N be the reflection of point H across point G. Let Z be the line through points N and E. Lines Z and Y intersect at point P. Find the distance between points P and H. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.47", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center X and circumradius 12. Let U be the midpoint of segment GH. Let Y be the midpoint of segment HU. Line T is the perpendicular bisector of segment YE. Point I is the reflection of point U across line T. Find the length of segment CI. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18.28", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center K and circumradius 2. Let I be the perpendicular bisector of segment AB. Line Y passes through point K perpendicular to line I. Line R is the reflection of line Y across point B. Line L is the reflection of line Y across line R. Let J be the line through points E and K. Point H is the reflection of point B across line L. Point G is the reflection of point H across line J. What is the length of segment KG? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.17", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center Y and circumradius 5. Let Q be the reflection of point Y across point E. Let O be the rotation of point C by angle pi/4 around point B. Point W is the midpoint of segment FD. Let S be the line through point Q parallel to segment OW. Point K is on line S. Let R be the reflection of point K across point B. Let I be the reflection of point R across point F. Line J passes through point K perpendicular to segment OW. Point V is the reflection of point I across line J. What is the distance between points V and K? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center Q and circumradius 3. Let P be the midpoint of segment DA. Let Z be the circumcircle of polygon ABCDE. Let Y be the inversion of point P in circle Z. Point L is the midpoint of segment BY. What is the distance between points P and L? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.85", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center L and circumradius 7. Let K be the midpoint of segment CB. Point J is the reflection of point F across segment CB. Let P be the midpoint of segment LJ. Point U is the reflection of point K across point F. Find the distance between points U and P. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27.78", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center V and circumradius 5. Let O be the perpendicular bisector of segment CH. Let U be the midpoint of segment HE. Line K passes through point D parallel to line O. Point Z is the reflection of point U across line K. Find the length of segment ZG. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.22", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center X and circumradius 3. Let J be the perpendicular bisector of segment HG. Let T be the reflection of point F across point G. Let K be the line through point H perpendicular to line J. Point S is the reflection of point X across line K. Find the measure of angle XST, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.39", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFG be a regular hexagon with center C and circumradius 6. Line P passes through points D and A. Line Q passes through points C and E. Line L is the reflection of line P across line Q. Point Y is the reflection of point D across line L. What is the distance between points G and Y? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDFG is a regular hexagon with center E and circumradius 5. Let M be the midpoint of segment AB. Let Z be the reflection of point E across point A. Point O is the reflection of point E across segment ZM. What is the distance between points O and Z? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.0", "dataset": "MATH"}
{"messages": [{"content": "ABCEFGHIJK is a regular 10-gon with center D and circumradius 8. Line M is the angle bisector of angle GHD. Line Z passes through point G parallel to line M. Let S be the circumcircle of polygon ABCEFGHIJK. Let Y be the reflection of line Z across line M. Point V is the reflection of point G across line Y. Let Q be the inversion of point V in circle S. What is the distance between points C and Q? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19.03", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center F and circumradius 11. Point E is the reflection of point F across point A. Point G is the reflection of point E across point F. Point V is the reflection of point G across segment EB. What is the distance between points C and V? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35.81", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center V and circumradius 1. Point S is the midpoint of segment EG. Let T be the circumcircle of polygon ABCDEFGHIJ. Let R be the reflection of circle T through point F. Let Q be the inversion of point G in circle R. What is the distance between points Q and S? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.61", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center L and circumradius 5. Point S is the midpoint of segment BD. Line K passes through points A and S. Let O be the midpoint of segment CB. Let M be the reflection of line K through point O. Point X is the reflection of point B across line M. Find the length of segment XB. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.27", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFHIJKLM be a regular dodecagon with center G and circumradius 2. Let W be the circumcircle of polygon ABCDEFHIJKLM. Line N is the angle bisector of angle EFG. Point Q is the reflection of point E across line N. Point S is the inversion of point Q with respect to circle W. Find the distance between points E and S. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.61", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEFG be a regular hexagon with center B and circumradius 2. Let S be the circumcircle of polygon ACDEFG. Let O be the midpoint of segment GE. Point Y is the midpoint of segment BD. Let P be the inversion of point Y in circle S. Find the length of segment PO. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.58", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center Z and circumradius 9. Point R is the reflection of point E across point D. Let I be the circumcircle of polygon ABCDE. Line X is the perpendicular bisector of segment RC. Circle Y is the reflection of circle I across line X. Point O is the inversion of point C with respect to circle Y. Find the length of segment OC. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.82", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center G and circumradius 1. Let L be the line through points E and D. Let R be the reflection of line L through point G. Point K is the reflection of point C across point G. Point Z is the reflection of point C across line R. What is the length of segment ZK? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.5", "dataset": "MATH"}
{"messages": [{"content": "ABCDF is a regular 5-gon with center E and circumradius 7. Line P is the perpendicular bisector of segment BA. Point R is the reflection of point F across segment BA. Point U is the reflection of point R across line P. What is the measure of angle RUD, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.26", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKM is a regular dodecagon with center L and circumradius 4. Let R be the line through points D and C. Let O be the reflection of line R through point A. Let U be the line through point J perpendicular to line O. Point T is the reflection of point G across line U. Find the distance between points T and E. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.58", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center S and circumradius 11. Line R is the perpendicular bisector of segment AS. Let V be the circumcircle of polygon ABCDEFG. Circle L is the reflection of circle V across line R. Let H be the reflection of circle L through point B. Point Z is the inversion of point A with respect to circle H. Find the length of segment ZD. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.78", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center Q and circumradius 1. Let X be the midpoint of segment QG. Line O is the angle bisector of angle CFX. Let M be the reflection of line O through point X. Point U is the reflection of point Q across line M. Find the length of segment AU. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.03", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center Y and circumradius 3. Point W is obtained by rotating point F by angle pi/4 about point B. Let L be the midpoint of segment CB. Let T be the line through points G and L. Let Q be the rotation of point B by angle pi/4 around point F. Point N is the midpoint of segment YQ. Line Z is the perpendicular bisector of segment NE. Let S be the line through point F parallel to line Z. Lines T and S intersect at point U. Point P is the reflection of point W across line Z. Point X is the reflection of point A across segment UH. What is the distance between points X and P? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.18", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center W and circumradius 5. Point T is the reflection of point W across segment BA. Point M is the midpoint of segment BA. Line Z passes through points D and M. Point P is the reflection of point B across line Z. What is the distance between points P and T? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.04", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEFG be a regular hexagon with center B and circumradius 5. Line W passes through points B and E. Let Z be the angle bisector of angle DEA. Point N is the midpoint of segment BD. Let J be the reflection of line W through point D. Line L is the reflection of line J across line Z. Point U is the reflection of point N across line L. What is the distance between points U and F? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.5", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFHIJKL be a regular 11-gon with center G and circumradius 5. Let U be the reflection of point J across point G. Let V be the angle bisector of angle UJH. Let N be the reflection of line V through point U. Point R is the reflection of point F across line N. What is the distance between points R and H? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.52", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center Y and circumradius 7. Let S be the rotation of point J by angle pi around point A. Let X be the midpoint of segment KA. Let Q be the reflection of point S across point I. Point Z is the midpoint of segment FX. Find the length of segment ZQ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28.26", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center W and circumradius 4. Let Q be the rotation of point I by angle 5pi/12 around point K. Let S be the line through points J and A. Let N be the reflection of point W across point A. Let T be the line through point N parallel to line S. Point U is the reflection of point Q across line T. Let O be the line through point W parallel to segment CH. Point Z is the reflection of point C across point B. Point V is the reflection of point U across line O. What is the length of segment VZ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.96", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center Q and circumradius 2. Point Y is the reflection of point F across point G. Point T is the midpoint of segment YE. Let U be the midpoint of segment GT. Point M is the reflection of point B across segment TU. Find the length of segment MB. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.7", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center X and circumradius 11. Line W is the angle bisector of angle AHB. Let J be the line through point G parallel to line W. Segment AE and line J intersect at point Z. Point N is the reflection of point D across line W. Let Y be the reflection of point N across point Z. What is the length of segment YE? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.74", "dataset": "MATH"}
{"messages": [{"content": "ABCDFGHI is a regular octagon with center E and circumradius 2. Let L be the angle bisector of angle CFE. Line T is the reflection of line L across point C. Let Z be the reflection of line L across line T. Let S be the reflection of point B across point C. Point N is the reflection of point E across line Z. What is the length of segment NS? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.27", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center Q and circumradius 6. Let H be the perpendicular bisector of segment DC. Let V be the reflection of line H through point C. Point O is the reflection of point E across line V. Point J is the reflection of point O across point D. What is the length of segment JD? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21.63", "dataset": "MATH"}
{"messages": [{"content": "Let ABDE be a square with center C and circumradius 6. Let L be the reflection of point D across point B. Point M is the reflection of point C across point E. Point T is the midpoint of segment LM. What is the distance between points L and T? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.37", "dataset": "MATH"}
{"messages": [{"content": "ABCDFG is a regular hexagon with center E and circumradius 12. Let R be the rotation of point C by angle 5pi/12 around point B. Let O be the line through points D and F. Point J is the reflection of point C across line O. Line P is the angle bisector of angle JCE. Point V is the reflection of point R across line P. What is the distance between points E and V? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.08", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center G and circumradius 9. Let I be the line through points C and D. Point R is the midpoint of segment FA. Point T is the reflection of point G across line I. Let Q be the midpoint of segment TR. What is the length of segment CQ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.95", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFGHI be a regular octagon with center C and circumradius 6. Point X is the midpoint of segment GD. Line Y is the perpendicular bisector of segment IC. Line J passes through point D parallel to line Y. Line M passes through points D and B. Line T passes through point C parallel to line M. Lines T and J intersect at point L. What is the measure of angle XLE, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.96", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center N and circumradius 5. Point Z is obtained by rotating point A by angle pi/6 about point F. Point V is the reflection of point N across segment FZ. Point T is the intersection of segments VG and FZ. Find the length of segment TC. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.15", "dataset": "MATH"}
{"messages": [{"content": "Let ACDE be a square with center B and circumradius 1. Line K is the perpendicular bisector of segment EC. Point U is the midpoint of segment CD. Let H be the line through point D perpendicular to line K. Let P be the reflection of point A across point E. Point J is the reflection of point U across line H. Find the length of segment JP. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.55", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center P and circumradius 3. Line Z passes through points H and C. Let S be the midpoint of segment BG. Point Y is the reflection of point S across line Z. Point U is the midpoint of segment YE. What is the distance between points U and Y? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.34", "dataset": "MATH"}
{"messages": [{"content": "Let ABCEFGHIJKLM be a regular dodecagon with center D and circumradius 7. Let S be the perpendicular bisector of segment EI. Line T passes through points L and I. Let W be the reflection of line T across line S. Lines W and T intersect at point N. Find the measure of angle HDN, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.62", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center W and circumradius 7. Let K be the line through points B and E. Point S is the reflection of point A across line K. Point N is the midpoint of segment SW. Point T is the midpoint of segment DN. What is the distance between points H and T? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.37", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center Z and circumradius 5. Point R is the midpoint of segment AZ. Let O be the circumcircle of polygon ABCDE. Point W is the midpoint of segment DR. Point S is the midpoint of segment DW. Circle L is the reflection of circle O across point S. Let X be the inversion of point B in circle L. Let J be the midpoint of segment ZX. What is the length of segment JA? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.89", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center X and circumradius 8. Point P is obtained by rotating point X by angle pi/2 about point D. Point M is the midpoint of segment XE. Point Y is the reflection of point P across point M. Point T is the reflection of point D across segment MY. What is the length of segment TP? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDFGHIJKLM be a regular dodecagon with center E and circumradius 1. Line Q is the perpendicular bisector of segment BI. Let X be the reflection of line Q through point J. Let P be the reflection of point E across point D. Let W be the reflection of point P across point J. Point Z is the reflection of point L across line X. Point T is the reflection of point W across point Z. What is the distance between points T and F? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.2", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center Q and circumradius 3. Let Z be the midpoint of segment DC. Let T be the perpendicular bisector of segment ZE. Point J is on line T. Point G is the reflection of point D across segment JE. What is the length of segment GE? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.0", "dataset": "MATH"}
{"messages": [{"content": "MI = 3. ABCDEFGHJKLN is a regular dodecagon with center I and circumradius 1. Line O is the angle bisector of angle BCH. Line P passes through points E and J. Line V is the reflection of line P across line O. Point S is the reflection of point L across line V. Point Y is the midpoint of segment SG. What is the length of segment YF? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.87", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGHIJ is a regular 9-gon with center F and circumradius 3. Let Z be the perpendicular bisector of segment AG. Point Q lies on line Z at distance 10 from point G. Let O be the reflection of line Z through point H. Line W is the reflection of line O across point B. Let R be the reflection of line W through point G. Point U is the reflection of point Q across line R. What is the distance between points A and U? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34.37", "dataset": "MATH"}
{"messages": [{"content": "ABCDF is a regular 5-gon with center E and circumradius 7. Line H passes through points B and D. Let V be the reflection of line H through point F. Point L is the reflection of point A across line H. Point R is the reflection of point L across point D. Let J be the reflection of point F across point R. Point Q is the reflection of point J across line V. Point I is the reflection of point Q across segment RA. Find the length of segment IQ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31.3", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDFG be a regular hexagon with center E and circumradius 8. Let M be the reflection of point A across point C. Point Y is the reflection of point G across segment MA. Let W be the reflection of point M across point A. Find the distance between points Y and W. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28.84", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center Y and circumradius 10. Point T is the midpoint of segment BY. Point E is the midpoint of segment CD. Let N be the perpendicular bisector of segment EY. Line O passes through point T perpendicular to line N. Point Z is the reflection of point A across line O. Let U be the midpoint of segment AZ. Let H be the midpoint of segment DE. Find the distance between points H and U. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.81", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center N and circumradius 6. Let Z be the line through points D and C. Let M be the reflection of line Z through point F. Point X is the reflection of point F across line M. Let W be the midpoint of segment DN. What is the length of segment WX? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28.62", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center L and circumradius 12. Point T is the midpoint of segment AI. Point X is the reflection of point C across segment TH. What is the distance between points D and X? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38.24", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center Y and circumradius 2. Let U be the rotation of point C by angle pi around point D. Point Z is the reflection of point A across point D. Let W be the perpendicular bisector of segment ZY. Line X is the reflection of line W across point C. Line Q is the reflection of line W across line X. Point J is the reflection of point D across line Q. Find the distance between points U and J. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.1", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center W and circumradius 9. Point X is the reflection of point A across point B. Line N passes through points A and W. Let Y be the line through points X and E. Line S is the reflection of line N across line Y. Point M is the intersection of lines S and N. Let I be the midpoint of segment DM. Point H is the midpoint of segment DI. Find the length of segment FH. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.31", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center Q and circumradius 11. Let P be the reflection of point A across point E. Let M be the circumcircle of polygon ABCDE. Let U be the angle bisector of angle QPD. Let K be the reflection of circle M across line U. Let W be the inversion of point A in circle K. Find the distance between points C and W. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.83", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center V and circumradius 1. Line O passes through points B and E. Let H be the angle bisector of angle CDA. Let J be the reflection of line H across line O. Line T passes through point C perpendicular to line J. Point U is the reflection of point D across line T. Point Z is the midpoint of segment UA. What is the distance between points A and Z? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.54", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center Q and circumradius 6. Line N is the perpendicular bisector of segment HF. Let O be the reflection of line N through point H. Point Z is the reflection of point A across line O. What is the length of segment ZH? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center V and circumradius 7. Let O be the circumcircle of polygon ABCDE. Circle N is the reflection of circle O across point D. Let P be the line through points V and D. Let Y be the inversion of point E in circle N. Let S be the perpendicular bisector of segment DY. Line W passes through point C parallel to line S. Point G is the reflection of point C across point E. Lines P and W intersect at point Q. Find the distance between points G and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.2", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center I and circumradius 9. Point M is the midpoint of segment GI. Let S be the circumcircle of polygon ABCDEFG. Let T be the angle bisector of angle IGE. Circle W is the reflection of circle S across line T. Let P be the reflection of circle W through point E. Point Z is the inversion of point M with respect to circle P. What is the length of segment AZ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.22", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center K and circumradius 9. Line J passes through points A and C. Line U is the angle bisector of angle ACD. Line W passes through point F parallel to line U. Lines W and J intersect at point T. What is the measure of angle CTB, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.83", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center M and circumradius 9. Line R is the perpendicular bisector of segment DC. Let Y be the reflection of point E across point B. Let Z be the line through point C perpendicular to line R. Point V is the midpoint of segment YC. Point T is the reflection of point A across line Z. What is the distance between points V and T? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22.5", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center Z and circumradius 10. Line M passes through points C and A. Let Q be the line through point D perpendicular to segment GA. Point I is the reflection of point Z across line M. Line X is the reflection of line M across line Q. Line H passes through points Z and I. Line N is the reflection of line H across line X. Let J be the reflection of line Q across line N. Lines M and J intersect at point O. Find the distance between points O and Z. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.19", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center N and circumradius 10. Let J be the midpoint of segment ND. Let R be the perpendicular bisector of segment NB. Point I is the reflection of point J across line R. Point V is the midpoint of segment BI. What is the length of segment IV? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.5", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFGHIJ be a regular 9-gon with center C and circumradius 9. Point Z is the reflection of point F across point C. Line T passes through points E and A. Point Q is the reflection of point Z across line T. Let Y be the line through points A and Z. Point K is the reflection of point B across line Y. What is the distance between points Q and K? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.13", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center U and circumradius 5. Let O be the reflection of point F across point J. Let M be the midpoint of segment IK. Let W be the midpoint of segment MG. What is the distance between points W and O? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.39", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJK be a regular 11-gon with center V and circumradius 10. Let L be the line through points A and C. Point O is the reflection of point E across line L. Line W passes through points D and C. Point R is the reflection of point F across line W. What is the distance between points R and O? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.29", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHJ is a regular 9-gon with center I and circumradius 2. Let N be the rotation of point F by angle 2pi/3 around point B. Point L is the reflection of point N across point E. Let U be the midpoint of segment JI. Point Y is the midpoint of segment AU. What is the distance between points L and Y? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.36", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center U and circumradius 4. Let Z be the circumcircle of polygon ABCDEFGHIJKL. Let T be the reflection of point A across point I. Circle X is the reflection of circle Z across point L. Point Y is the inversion of point B with respect to circle X. What is the distance between points Y and T? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.63", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center Y and circumradius 7. Point J is the midpoint of segment AF. Point U is obtained by rotating point C by angle 7pi/12 about point J. Line Z passes through points U and C. Line G passes through point B parallel to line Z. Point M is the intersection of line Z and segment EJ. Point T is the reflection of point J across line G. Point S is the reflection of point M across segment CJ. Find the length of segment TS. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.65", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHI be a regular 9-gon with center P and circumradius 7. Point J is the reflection of point D across point G. Let V be the line through point I perpendicular to segment CB. Let N be the reflection of line V through point J. Point U is the reflection of point G across line N. Find the distance between points E and U. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18.29", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center W and circumradius 8. Point P is the midpoint of segment JH. Let V be the reflection of point C across point P. Point K is the midpoint of segment JB. What is the distance between points V and K? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.52", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center I and circumradius 7. Let L be the circumcircle of polygon ABCDEFG. Circle N is the reflection of circle L across point F. Point Y is the inversion of point A with respect to circle N. Let V be the perpendicular bisector of segment IC. Let K be the line through point C perpendicular to line V. Point R is the reflection of point Y across line K. Let O be the midpoint of segment RB. Let X be the midpoint of segment DB. Point S is the midpoint of segment XA. Find the length of segment OS. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.76", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center N and circumradius 6. Point Z is the midpoint of segment DA. Point L is the midpoint of segment ZJ. Point O is the reflection of point D across point L. Find the distance between points I and O. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.96", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJK is a regular 11-gon with center U and circumradius 8. Let Y be the line through points H and K. Line Q passes through points C and B. Line V is the reflection of line Y across point J. Point S is the intersection of lines V and Q. What is the distance between points S and B? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.78", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center K and circumradius 9. Let I be the circumcircle of polygon ABCDEF. Let N be the midpoint of segment BF. Point P is the reflection of point N across point E. Point Z is the inversion of point P with respect to circle I. Find the length of segment FZ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.56", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center O and circumradius 1. Point J is the midpoint of segment FE. Point W is the reflection of point C across segment DB. Point S is the midpoint of segment WF. Point Y is the reflection of point J across point O. What is the distance between points S and Y? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.31", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center T and circumradius 2. Let N be the line through points T and G. Line Y is the perpendicular bisector of segment TF. Line U is the reflection of line Y across line N. Point K is the reflection of point A across line U. Find the length of segment KC. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center O and circumradius 5. Line X passes through points B and F. Let G be the perpendicular bisector of segment OD. Point M is the reflection of point D across line X. Let R be the angle bisector of angle ABM. Line J is the reflection of line G across line R. Lines X and J intersect at point Q. What is the length of segment QM? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.97", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center P and circumradius 12. Let Y be the perpendicular bisector of segment GB. Line L is the reflection of line Y across point B. Line R passes through point G perpendicular to line L. Point Z is on line R. Point N is the reflection of point Z across point B. Find the measure of angle EPN, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.89", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHI be a regular 9-gon with center U and circumradius 12. Point Y is obtained by rotating point I by angle pi/2 about point A. Point V is the midpoint of segment EI. Let K be the line through point V parallel to segment IA. Line J passes through point H perpendicular to line K. Point R is the reflection of point Y across line J. Find the length of segment RF. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33.43", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center Y and circumradius 10. Line X is the angle bisector of angle AHJ. Point M is the reflection of point G across line X. Point Z is the midpoint of segment CB. Point P is the reflection of point M across segment LC. What is the distance between points Z and P? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22.08", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center N and circumradius 3. Point T is obtained by rotating point C by angle 2pi/3 about point A. Let W be the line through point H parallel to segment NF. Let Y be the line through point T perpendicular to line W. Point S is the reflection of point A across line Y. Find the distance between points S and N. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.64", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center M and circumradius 1. Line O is the perpendicular bisector of segment GD. Point W is the midpoint of segment FM. Let P be the midpoint of segment WG. Let Q be the line through point P perpendicular to line O. Point X is the reflection of point B across line Q. What is the length of segment XP? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.63", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center L and circumradius 5. Let R be the circumcircle of polygon ABCDEFGHIJ. Line Z is the perpendicular bisector of segment CL. Point Q is the midpoint of segment BD. Let U be the reflection of circle R across line Z. Let W be the inversion of point Q in circle U. Find the distance between points C and W. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26.18", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center Z and circumradius 1. Point V is the reflection of point A across point L. Point Q is the reflection of point C across point G. Let N be the angle bisector of angle ZKJ. Line O passes through point H parallel to line N. Let R be the reflection of line O across line N. Point W is the reflection of point Q across line R. What is the distance between points V and W? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.41", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center M and circumradius 12. Line H passes through point A perpendicular to segment DM. Let J be the line through point D parallel to line H. Point Y is the reflection of point M across line J. Let L be the reflection of point M across point Y. What is the distance between points F and L? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54.99", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center K and circumradius 4. Line W is the angle bisector of angle BAC. Let M be the midpoint of segment KD. Line U is the reflection of line W across point B. Point Z is the reflection of point D across line U. Point O is the reflection of point D across point Z. Point P is the reflection of point O across segment DM. What is the length of segment PA? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36.44", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGHIJK is a regular 10-gon with center F and circumradius 6. Line S is the angle bisector of angle IGB. Let O be the reflection of line S through point J. Let V be the reflection of line O through point E. Line X is the reflection of line V across point H. Point M is the reflection of point D across line X. Find the distance between points M and H. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42.54", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center H and circumradius 1. Let L be the reflection of point E across point A. Point Q is the midpoint of segment LC. Let U be the reflection of point L across point E. Point X is the midpoint of segment QL. What is the distance between points X and U? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.29", "dataset": "MATH"}
{"messages": [{"content": "Let ABCEFGHI be a regular octagon with center D and circumradius 4. Point V is the midpoint of segment EA. Line J is the angle bisector of angle FGV. Let N be the line through point F parallel to line J. Line X is the reflection of line J across line N. Point P is the midpoint of segment BF. Point O is the reflection of point P across line X. Find the distance between points A and O. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.24", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center L and circumradius 7. Let J be the reflection of point A across point E. Let M be the circumcircle of polygon ABCDEFGH. Circle K is the reflection of circle M across point G. Let X be the inversion of point B in circle K. What is the length of segment JX? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24.54", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHJK is a regular 10-gon with center I and circumradius 8. Let R be the angle bisector of angle GKC. Let Q be the line through point F parallel to line R. Let M be the reflection of line Q across line R. Point Y is the reflection of point C across line M. Find the distance between points J and Y. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.06", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHJKLM be a regular dodecagon with center I and circumradius 1. Let Z be the line through points G and F. Line Q passes through point I parallel to line Z. Line W is the reflection of line Z across line Q. Point N is the reflection of point G across line W. Find the distance between points H and N. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.63", "dataset": "MATH"}
{"messages": [{"content": "ABDEFGHIJKLM is a regular dodecagon with center C and circumradius 11. Let N be the reflection of point D across point K. Point P is the midpoint of segment NG. Let T be the midpoint of segment LJ. Find the distance between points P and T. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.28", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDFGHIJK be a regular 10-gon with center E and circumradius 8. Line S passes through points H and G. Line Y passes through point J parallel to line S. Point R is the reflection of point H across point A. Let X be the midpoint of segment GR. Line Y and segment HK intersect at point L. Find the length of segment XL. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.07", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center Y and circumradius 11. Line Z is the angle bisector of angle FEA. Point S is the reflection of point D across line Z. Point U is the reflection of point G across point S. Point L is the midpoint of segment HC. Find the length of segment LU. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37.88", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center L and circumradius 1. Let Q be the circumcircle of polygon ABCDEF. Circle X is the reflection of circle Q across point E. Point W is the inversion of point F with respect to circle X. Point O is the inversion of point W with respect to circle Q. What is the length of segment OW? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.87", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center G and circumradius 3. Let O be the midpoint of segment BD. Point Z is the reflection of point A across segment DB. Point H is the reflection of point C across point Z. Find the length of segment HO. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.87", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center U and circumradius 6. Let Q be the reflection of point K across point A. Let X be the midpoint of segment UB. Let Z be the midpoint of segment FQ. Find the length of segment ZX. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.69", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center T and circumradius 9. Point Z is the reflection of point G across point B. Point L is the midpoint of segment FZ. Point K is the midpoint of segment DZ. Find the distance between points K and L. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.29", "dataset": "MATH"}
{"messages": [{"content": "ABCDFGHIJK is a regular 10-gon with center E and circumradius 9. Let X be the line through points B and J. Point N is the intersection of segment AD and line X. Point P is the reflection of point J across point A. Let S be the midpoint of segment NB. Find the length of segment PS. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.79", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center U and circumradius 2. Let T be the reflection of point G across point B. Let S be the midpoint of segment GB. Point R is the reflection of point T across point S. Find the length of segment RA. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.4", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center S and circumradius 12. Let I be the perpendicular bisector of segment BS. Point P is the reflection of point E across line I. Line X passes through point D perpendicular to line I. Line L passes through point E perpendicular to line X. Point W is the reflection of point C across line L. What is the length of segment PW? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "48.49", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFGHIJKLM be a regular dodecagon with center C and circumradius 5. Let Q be the circumcircle of polygon ABDEFGHIJKLM. Point T is the reflection of point B across point K. Line V passes through points C and F. Let Y be the line through point T perpendicular to line V. Point Z is the reflection of point G across line Y. Point O is the inversion of point Z with respect to circle Q. Find the distance between points A and O. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.66", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIK is a regular 10-gon with center J and circumradius 3. Let P be the circumcircle of polygon ABCDEFGHIK. Let N be the perpendicular bisector of segment JE. Circle R is the reflection of circle P across point H. Point Q is the reflection of point D across line N. Circle T is the reflection of circle R across line N. Let S be the inversion of point Q in circle T. What is the length of segment QS? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.56", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKM be a regular dodecagon with center L and circumradius 6. Point Q is the midpoint of segment HJ. Let O be the midpoint of segment EC. Point T is the reflection of point B across point O. Find the distance between points Q and T. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.17", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center O and circumradius 2. Line I passes through point C parallel to segment OA. Point Z is the reflection of point D across line I. Point T is the midpoint of segment ZE. Find the distance between points O and T. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.73", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center X and circumradius 12. Let T be the circumcircle of polygon ABCDEFGHIJKL. Circle Y is the reflection of circle T across point E. Circle V is the reflection of circle T across point K. Point S is the inversion of point F with respect to circle V. Let Q be the inversion of point S in circle Y. Find the distance between points G and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.99", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center Z and circumradius 1. Let T be the midpoint of segment CE. Let U be the angle bisector of angle TFB. Let S be the line through point C perpendicular to line U. Point R is the reflection of point F across line S. What is the length of segment BR? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.88", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center G and circumradius 11. Point L is the reflection of point A across point D. Let V be the perpendicular bisector of segment CF. Let M be the line through point L perpendicular to line V. Point X is the reflection of point D across line M. What is the distance between points G and X? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47.95", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center N and circumradius 10. Point V is the midpoint of segment HF. Let R be the perpendicular bisector of segment AV. Point J is the reflection of point G across line R. Find the distance between points G and J. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.63", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center P and circumradius 8. Point R is the midpoint of segment AF. Let K be the perpendicular bisector of segment BR. Let Z be the reflection of line K through point A. Point H is the reflection of point R across point D. Point I is the reflection of point H across line Z. What is the distance between points I and E? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18.14", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center G and circumradius 11. Point R is obtained by rotating point G by angle pi/4 about point B. Let T be the midpoint of segment ER. Let M be the circumcircle of polygon ABCDEF. Let O be the midpoint of segment FT. Point V is the reflection of point D across point B. Circle U is the reflection of circle M across point O. Let X be the inversion of point F in circle U. What is the length of segment VX? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24.31", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center U and circumradius 1. Point Q is the midpoint of segment UE. Point G is the midpoint of segment QC. Point T is the reflection of point G across point C. Point X is the reflection of point T across segment AQ. Find the distance between points T and X. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.93", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center Y and circumradius 6. Let T be the perpendicular bisector of segment DB. Line Z is the angle bisector of angle HGA. Line L is the reflection of line Z across line T. Let W be the rotation of point Y by angle 11pi/12 around point A. Let R be the circumcircle of polygon ABCDEFGH. Circle S is the reflection of circle R across point W. Let O be the reflection of circle S across line L. Point U is the inversion of point G with respect to circle O. Find the length of segment UH. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.26", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center K and circumradius 2. Point N is obtained by rotating point B by angle 11pi/12 about point C. Let W be the perpendicular bisector of segment NE. Line Q passes through point I perpendicular to line W. Let M be the line through point C parallel to line Q. Line M and segment KD intersect at point Y. Find the distance between points Y and A. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.44", "dataset": "MATH"}
{"messages": [{"content": "ABDEFG is a regular hexagon with center C and circumradius 8. Let K be the line through points A and D. Point X is the reflection of point F across point B. Let L be the reflection of point G across point D. Point U is the reflection of point X across segment EB. Point T is the reflection of point L across line K. What is the length of segment TU? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40.0", "dataset": "MATH"}
{"messages": [{"content": "ACDEFGHIJKLM is a regular dodecagon with center B and circumradius 11. Line O passes through points A and F. Point S is the reflection of point C across point D. Line W is the perpendicular bisector of segment KS. Let Y be the reflection of line W across line O. Point Z is the intersection of lines Y and O. What is the length of segment ZI? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.24", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFHI be a regular octagon with center G and circumradius 4. Line X passes through points I and G. Point M is the reflection of point I across point F. Point U is the reflection of point M across line X. Let T be the midpoint of segment UC. What is the length of segment TM? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.43", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center K and circumradius 4. Let Y be the midpoint of segment CK. Let W be the reflection of point B across point A. Point R is the reflection of point Y across segment DW. What is the length of segment CR? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.56", "dataset": "MATH"}
{"messages": [{"content": "ABCEFGHI is a regular octagon with center D and circumradius 12. Line X is the angle bisector of angle FBI. Line M is the reflection of line X across point H. Point W is the reflection of point D across line M. Point Y is the midpoint of segment WG. Find the length of segment YC. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22.88", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center S and circumradius 8. Line O passes through point D perpendicular to segment DJ. Point Z is the reflection of point S across line O. Let K be the reflection of point I across point B. Find the length of segment KZ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19.98", "dataset": "MATH"}
{"messages": [{"content": "ABCDFGHI is a regular octagon with center E and circumradius 7. Let J be the line through points I and C. Point T is the reflection of point B across line J. Let M be the midpoint of segment AT. Point R is the reflection of point D across point M. Find the distance between points R and A. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.1", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGHIJ is a regular 9-gon with center F and circumradius 9. Let O be the reflection of point E across point H. Let W be the angle bisector of angle CBG. Point Q is the reflection of point F across line W. Let Y be the reflection of line W through point Q. Point T is the reflection of point F across line Y. What is the length of segment OT? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "53.38", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGIJ is a regular 9-gon with center H and circumradius 10. Let V be the line through points H and E. Line K is the perpendicular bisector of segment CH. Let N be the reflection of line K through point D. Point U is the reflection of point A across line N. Point Y is the intersection of lines V and N. What is the length of segment YU? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "68.92", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHI be a regular 9-gon with center U and circumradius 5. Point S is the midpoint of segment UD. Point O is the reflection of point I across point D. Line Q passes through point O perpendicular to segment ID. Line P is the reflection of line Q across point B. Line Y is the reflection of line P across line Q. Point X is the reflection of point A across line Y. Find the distance between points X and S. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "89.45", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center Y and circumradius 10. Point U is obtained by rotating point B by angle 3pi/4 about point F. Point K is the reflection of point U across point F. Point Z is the reflection of point U across point B. Line N is the perpendicular bisector of segment BK. Point W is the reflection of point A across line N. What is the length of segment ZW? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38.46", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center F and circumradius 1. Let L be the angle bisector of angle ACF. Point W is the midpoint of segment ED. Point H is the reflection of point W across line L. Point X is the midpoint of segment DC. Point J is the midpoint of segment HX. Find the distance between points H and J. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.85", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center T and circumradius 3. Point U is the midpoint of segment BD. Point N is the reflection of point A across segment DU. Point Z is the midpoint of segment ND. Let R be the midpoint of segment ZE. Find the length of segment RN. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.11", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center K and circumradius 3. Let P be the line through points C and B. Line L is the angle bisector of angle KFG. Line Y is the reflection of line L across line P. Point V is the reflection of point B across line Y. What is the distance between points H and V? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.94", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center P and circumradius 12. Let X be the circumcircle of polygon ABCDEFGH. Point O is the midpoint of segment CB. Let K be the inversion of point O in circle X. Let Z be the midpoint of segment KF. What is the distance between points Z and E? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.39", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center M and circumradius 8. Let F be the angle bisector of angle CBM. Let S be the line through point M perpendicular to line F. Let Z be the reflection of line S through point C. Let Q be the line through point E parallel to line Z. Line N is the reflection of line F across point E. Line Z and segment AE intersect at point O. Lines Q and N intersect at point W. What is the length of segment WO? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.0", "dataset": "MATH"}
{"messages": [{"content": "ACDEF is a regular 5-gon with center B and circumradius 6. Let O be the circumcircle of polygon ACDEF. Let Y be the reflection of circle O through point D. Let K be the reflection of point F across point E. Let U be the inversion of point A in circle Y. Let R be the line through point D parallel to segment CU. Point V is the reflection of point K across line R. What is the length of segment VK? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.42", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center K and circumradius 3. Let P be the circumcircle of polygon ABCDEF. Point N is the reflection of point A across segment CE. Let R be the reflection of point C across point N. Point S is the inversion of point R with respect to circle P. Find the length of segment SD. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.2", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center Y and circumradius 2. Let Z be the reflection of point D across point Y. Point S is the reflection of point A across segment BZ. Let W be the midpoint of segment ZB. Find the distance between points W and S. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.77", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center H and circumradius 7. Let T be the midpoint of segment AB. Let E be the circumcircle of polygon ABCD. Let G be the inversion of point T in circle E. Point X is the reflection of point G across point H. Point Q is the reflection of point A across point X. Find the distance between points Q and X. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.65", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center L and circumradius 12. Let Q be the rotation of point G by angle pi/4 around point F. Point T is the midpoint of segment BJ. Let X be the circumcircle of polygon ABCDEFGHIJ. Let R be the midpoint of segment JF. Let N be the inversion of point T in circle X. Let Y be the line through point I perpendicular to segment RA. Let P be the line through point Q parallel to line Y. Point U is the reflection of point N across line P. What is the distance between points J and U? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34.47", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEGHIJ be a regular 9-gon with center F and circumradius 5. Point L is the midpoint of segment FJ. Let P be the midpoint of segment LC. Point W is the midpoint of segment PI. Let U be the line through points W and A. Segment BH and line U intersect at point N. What is the distance between points N and W? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.3", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center N and circumradius 3. Let F be the rotation of point N by angle 7pi/12 around point B. Let M be the perpendicular bisector of segment NC. Let U be the perpendicular bisector of segment AC. Point Q is the reflection of point F across line M. Line L passes through point Q parallel to line U. Point K is the reflection of point F across line L. What is the distance between points K and Q? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.8", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center X and circumradius 3. Let W be the line through point H perpendicular to segment GE. Point T is the reflection of point C across line W. Let J be the midpoint of segment FT. Point R is the reflection of point J across segment GE. What is the distance between points R and G? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.57", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center I and circumradius 6. Let O be the angle bisector of angle ACD. Let U be the circumcircle of polygon ABCDEFGH. Circle L is the reflection of circle U across line O. Let T be the inversion of point B in circle L. Let S be the midpoint of segment BH. What is the distance between points S and T? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.73", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center P and circumradius 2. Line O is the angle bisector of angle HJP. Point U is the reflection of point C across line O. Let Z be the midpoint of segment KU. Point R is the midpoint of segment BZ. What is the distance between points R and A? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.83", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEG be a regular hexagon with center F and circumradius 2. Let T be the line through points B and G. Let M be the reflection of point G across point D. Let W be the reflection of point M across point E. Point K is the reflection of point W across line T. What is the length of segment KA? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIK be a regular 10-gon with center J and circumradius 8. Line R is the perpendicular bisector of segment AH. Let Y be the line through point F parallel to line R. Point P is the midpoint of segment CG. Point X is the midpoint of segment DG. Point S is the reflection of point X across line Y. Find the distance between points P and S. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.53", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center N and circumradius 9. Line Y is the angle bisector of angle HND. Line O is the reflection of line Y across point G. Point L is the reflection of point E across line O. Point I is the intersection of segment LF and line O. Find the distance between points I and B. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21.19", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center K and circumradius 9. Line U passes through points A and E. Let S be the reflection of point D across point E. Line Y passes through point F perpendicular to line U. Point R is the reflection of point S across line Y. Let P be the midpoint of segment SF. What is the length of segment PR? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.91", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center M and circumradius 9. Let P be the circumcircle of polygon ABCDEFG. Point J is the midpoint of segment BA. Let O be the inversion of point J in circle P. What is the distance between points O and J? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.88", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEGHIJKLM be a regular dodecagon with center F and circumradius 9. Point W is the reflection of point E across point J. Let O be the line through point I parallel to segment LM. Point S is the reflection of point W across line O. What is the length of segment MS? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34.63", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center Q and circumradius 4. Point L is the reflection of point D across point A. Point Y is the reflection of point L across segment CE. Let Z be the midpoint of segment CE. What is the distance between points Z and Y? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.62", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIKLM be a regular dodecagon with center J and circumradius 12. Let Z be the line through points M and A. Point X is the reflection of point I across point B. Let Q be the line through point F parallel to line Z. Point U is the reflection of point X across line Q. What is the length of segment UX? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "63.34", "dataset": "MATH"}
{"messages": [{"content": "Let ABCEFGHIJK be a regular 10-gon with center D and circumradius 8. Let S be the line through points A and J. Point T is the midpoint of segment CE. Let M be the line through point F perpendicular to line S. Point O is the reflection of point T across line M. What is the length of segment OI? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.09", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHJKL be a regular 11-gon with center I and circumradius 11. Let Q be the rotation of point A by angle 2pi/3 around point I. Point M is the reflection of point L across segment JD. Let Y be the midpoint of segment JQ. Find the length of segment MY. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.3", "dataset": "MATH"}
{"messages": [{"content": "ABCDF is a regular 5-gon with center E and circumradius 12. Let Y be the midpoint of segment BD. Point Q is the reflection of point Y across point C. Let K be the circumcircle of polygon ABCDF. Point R is the inversion of point Y with respect to circle K. Find the length of segment RQ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18.54", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center Z and circumradius 5. Let N be the line through points E and F. Point V is the reflection of point A across line N. Point T is the reflection of point V across segment FD. What is the distance between points V and T? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.07", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center T and circumradius 3. Let X be the circumcircle of polygon ABCDEFGHIJKL. Let Z be the reflection of point D across point J. Point W is the reflection of point J across point E. Line N passes through point L parallel to segment DZ. Circle V is the reflection of circle X across line N. Point Y is the inversion of point W with respect to circle V. What is the length of segment GY? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.67", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center T and circumradius 12. Let U be the midpoint of segment AJ. Let X be the reflection of point U across point L. Let O be the midpoint of segment JX. Point S is the midpoint of segment OF. Let Z be the reflection of point K across point E. Find the distance between points S and Z. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36.61", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center S and circumradius 10. Let X be the line through points A and B. Point Q is the reflection of point A across point B. Line U is the reflection of line X across point C. Line W passes through point H parallel to line U. Point P is the reflection of point J across line W. What is the length of segment PQ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31.62", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center S and circumradius 3. Line J is the perpendicular bisector of segment FG. Line W passes through point A perpendicular to line J. Point Z is the reflection of point B across point E. Line M is the reflection of line W across point B. Point R is the reflection of point Z across line M. Point X is the midpoint of segment RD. What is the distance between points D and X? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.65", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center P and circumradius 9. Let Z be the reflection of point B across point J. Let Q be the midpoint of segment DA. Point V is the midpoint of segment EQ. What is the length of segment ZV? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29.69", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFGHIJKLM be a regular dodecagon with center C and circumradius 3. Let Y be the perpendicular bisector of segment HE. Line S passes through point J parallel to line Y. Point R is the reflection of point M across line S. Let W be the reflection of point R across point D. What is the distance between points W and F? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.69", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center T and circumradius 10. Line Y passes through points D and A. Let V be the reflection of line Y through point B. Point X is the reflection of point T across line V. Line R passes through points I and X. Point U is the reflection of point J across line R. Find the distance between points U and T. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.97", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFHIJKLM is a regular dodecagon with center G and circumradius 4. Point V is the midpoint of segment KE. Line U is the angle bisector of angle IHV. Point W is the reflection of point E across line U. Point O is the reflection of point B across point W. What is the distance between points V and O? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.96", "dataset": "MATH"}
{"messages": [{"content": "ABDEFGH is a regular 7-gon with center C and circumradius 8. Let Q be the circumcircle of polygon ABDEFGH. Point R is the midpoint of segment BH. Let I be the reflection of circle Q through point F. Point W is the inversion of point R with respect to circle I. What is the length of segment EW? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.3", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJL is a regular 11-gon with center K and circumradius 12. Let Q be the circumcircle of polygon ABCDEFGHIJL. Line S passes through points E and H. Let M be the line through point G perpendicular to line S. Let T be the reflection of circle Q across line M. Point X is the reflection of point H across point F. Let W be the inversion of point X in circle T. What is the length of segment JW? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.66", "dataset": "MATH"}
{"messages": [{"content": "ABDEFGHI is a regular octagon with center C and circumradius 8. Let T be the perpendicular bisector of segment AC. Line R is the reflection of line T across point C. Line L is the reflection of line R across line T. Point Y is the reflection of point I across line L. Find the length of segment CY. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19.2", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFHI be a regular octagon with center G and circumradius 3. Point R is the midpoint of segment IH. Point K is the midpoint of segment DR. Point O is the midpoint of segment CA. What is the distance between points O and K? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.66", "dataset": "MATH"}
{"messages": [{"content": "ABCDFGHIJKLM is a regular dodecagon with center E and circumradius 5. Line O passes through point K parallel to segment HG. Line P passes through point C perpendicular to line O. Point Y is the reflection of point J across line P. Point X is the reflection of point Y across segment HG. Find the length of segment XK. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21.6", "dataset": "MATH"}
{"messages": [{"content": "ACDEFG is a regular hexagon with center B and circumradius 3. Line I is the angle bisector of angle DBE. Point N is obtained by rotating point G by angle 5pi/12 about point B. Let M be the perpendicular bisector of segment AB. Point H lies on line I at distance 3 from point A. Let V be the line through point N perpendicular to line M. Line S passes through point H perpendicular to line V. Let P be the line through point D parallel to line S. Point Y is the reflection of point A across line P. Point Z is the reflection of point N across point G. Find the length of segment ZY. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.54", "dataset": "MATH"}
{"messages": [{"content": "Let BCDEFGHI be a regular octagon with center A and circumradius 5. Let O be the line through points F and I. Let Q be the line through point H parallel to line O. Let R be the line through point B parallel to line O. Point S is the reflection of point C across line R. Point X is the reflection of point D across line Q. Find the distance between points S and X. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.61", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center Z and circumradius 9. Line I passes through points B and Z. Let H be the reflection of point D across point B. Let Y be the line through point B perpendicular to line I. Line P passes through points Z and C. Let T be the reflection of line Y through point H. Lines T and P intersect at point V. What is the length of segment VD? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "132.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center Y and circumradius 4. Line N is the perpendicular bisector of segment YB. Point Z is the reflection of point A across line N. Point T is the reflection of point Z across segment DY. Let H be the midpoint of segment ET. Point F is the reflection of point D across point H. Point G is the reflection of point D across segment EA. Find the length of segment GF. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.47", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center T and circumradius 11. Point X is the midpoint of segment BG. Line Q passes through points X and A. Point M is the intersection of line Q and segment FD. Let J be the reflection of point E across point H. Find the length of segment MJ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36.15", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center W and circumradius 12. Let R be the reflection of point G across point L. Let Y be the line through points D and G. Segment ER and line Y intersect at point Q. Find the distance between points Q and H. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.43", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJK is a regular 11-gon with center W and circumradius 10. Let V be the line through points E and F. Let L be the line through point C perpendicular to line V. Let P be the line through points E and A. Let S be the reflection of line P across line L. Point M is the reflection of point A across line S. Find the length of segment ME. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28.8", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center M and circumradius 1. Line Z is the perpendicular bisector of segment DM. Let I be the midpoint of segment BA. Let H be the circumcircle of polygon ABCD. Let K be the reflection of circle H across line Z. Point X is the inversion of point I with respect to circle K. Find the length of segment XB. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.41", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEF be a regular 5-gon with center B and circumradius 10. Point M is the midpoint of segment FB. Line W passes through point A perpendicular to segment CB. Point L is the reflection of point M across line W. Find the distance between points C and L. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.95", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFH be a regular 7-gon with center G and circumradius 4. Point W is obtained by rotating point D by angle 3pi/4 about point B. Let N be the reflection of point G across point D. Line Y is the perpendicular bisector of segment BC. Line K is the reflection of line Y across point N. Point O is the reflection of point W across line K. What is the length of segment NO? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31.04", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center M and circumradius 12. Let X be the line through point G parallel to segment DH. Line L is the reflection of line X across point F. Point R is the reflection of point F across line L. Find the distance between points G and R. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.53", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKM is a regular dodecagon with center L and circumradius 11. Let V be the line through points D and A. Point Q is obtained by rotating point M by angle pi/2 about point F. Let W be the circumcircle of polygon ABCDEFGHIJKM. Point X is the midpoint of segment JL. Circle U is the reflection of circle W across line V. Let P be the inversion of point H in circle U. Let R be the line through points P and X. Point T is the reflection of point C across line R. Find the distance between points T and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19.11", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJK is a regular 11-gon with center Z and circumradius 6. Line V passes through point A perpendicular to segment JD. Let W be the line through point H perpendicular to line V. Point S is the reflection of point D across line W. Find the distance between points S and Z. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.46", "dataset": "MATH"}
{"messages": [{"content": "BCDEFGHIJK is a regular 10-gon with center A and circumradius 1. Let U be the line through points H and C. Let O be the midpoint of segment EA. Line V passes through point G parallel to line U. Point Z is the reflection of point O across line V. Point Y is the reflection of point A across segment DH. What is the length of segment YZ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.36", "dataset": "MATH"}
{"messages": [{"content": "EY = 2. Let ABCDFGHIJ be a regular 9-gon with center Y and circumradius 9. Point N is on segment YJ. Let S be the circumcircle of polygon ABCDFGHIJ. Point M is the inversion of point N with respect to circle S. What is the measure of angle HYM, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.4", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDFGHI be a regular octagon with center E and circumradius 9. Let V be the angle bisector of angle ECG. Point O is the midpoint of segment IA. Point X is the reflection of point I across line V. Point N is the reflection of point H across point X. What is the length of segment NO? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31.9", "dataset": "MATH"}
{"messages": [{"content": "ABCEFGHIJKLM is a regular dodecagon with center D and circumradius 9. Point S is obtained by rotating point F by angle 7pi/12 about point A. Let Z be the rotation of point L by angle 5pi/6 around point G. Let R be the line through points M and D. Point X is the reflection of point Z across point D. Point N is the midpoint of segment SZ. Point Q is the reflection of point X across line R. What is the distance between points N and Q? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28.42", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center L and circumradius 2. Let P be the circumcircle of polygon ABCDE. Line O is the angle bisector of angle LCA. Circle M is the reflection of circle P across line O. Let W be the perpendicular bisector of segment LC. Circle R is the reflection of circle M across point E. Point U is the inversion of point L with respect to circle R. Point G is the reflection of point U across line W. What is the distance between points G and B? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.89", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center H and circumradius 5. Let L be the vector from point C to point A. Point Q is the translation of point F by vector L. Let Z be the reflection of point Q across point D. Let K be the circumcircle of polygon ABCDEF. Let I be the midpoint of segment QA. Let W be the reflection of point D across point I. Let T be the reflection of circle K through point I. Let O be the midpoint of segment WE. Point R is the inversion of point Z with respect to circle T. Point J is the reflection of point I across segment OZ. What is the distance between points R and J? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.34", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center M and circumradius 12. Let L be the reflection of point F across point C. Let O be the reflection of point D across point B. Point Y is the midpoint of segment LM. What is the length of segment YO? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33.41", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center V and circumradius 3. Let O be the angle bisector of angle LAI. Point N is the midpoint of segment HG. Let W be the perpendicular bisector of segment NV. Point M is the reflection of point J across line O. Let Y be the reflection of line W through point M. Point S is the reflection of point M across line Y. Find the distance between points S and C. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.92", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJK is a regular 11-gon with center X and circumradius 6. Let U be the line through points I and C. Point N is the reflection of point A across point H. Segment JD and line U intersect at point R. Find the length of segment RN. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.31", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center J and circumradius 8. Point X is the reflection of point A across point D. Let S be the circumcircle of polygon ABCDE. Let Z be the inversion of point X in circle S. Point T is the midpoint of segment AD. What is the length of segment ZT? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.52", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center I and circumradius 10. Line Q is the perpendicular bisector of segment IC. Line T is the reflection of line Q across point G. Line P passes through points D and A. Lines T and P intersect at point M. Let L be the perpendicular bisector of segment MI. Point K is the reflection of point A across line L. What is the distance between points K and A? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38.06", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center W and circumradius 11. Let O be the line through points D and B. Point X is the reflection of point H across line O. Point S is the reflection of point F across line O. Point K is the reflection of point S across segment WX. What is the length of segment AK? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.77", "dataset": "MATH"}
{"messages": [{"content": "FU = 3. Let ABCDE be a regular 5-gon with center U and circumradius 2. Point I is the midpoint of segment CD. Point R is the midpoint of segment IC. Point O is the midpoint of segment IR. Point G is the reflection of point D across point E. Find the length of segment GO. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.34", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center P and circumradius 3. Let V be the reflection of point D across point F. Line X passes through point G perpendicular to segment BV. Point S is the reflection of point J across line X. What is the distance between points E and S? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.16", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center L and circumradius 8. Let H be the midpoint of segment FD. Let K be the midpoint of segment DC. Point Z is the reflection of point K across segment BH. What is the measure of angle BAZ, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.09", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center P and circumradius 1. Let K be the midpoint of segment GP. Line X is the perpendicular bisector of segment AP. Point N is obtained by rotating point A by angle 2pi/3 about point P. Point T is the reflection of point D across point H. Line J passes through points K and E. Line Z passes through point T perpendicular to line J. Let W be the reflection of line Z across line X. Let V be the reflection of line W through point F. Line V and segment BN intersect at point O. Point L is the reflection of point O across line X. What is the length of segment LP? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.61", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center N and circumradius 9. Line Q is the perpendicular bisector of segment DC. Line J is the reflection of line Q across point D. Point T is obtained by rotating point N by angle 5pi/6 about point A. Point H is the reflection of point T across line J. Let P be the line through points N and H. Let U be the midpoint of segment DH. Point E is the reflection of point D across line P. Find the length of segment EU. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.88", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center P and circumradius 1. Let W be the circumcircle of polygon ABCDEF. Point Q is the midpoint of segment DF. Point V is the inversion of point Q with respect to circle W. Point O is the reflection of point B across point F. What is the length of segment VO? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.73", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center R and circumradius 7. Line Y is the perpendicular bisector of segment ER. Let O be the midpoint of segment AR. Let K be the line through point D parallel to line Y. Point J is the reflection of point O across line K. Let T be the midpoint of segment OD. Let G be the reflection of point R across point T. Find the distance between points J and G. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.66", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center Z and circumradius 12. Let X be the angle bisector of angle DBC. Point Q is the midpoint of segment DC. Line O is the perpendicular bisector of segment CQ. Point J is the reflection of point Z across line O. Point W is the reflection of point A across line X. What is the distance between points J and W? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.5", "dataset": "MATH"}
{"messages": [{"content": "Let ACDE be a square with center B and circumradius 10. Let W be the rotation of point E by angle 2pi/3 around point A. Let X be the circumcircle of polygon ACDE. Let K be the line through points D and B. Let Y be the line through points C and W. Let J be the midpoint of segment DE. Let Q be the reflection of line Y across line K. Lines K and Q intersect at point R. Let T be the inversion of point J in circle X. What is the length of segment TR? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29.09", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center N and circumradius 6. Let W be the rotation of point E by angle pi/2 around point B. Let Z be the perpendicular bisector of segment CW. Let S be the line through point N parallel to line Z. Line M passes through points C and E. Let H be the reflection of line S across line M. Point X is the reflection of point N across line H. Find the distance between points X and D. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.17", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center O and circumradius 4. Line Z passes through point D perpendicular to segment OI. Let N be the line through point H parallel to line Z. Point Q is the reflection of point G across line N. What is the measure of angle ACQ, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.94", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center J and circumradius 7. Let Q be the circumcircle of polygon ABCDEF. Point X is obtained by rotating point J by angle pi/12 about point F. Point M is the reflection of point E across segment CB. Point H is the inversion of point X with respect to circle Q. Point T is the reflection of point E across segment HB. What is the length of segment TM? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18.77", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center V and circumradius 6. Let Y be the circumcircle of polygon ABCDEF. Let K be the reflection of circle Y through point B. Point H is the reflection of point E across point A. Let M be the reflection of circle K through point H. Point Q is the inversion of point E with respect to circle M. Find the distance between points Q and H. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.44", "dataset": "MATH"}
{"messages": [{"content": "ACDEFGH is a regular 7-gon with center B and circumradius 10. Line Y is the perpendicular bisector of segment GB. Let Q be the circumcircle of polygon ACDEFGH. Let V be the reflection of circle Q through point E. Point R is the reflection of point A across line Y. Point W is the inversion of point B with respect to circle V. What is the distance between points W and R? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28.92", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center N and circumradius 4. Point V is the midpoint of segment CE. Let Y be the reflection of point F across point E. Point K is the reflection of point N across point D. Point R is the reflection of point K across point E. Line M passes through points V and F. Let O be the line through point R parallel to line M. Point S is the reflection of point Y across line O. What is the length of segment SN? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.45", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center P and circumradius 3. Let G be the reflection of point E across point C. Point S is the reflection of point G across segment DE. Point Z is the reflection of point S across point D. Point W is the reflection of point Z across segment DE. What is the length of segment WE? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center H and circumradius 3. Line Y is the perpendicular bisector of segment AB. Let I be the line through point D perpendicular to line Y. Point V is the reflection of point H across line I. Point O is the reflection of point H across segment BV. Point J is the midpoint of segment BV. Let N be the midpoint of segment DJ. Find the length of segment ON. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.16", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGIJKLM be a regular dodecagon with center H and circumradius 7. Let P be the line through points E and G. Let N be the line through point A parallel to line P. Let Z be the midpoint of segment LK. Point Q is the reflection of point Z across line N. What is the length of segment KQ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.05", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center T and circumradius 11. Line O passes through points T and A. Point F is the midpoint of segment AD. Point Y is the reflection of point B across point D. Point S is the reflection of point Y across line O. What is the length of segment SF? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38.89", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center P and circumradius 3. Let S be the midpoint of segment FB. Let M be the angle bisector of angle HAI. Let Q be the line through point C parallel to line M. Line R passes through point B perpendicular to line Q. Let Z be the line through point S perpendicular to line R. Point W is the reflection of point A across line Z. What is the distance between points E and W? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.71", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center U and circumradius 4. Let N be the circumcircle of polygon ABCDEF. Line L passes through points E and D. Let O be the line through point F perpendicular to line L. Circle W is the reflection of circle N across line O. Let I be the inversion of point D in circle W. What is the measure of angle AID, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.05", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center H and circumradius 10. Let X be the perpendicular bisector of segment HD. Let U be the reflection of line X through point E. Point W is the reflection of point H across line U. Point T is the midpoint of segment EC. Let N be the midpoint of segment WD. What is the distance between points N and T? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.09", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center S and circumradius 7. Line I is the angle bisector of angle ACB. Line K is the reflection of line I across point S. Let V be the line through points B and C. Line G is the reflection of line K across line V. Point U is the reflection of point C across line G. Find the distance between points U and D. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25.24", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center I and circumradius 2. Line O is the angle bisector of angle DCI. Let M be the reflection of line O through point A. Point G is the reflection of point D across line M. Let V be the reflection of point B across point D. Let J be the midpoint of segment CG. Point K is the midpoint of segment DJ. Find the distance between points V and K. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.42", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center E and circumradius 11. Point F is the reflection of point A across point D. Point M is the reflection of point C across point B. Let W be the perpendicular bisector of segment BD. Let X be the line through point M parallel to line W. Point N is the reflection of point F across line X. What is the distance between points N and C? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "66.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDF is a regular 5-gon with center E and circumradius 10. Let X be the midpoint of segment EA. Line J passes through point C parallel to segment FB. Point T is obtained by rotating point X by angle pi/2 about point D. Point M is the reflection of point A across point C. Point P is the reflection of point T across line J. Point Z is the reflection of point M across segment FB. Let I be the line through points Z and P. Point U is the reflection of point X across line I. What is the length of segment UD? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24.52", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center M and circumradius 1. Point Q is the reflection of point B across segment DF. Line Z is the perpendicular bisector of segment DB. Let T be the perpendicular bisector of segment FQ. Let R be the reflection of line Z across line T. Point H is the reflection of point M across line R. Find the distance between points F and H. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.61", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center M and circumradius 5. Let T be the angle bisector of angle AGB. Point X is the reflection of point C across line T. Point N is the reflection of point X across segment AF. Find the distance between points N and H. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.21", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center M and circumradius 4. Let U be the circumcircle of polygon ABCDEF. Line I is the perpendicular bisector of segment MF. Let W be the reflection of circle U across line I. Let O be the inversion of point C in circle W. Point R is the reflection of point D across line I. Find the length of segment RO. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.29", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFH is a regular 7-gon with center G and circumradius 5. Let X be the rotation of point D by angle 3pi/4 around point B. Let O be the midpoint of segment XB. Let J be the angle bisector of angle FOA. Line M is the reflection of line J across point F. Point I is the reflection of point H across line M. What is the length of segment GI? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.87", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEF be a regular 5-gon with center B and circumradius 10. Point P is obtained by rotating point C by angle pi/3 about point A. Line W is the perpendicular bisector of segment FA. Point V is the reflection of point P across segment FA. Let Z be the line through point V parallel to line W. Point H is the intersection of line Z and segment FA. What is the distance between points V and H? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.74", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHJKLM be a regular dodecagon with center I and circumradius 10. Line V passes through points E and C. Let U be the angle bisector of angle JDB. Line S is the reflection of line U across line V. Lines S and V intersect at point Z. Find the length of segment ZI. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.68", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center X and circumradius 8. Let U be the angle bisector of angle HAD. Point T is the reflection of point E across point D. Point R is the reflection of point T across line U. Line Y passes through point R parallel to segment GF. Line K passes through point A parallel to line Y. Point N is the reflection of point R across line K. What is the length of segment NT? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22.63", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center S and circumradius 12. Let R be the circumcircle of polygon ABCDEFGHIJKL. Let N be the midpoint of segment EJ. Let M be the reflection of circle R through point K. Point O is the reflection of point N across point L. Point Q is the inversion of point O with respect to circle M. What is the distance between points L and Q? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.56", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFG be a regular hexagon with center C and circumradius 6. Point Q is obtained by rotating point F by angle pi/12 about point E. Let Z be the angle bisector of angle BAE. Let T be the line through point Q perpendicular to line Z. Point H is the reflection of point C across line T. Line P is the angle bisector of angle HCQ. Point N is the reflection of point G across line P. What is the length of segment NH? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.36", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center I and circumradius 7. Point Q is the midpoint of segment AC. Point R is the midpoint of segment QD. Point K is the midpoint of segment HF. What is the distance between points K and R? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.2", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center H and circumradius 6. Let J be the line through points E and C. Point R is the reflection of point F across line J. Point Y is the reflection of point G across segment CB. What is the measure of angle RYB, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.45", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center R and circumradius 12. Line J passes through points B and D. Line P passes through point R perpendicular to segment FD. Let T be the reflection of line P across line J. Point Z is the intersection of lines J and T. Point L is the midpoint of segment FD. What is the distance between points L and Z? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41.11", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center U and circumradius 2. Let I be the reflection of point A across point C. Point Q is the reflection of point E across point I. Let J be the midpoint of segment ID. Line V passes through point Q perpendicular to segment CJ. Point Z is the reflection of point C across line V. What is the distance between points I and Z? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.17", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center P and circumradius 3. Let Q be the rotation of point B by angle 3pi/4 around point C. Point U is obtained by rotating point E by angle pi about point A. Let S be the rotation of point Q by angle 5pi/6 around point U. Let Z be the midpoint of segment AF. Let T be the line through point U perpendicular to segment DU. Point L is the reflection of point Z across line T. What is the distance between points S and L? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.74", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center L and circumradius 11. Point S is obtained by rotating point D by angle 5pi/6 about point C. Line I passes through point C perpendicular to segment BS. Line R passes through point A parallel to line I. Point N is the reflection of point C across line R. What is the length of segment NE? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.86", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJK is a regular 11-gon with center P and circumradius 4. Line N passes through points B and D. Point R is the intersection of segment FC and line N. Let M be the midpoint of segment PI. Find the length of segment MR. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.4", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center W and circumradius 1. Let M be the rotation of point C by angle 5pi/12 around point E. Let R be the perpendicular bisector of segment MF. Let J be the line through point W perpendicular to segment EB. Let S be the reflection of line J across line R. Line S and segment MB intersect at point Y. What is the length of segment YC? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.44", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center H and circumradius 6. Point Q is the reflection of point A across point F. Let U be the reflection of point B across point Q. Let L be the circumcircle of polygon ABCDEF. Point P is the inversion of point U with respect to circle L. Find the length of segment PD. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center W and circumradius 1. Point U is the reflection of point D across point W. Point O is the midpoint of segment CD. Let K be the line through points B and E. Point P is the reflection of point D across point U. Line R is the reflection of line K across point O. Let S be the reflection of point P across point O. Point V is the reflection of point P across line R. Point G is the reflection of point S across point V. Find the length of segment UG. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.44", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center P and circumradius 3. Point V is the reflection of point F across point B. Line Q passes through points V and P. Point M is the reflection of point B across line Q. Point Y is the midpoint of segment PA. Find the length of segment MY. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.54", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center M and circumradius 7. Let N be the reflection of point E across point A. Let K be the circumcircle of polygon ABCDEF. Line I is the angle bisector of angle MNC. Let X be the reflection of circle K across line I. Let G be the inversion of point F in circle X. What is the length of segment NG? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.56", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center I and circumradius 11. Let T be the circumcircle of polygon ABCDEF. Let P be the reflection of circle T through point D. Point K is the inversion of point A with respect to circle P. Point J is the reflection of point B across point F. Find the distance between points K and J. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "37.21", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFHIJ be a regular 9-gon with center G and circumradius 10. Point Q is the midpoint of segment FG. Let L be the line through points I and E. Line X passes through point D perpendicular to line L. Point R is the reflection of point Q across line X. Point P is the reflection of point R across line L. What is the distance between points P and B? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.61", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center X and circumradius 7. Let V be the midpoint of segment XC. Point O is the reflection of point X across point F. Let Z be the circumcircle of polygon ABCDEFGHIJ. Let Q be the inversion of point V in circle Z. Let T be the reflection of point O across point Q. Find the length of segment BT. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "28.21", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIKLM be a regular dodecagon with center J and circumradius 1. Let N be the angle bisector of angle GDJ. Point S is the reflection of point B across line N. Line O is the angle bisector of angle CSF. Point R is the reflection of point A across line O. Find the length of segment SR. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.33", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center O and circumradius 12. Let W be the circumcircle of polygon ABCDEFGHI. Line V passes through points C and D. Circle S is the reflection of circle W across line V. Point J is the inversion of point G with respect to circle S. Find the distance between points A and J. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24.07", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center S and circumradius 11. Let N be the midpoint of segment CA. Point M is the midpoint of segment CF. Line X passes through point E perpendicular to segment FN. Let P be the reflection of point H across point M. Point R is the reflection of point D across line X. Find the length of segment PR. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.58", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center F and circumradius 2. Let L be the rotation of point B by angle pi/6 around point A. Let G be the reflection of point B across point F. Let N be the vector from point B to point G. Line W is the perpendicular bisector of segment EC. Point V is the translation of point F by vector N. Let R be the reflection of point V across point C. Let Z be the line through point D perpendicular to line W. Line I passes through point L parallel to line Z. Line I and segment CR intersect at point U. Let O be the reflection of point U across point D. What is the length of segment UO? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.74", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center R and circumradius 12. Point Z is the reflection of point A across segment GB. Let K be the angle bisector of angle BAZ. Point W is the reflection of point D across line K. What is the distance between points W and G? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29.45", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center O and circumradius 3. Line T passes through points A and F. Let Q be the perpendicular bisector of segment EF. Let J be the midpoint of segment EF. Let Y be the circumcircle of polygon ABCDEF. Line R is the reflection of line T across line Q. Circle S is the reflection of circle Y across line R. Let H be the inversion of point J in circle S. What is the length of segment OH? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.61", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHI be a regular 9-gon with center W and circumradius 4. Let Z be the midpoint of segment GW. Point L is the midpoint of segment ZE. Let T be the midpoint of segment AL. Let N be the reflection of point T across point G. What is the distance between points N and W? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.36", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center O and circumradius 1. Point G is the reflection of point C across point B. Let L be the perpendicular bisector of segment AD. Line I passes through point G perpendicular to line L. Point N is the reflection of point E across line I. Find the measure of angle END, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.31", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center R and circumradius 1. Point Z is the reflection of point D across point B. Let S be the circumcircle of polygon ABCDEFGHIJKL. Circle W is the reflection of circle S across point D. Let P be the inversion of point K in circle W. What is the length of segment PZ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.36", "dataset": "MATH"}
{"messages": [{"content": "Let ABCEFG be a regular hexagon with center D and circumradius 5. Let N be the line through points G and B. Line Z is the reflection of line N across point A. Point J is the reflection of point A across line Z. Let I be the reflection of point B across point J. Find the length of segment IJ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.66", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center N and circumradius 12. Point K is the reflection of point A across point B. Let Y be the midpoint of segment AN. Point V is the midpoint of segment YK. Find the length of segment VD. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18.25", "dataset": "MATH"}
{"messages": [{"content": "BCDEFG is a regular hexagon with center A and circumradius 12. Let O be the midpoint of segment FD. Point Q is the reflection of point O across point B. Let M be the reflection of point E across point C. What is the length of segment MQ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21.63", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGHIJKLM is a regular dodecagon with center F and circumradius 12. Point X is the midpoint of segment KB. Let Z be the angle bisector of angle HXD. Point Q is the reflection of point C across line Z. Let U be the midpoint of segment GQ. What is the distance between points G and U? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.1", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEF be a regular 5-gon with center C and circumradius 10. Let Y be the circumcircle of polygon ABDEF. Point T is the midpoint of segment FE. Point V is the inversion of point T with respect to circle Y. Point W is the reflection of point V across point E. Find the distance between points T and W. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.51", "dataset": "MATH"}
{"messages": [{"content": "ABCDFGHIJKLM is a regular dodecagon with center E and circumradius 5. Let N be the midpoint of segment GD. Let Y be the line through point N perpendicular to segment NJ. Point P is the reflection of point H across line Y. What is the length of segment KP? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.56", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center W and circumradius 3. Line Q passes through points G and I. Point L is the reflection of point A across line Q. Let K be the midpoint of segment LW. What is the distance between points F and K? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.68", "dataset": "MATH"}
{"messages": [{"content": "Let ABCEFGH be a regular 7-gon with center D and circumradius 10. Line N passes through points G and F. Let P be the circumcircle of polygon ABCEFGH. Circle L is the reflection of circle P across line N. Let X be the inversion of point E in circle L. Find the distance between points X and F. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.68", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGIJKLM be a regular dodecagon with center H and circumradius 11. Let Q be the angle bisector of angle BHE. Point U is the midpoint of segment CB. Let V be the line through point D parallel to line Q. Let S be the reflection of point H across point U. Point R is the reflection of point L across segment CI. Point P is the reflection of point A across line V. What is the measure of angle SPR, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.79", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center X and circumradius 4. Line I is the perpendicular bisector of segment FE. Line R is the reflection of line I across point G. Let H be the reflection of line R through point E. Point K is the reflection of point E across line H. Find the distance between points K and D. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.12", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center T and circumradius 10. Point H is obtained by rotating point T by angle pi/12 about point A. Let I be the midpoint of segment FH. Line M is the perpendicular bisector of segment TF. Point L is the reflection of point C across line M. Let W be the reflection of point I across point L. Let Z be the reflection of point W across point H. What is the length of segment ZE? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34.52", "dataset": "MATH"}
{"messages": [{"content": "ABCDEG is a regular hexagon with center F and circumradius 8. Point W is the midpoint of segment BF. Let S be the midpoint of segment WD. Let H be the circumcircle of polygon ABCDEG. Point O is the inversion of point S with respect to circle H. Find the length of segment AO. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25.72", "dataset": "MATH"}
{"messages": [{"content": "ABDEFGHI is a regular octagon with center C and circumradius 12. Let N be the line through point E perpendicular to segment FE. Let Z be the line through point F parallel to line N. Let O be the perpendicular bisector of segment FE. Point J is the reflection of point B across line Z. Point X is the reflection of point J across line O. Point R is the reflection of point X across point A. Find the length of segment RE. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32.67", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center R and circumradius 9. Point Z is the reflection of point A across point D. Point O is the midpoint of segment CZ. Let Y be the perpendicular bisector of segment RZ. Point H is the reflection of point O across line Y. Find the length of segment HA. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.08", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center I and circumradius 12. Let E be the rotation of point D by angle 11pi/12 around point I. Point V is the reflection of point B across segment EC. Point L is the midpoint of segment EV. What is the distance between points L and E? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.57", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center I and circumradius 3. Point Z is the reflection of point H across point A. Let R be the midpoint of segment DC. Line Q is the angle bisector of angle RBE. Point O is the reflection of point R across line Q. Point P is the reflection of point G across point O. What is the distance between points P and Z? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.17", "dataset": "MATH"}
{"messages": [{"content": "ABCDFGHI is a regular octagon with center E and circumradius 9. Point Y is the midpoint of segment DC. Let N be the midpoint of segment CI. Let W be the line through point I perpendicular to segment AF. Point J is the reflection of point Y across line W. What is the length of segment NJ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.23", "dataset": "MATH"}
{"messages": [{"content": "BCDEF is a regular 5-gon with center A and circumradius 1. Line I passes through point E perpendicular to segment BD. Point R is the reflection of point A across line I. Let Y be the midpoint of segment BD. Find the distance between points R and Y. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.22", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center M and circumradius 5. Line X is the angle bisector of angle CEM. Point J is the reflection of point M across line X. Let W be the line through points A and J. Point S is the reflection of point M across line W. Find the distance between points J and S. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.59", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center R and circumradius 4. Let Q be the rotation of point H by angle 5pi/12 around point A. Line O is the perpendicular bisector of segment EQ. Point Y is the reflection of point D across line O. Let X be the perpendicular bisector of segment YA. Point T is the reflection of point J across line X. Find the distance between points T and Y. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.47", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center R and circumradius 2. Let X be the midpoint of segment GA. Let O be the perpendicular bisector of segment HD. Point U is the reflection of point X across line O. Find the length of segment CU. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.41", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFHIJKLM is a regular dodecagon with center G and circumradius 4. Line O is the angle bisector of angle FIH. Point Q is the reflection of point D across line O. Let T be the midpoint of segment JQ. What is the measure of angle TJF, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.52", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center P and circumradius 12. Let Y be the angle bisector of angle DBE. Let T be the rotation of point G by angle 2pi/3 around point H. Let Q be the line through point P perpendicular to segment GB. Let W be the reflection of line Q across line Y. Point Z is the reflection of point G across line W. What is the distance between points T and Z? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36.66", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFGHIJKL be a regular 11-gon with center C and circumradius 2. Line O is the angle bisector of angle DGE. Let S be the circumcircle of polygon ABDEFGHIJKL. Circle Y is the reflection of circle S across line O. Let R be the inversion of point K in circle Y. Find the distance between points R and G. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.45", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIKLM is a regular dodecagon with center J and circumradius 4. Let U be the perpendicular bisector of segment CJ. Let P be the line through point L parallel to line U. Segment BI and line P intersect at point T. What is the measure of angle ITK, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.05", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center P and circumradius 1. Line W is the perpendicular bisector of segment CJ. Line T passes through points A and I. Line V is the reflection of line W across point I. Point U is the intersection of lines V and T. Let X be the midpoint of segment DA. What is the length of segment XU? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.99", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFG be a regular hexagon with center C and circumradius 5. Line S passes through point F parallel to segment ED. Let X be the midpoint of segment ED. Point N is the reflection of point X across line S. What is the length of segment EN? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.01", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center X and circumradius 11. Let N be the angle bisector of angle DXG. Let V be the line through point D parallel to line N. Let J be the reflection of point A across point B. Point L is the reflection of point J across line V. Find the length of segment LG. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22.92", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center J and circumradius 8. Let L be the angle bisector of angle AJE. Let W be the line through point B parallel to line L. Let G be the circumcircle of polygon ABCDEF. Let I be the reflection of circle G across line W. Let M be the inversion of point J in circle I. Find the distance between points M and F. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.22", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center O and circumradius 4. Let X be the line through points B and C. Line I passes through point C perpendicular to line X. Line P passes through point D parallel to line I. Let R be the circumcircle of polygon ABCDEFGH. Let W be the reflection of line P across line I. Circle N is the reflection of circle R across line W. Let U be the inversion of point C in circle N. Find the length of segment UF. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.49", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center I and circumradius 2. Point J is obtained by rotating point G by angle 5pi/12 about point A. Point X is the reflection of point D across point H. Line U passes through point I perpendicular to segment BJ. Point W is the reflection of point B across line U. Point T is the reflection of point W across segment EJ. Find the length of segment TX. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.01", "dataset": "MATH"}
{"messages": [{"content": "ABCEFGH is a regular 7-gon with center D and circumradius 5. Let R be the reflection of point G across point D. Let N be the line through points R and C. Point I lies on line N. Line U is the perpendicular bisector of segment AI. Point X is the reflection of point H across line U. Find the length of segment XI. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.34", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGIJKLM is a regular dodecagon with center H and circumradius 3. Line V is the angle bisector of angle JLD. Line W passes through point E perpendicular to line V. Point Z is the reflection of point A across line W. Let T be the midpoint of segment KZ. What is the length of segment MT? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.06", "dataset": "MATH"}
{"messages": [{"content": "BCDEFGHIJK is a regular 10-gon with center A and circumradius 6. Let V be the midpoint of segment GF. Line W passes through points D and H. Line U passes through point V perpendicular to line W. Point S is the reflection of point D across line U. What is the distance between points A and S? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.42", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGHIJKLM is a regular dodecagon with center F and circumradius 3. Let X be the angle bisector of angle BMD. Line O passes through point G parallel to line X. Point Q is the reflection of point B across line O. Point T is the reflection of point Q across line X. Find the distance between points G and T. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.57", "dataset": "MATH"}
{"messages": [{"content": "Let ABCEFGHI be a regular octagon with center D and circumradius 2. Point N is the midpoint of segment FB. Let R be the line through point G parallel to segment AD. Line O passes through points C and E. Lines R and O intersect at point X. Find the length of segment NX. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.23", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center O and circumradius 12. Let Y be the angle bisector of angle HOE. Line S is the reflection of line Y across point H. Point W is the reflection of point I across line S. Let N be the reflection of point L across point A. What is the length of segment NW? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34.91", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFHIJ be a regular 9-gon with center G and circumradius 1. Point U is the reflection of point C across point B. Line O is the perpendicular bisector of segment HC. Line M is the perpendicular bisector of segment UD. Let S be the reflection of line M across line O. Line S and segment UD intersect at point V. What is the length of segment HV? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.74", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center R and circumradius 7. Let W be the perpendicular bisector of segment JR. Line K is the angle bisector of angle JGR. Line T is the reflection of line W across line K. Let L be the reflection of line T through point I. Line U is the reflection of line L across line W. Point Y is the reflection of point A across line U. Point X is the reflection of point Y across line L. Point N is the reflection of point X across line U. What is the length of segment NJ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.09", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center M and circumradius 8. Let N be the perpendicular bisector of segment CB. Line O passes through point A parallel to line N. Let Y be the midpoint of segment AD. Point W is the reflection of point Y across point B. Point T is the reflection of point D across line O. Find the distance between points T and W. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22.36", "dataset": "MATH"}
{"messages": [{"content": "Let BCDEFGHIJKLM be a regular dodecagon with center A and circumradius 7. Point Z is the midpoint of segment GC. Let U be the midpoint of segment GZ. Let V be the midpoint of segment UK. Find the length of segment BV. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.69", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center T and circumradius 8. Let K be the circumcircle of polygon ABCDEFG. Circle P is the reflection of circle K across point A. Point S is the midpoint of segment CT. Point N is the inversion of point E with respect to circle P. Let I be the line through points N and S. Let J be the reflection of line I through point B. Point O is the midpoint of segment TF. Point V is the reflection of point O across line J. Find the distance between points V and C. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.28", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center Q and circumradius 9. Point Z is the reflection of point C across point J. Point T is the midpoint of segment ZQ. Point S is the midpoint of segment QD. Find the length of segment ST. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.65", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center W and circumradius 4. Line X passes through points D and H. Point P is obtained by rotating point K by angle 7pi/12 about point E. Let Q be the angle bisector of angle IWP. Line S passes through point C parallel to line X. Let Z be the reflection of line Q across line S. Point R is the intersection of lines Z and X. Find the distance between points F and R. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.01", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center P and circumradius 12. Point V is the reflection of point D across point G. Point K is the reflection of point G across point V. Let M be the midpoint of segment ED. Let O be the midpoint of segment PD. Point T is the reflection of point M across segment FK. Find the distance between points O and T. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.1", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center S and circumradius 8. Let R be the line through points H and C. Point P is the reflection of point F across line R. Let V be the angle bisector of angle EHI. Point M is the reflection of point B across line V. What is the measure of angle MPA, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.52", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center H and circumradius 6. Point N is obtained by rotating point B by angle 5pi/6 about point D. Let I be the circumcircle of polygon ABCDEF. Let Q be the reflection of circle I through point D. Let P be the inversion of point N in circle Q. Let Y be the line through points F and D. Point V is the reflection of point P across line Y. Find the length of segment BV. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.59", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJK is a regular 11-gon with center Z and circumradius 4. Line W is the perpendicular bisector of segment BA. Line R is the perpendicular bisector of segment JB. Line M is the reflection of line W across line R. Point L is the intersection of segment KG and line M. What is the length of segment FL? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.29", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center X and circumradius 6. Let W be the midpoint of segment FX. Let Q be the midpoint of segment EW. Let K be the line through points C and Q. Point L is the reflection of point B across line K. What is the distance between points B and L? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.11", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center W and circumradius 3. Let Q be the reflection of point W across point C. Let G be the rotation of point E by angle 7pi/12 around point W. Let R be the rotation of point C by angle pi/4 around point D. Let U be the vector from point F to point D. Let Z be point R translated by vector U. Line V is the angle bisector of angle EZR. Line L passes through point D parallel to line V. Point P is the reflection of point W across segment QG. Let M be the line through points F and P. Lines M and L intersect at point T. What is the distance between points E and T? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.04", "dataset": "MATH"}
{"messages": [{"content": "BH = 11. Let ACDEFG be a regular hexagon with center H and circumradius 12. Let K be the rotation of point C by angle pi around point F. Let W be the reflection of point G across point C. Point M is the reflection of point K across point D. Let N be the midpoint of segment WC. What is the distance between points M and N? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31.18", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center Y and circumradius 5. Line W is the angle bisector of angle GIF. Let M be the circumcircle of polygon ABCDEFGHIJKL. Circle T is the reflection of circle M across line W. Let O be the inversion of point K in circle T. Find the length of segment OK. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.39", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJL be a regular 11-gon with center K and circumradius 10. Line N is the perpendicular bisector of segment KB. Point W is the reflection of point D across line N. Line R is the angle bisector of angle LWD. Line Z is the reflection of line N across line R. Let U be the reflection of line N across line Z. Point X is the reflection of point G across line U. Find the distance between points X and J. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19.64", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center S and circumradius 12. Let P be the line through points C and F. Point N is the reflection of point H across point J. Line U is the perpendicular bisector of segment HE. Let Y be the reflection of line P across line U. Point R is the reflection of point F across line Y. Line V passes through point N perpendicular to line P. Let Q be the line through point S parallel to line V. Point T is the intersection of lines Q and P. Point M is the midpoint of segment TC. Find the distance between points M and R. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22.69", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center V and circumradius 7. Let U be the line through points E and D. Let O be the reflection of line U through point A. Line P is the reflection of line U across line O. Point R is the reflection of point B across point F. Point S is the reflection of point D across line P. Find the length of segment SR. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "99.24", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEF be a regular 5-gon with center C and circumradius 1. Point L is the reflection of point A across point C. Point W is the reflection of point A across point L. Point Q is the midpoint of segment WC. Let K be the midpoint of segment WB. What is the distance between points K and Q? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.5", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center M and circumradius 7. Line N is the angle bisector of angle BCF. Let X be the reflection of line N through point E. Let R be the rotation of point M by angle 5pi/12 around point C. Point Z is the reflection of point R across line X. Point O is the midpoint of segment GI. Let P be the midpoint of segment OM. What is the distance between points Z and P? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29.65", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center Y and circumradius 6. Let G be the line through points E and F. Point Q is the reflection of point D across line G. Let M be the midpoint of segment QE. Let Z be the reflection of point Y across point M. What is the distance between points Z and D? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.87", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center N and circumradius 9. Let Y be the perpendicular bisector of segment CB. Line J is the reflection of line Y across point A. Point I is the reflection of point C across line J. Point Z is the midpoint of segment DF. Point T is the reflection of point I across point Z. Find the length of segment TA. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56.2", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGI is a regular octagon with center H and circumradius 8. Let Q be the perpendicular bisector of segment CB. Let X be the angle bisector of angle FGD. Line P is the reflection of line Q across line X. Let V be the line through point G perpendicular to line P. Let M be the reflection of line V through point H. Point Y is the reflection of point G across line M. Point L is the midpoint of segment YA. Find the length of segment LG. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.55", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center W and circumradius 6. Line T is the perpendicular bisector of segment EJ. Let S be the line through points C and E. Let M be the reflection of line T across line S. Point V is the intersection of lines S and M. What is the distance between points C and V? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.15", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center O and circumradius 8. Let W be the rotation of point A by angle pi/6 around point D. Let X be the line through points O and A. Point P is the reflection of point W across line X. Point Q is the reflection of point C across segment AP. Find the distance between points Q and B. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21.86", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center K and circumradius 3. Let V be the angle bisector of angle KAC. Point L is the reflection of point K across line V. Let P be the midpoint of segment LA. What is the distance between points B and P? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.08", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFHIJK is a regular 10-gon with center G and circumradius 2. Point N is obtained by rotating point K by angle 2pi/3 about point D. Point P is obtained by rotating point H by angle 5pi/12 about point K. Let T be the perpendicular bisector of segment GA. Line L passes through point P parallel to line T. Point O is the reflection of point F across line T. Point W is the reflection of point N across line L. Find the length of segment WO. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.35", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center P and circumradius 11. Point V is the reflection of point A across segment DB. Let M be the midpoint of segment KI. Let W be the reflection of point G across point V. What is the length of segment MW? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47.29", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center I and circumradius 9. Let P be the rotation of point E by angle pi/3 around point I. Point J is the midpoint of segment AB. Point Z is the midpoint of segment BP. Let X be the reflection of point Z across point A. Point G is the reflection of point X across segment BP. Find the distance between points J and G. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.15", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center M and circumradius 1. Let O be the perpendicular bisector of segment DM. Line H passes through point D perpendicular to line O. Point N is the reflection of point C across segment ED. Line Q passes through point D perpendicular to line H. Let Z be the line through points N and M. Let J be the reflection of line Z across line Q. Point Y is the intersection of line J and segment ED. What is the distance between points B and Y? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.84", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center I and circumradius 8. Point O is obtained by rotating point D by angle pi/6 about point F. Line P is the angle bisector of angle OBC. Point Y is the reflection of point O across line P. Point W is the reflection of point Y across segment OD. What is the distance between points D and W? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.28", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center N and circumradius 8. Line V passes through points H and C. Point J lies on line V. Line Z passes through points E and J. Point I is the reflection of point A across line Z. What is the distance between points E and I? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center P and circumradius 2. Line R is the angle bisector of angle PDH. Point W is the reflection of point G across point F. Point Z is the reflection of point B across line R. Point V is the reflection of point Z across point W. What is the distance between points V and W? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.54", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center Y and circumradius 9. Let M be the line through point C perpendicular to segment FB. Let I be the reflection of line M through point F. Point U is the reflection of point E across line I. What is the measure of angle HEU, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.96", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center P and circumradius 5. Let O be the midpoint of segment EP. Let I be the reflection of point O across point P. Point H is the midpoint of segment PC. What is the distance between points I and H? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.55", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center S and circumradius 4. Let U be the midpoint of segment JC. Point L is obtained by rotating point F by angle 3pi/4 about point H. Let Q be the perpendicular bisector of segment UF. Point Y is the reflection of point C across line Q. Line N is the angle bisector of angle CEY. Point T is the reflection of point U across line N. Find the length of segment TL. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.75", "dataset": "MATH"}
{"messages": [{"content": "Let ABCEFGHI be a regular octagon with center D and circumradius 5. Point U is the reflection of point I across point H. Let X be the angle bisector of angle EDA. Let O be the reflection of line X through point U. Let M be the line through point U parallel to line O. Line M and segment AC intersect at point J. Find the length of segment DJ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.23", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEGHIJ be a regular 9-gon with center F and circumradius 8. Point R is the reflection of point E across segment JB. Let Q be the line through point I perpendicular to segment JB. Point O is the reflection of point F across line Q. What is the length of segment OR? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27.07", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center V and circumradius 3. Let M be the midpoint of segment KH. Let Z be the line through points A and F. Point U lies on line Z. Find the measure of angle UFM, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.79", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center T and circumradius 6. Let Y be the circumcircle of polygon ABCDE. Let N be the angle bisector of angle DBE. Let M be the reflection of circle Y through point D. Line Z passes through point B perpendicular to line N. Let X be the inversion of point C in circle M. Point U is the reflection of point X across line Z. What is the length of segment CU? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.81", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center T and circumradius 1. Point O is the midpoint of segment BT. Let W be the perpendicular bisector of segment CD. Point Y is the reflection of point O across line W. What is the length of segment YO? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.71", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center Y and circumradius 2. Point O is the reflection of point H across point A. Point V is the midpoint of segment OF. Point Z is the midpoint of segment VG. What is the length of segment ZO? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.18", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center M and circumradius 12. Let S be the vector from point J to point I. Point L is the reflection of point H across point A. Point U is the translation of point G by vector S. Let K be the line through points U and M. Line R passes through point H perpendicular to line K. Let T be the line through point G parallel to segment LB. Let V be the midpoint of segment LB. Line Y is the reflection of line R across line T. Point Z is the reflection of point V across line K. Lines T and Y intersect at point N. What is the distance between points Z and N? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "30.6", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center L and circumradius 5. Point Q is the reflection of point J across point D. Line T is the angle bisector of angle BQI. Line K passes through point E perpendicular to line T. Point M is the reflection of point A across line K. What is the length of segment MC? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.93", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center T and circumradius 11. Let J be the reflection of point A across point D. Let V be the circumcircle of polygon ABCD. Let S be the inversion of point J in circle V. Let R be the reflection of point S across point C. Let X be the midpoint of segment CJ. Let L be the inversion of point R in circle V. Find the length of segment XL. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.34", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center P and circumradius 5. Point Y is on segment GE. Point N is the reflection of point H across segment CY. Find the length of segment NC. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.66", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center U and circumradius 7. Point X is the reflection of point D across point F. Line Q is the angle bisector of angle FEB. Line W passes through points X and B. Point I is on line W. Line Z is the reflection of line Q across point I. Point P is the reflection of point X across line Z. Find the measure of angle FXP, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.45", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center V and circumradius 6. Line Q passes through points F and B. Let M be the line through point C parallel to segment VE. Lines Q and M intersect at point H. What is the measure of angle ADH, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.05", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center R and circumradius 5. Let W be the perpendicular bisector of segment AI. Let V be the line through point F perpendicular to line W. Line M is the reflection of line V across point H. Point T is the reflection of point E across line M. What is the distance between points T and F? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.5", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center U and circumradius 6. Let I be the reflection of point B across point U. Let M be the midpoint of segment UB. Point L is the midpoint of segment EU. Let Y be the reflection of point L across point I. What is the distance between points M and Y? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.7", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center O and circumradius 6. Let W be the angle bisector of angle EHB. Line X passes through point E parallel to line W. Let Z be the line through point D parallel to segment BC. Lines X and Z intersect at point T. Find the measure of angle BDT, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.75", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJLM be a regular dodecagon with center K and circumradius 7. Let Q be the angle bisector of angle EAG. Line S is the perpendicular bisector of segment KC. Point T is the reflection of point M across line S. Point W is the reflection of point T across line Q. What is the measure of angle WTA, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.52", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center K and circumradius 8. Line L passes through points A and C. Let G be the circumcircle of polygon ABCDE. Let W be the rotation of point B by angle 5pi/12 around point E. Point Y is the reflection of point W across line L. Point J is the reflection of point E across line L. Let T be the reflection of circle G across line L. Point Q is the inversion of point Y with respect to circle T. Find the distance between points J and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.54", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center P and circumradius 8. Let S be the midpoint of segment PE. Let J be the midpoint of segment EC. Point I is the midpoint of segment SH. Find the length of segment JI. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.36", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center O and circumradius 4. Let R be the perpendicular bisector of segment AB. Let J be the angle bisector of angle ADH. Line T passes through point B parallel to line J. Point L is the reflection of point C across line J. Let X be the reflection of line T across line R. Line U is the reflection of line R across line T. Let Y be the line through point L parallel to line U. Lines X and Y intersect at point N. What is the distance between points N and F? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31.5", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center J and circumradius 8. Let F be the rotation of point E by angle pi/3 around point C. Point O is obtained by rotating point F by angle 7pi/12 about point J. Line V is the angle bisector of angle OAF. Let N be the line through point O perpendicular to line V. Let L be the midpoint of segment AC. Point M is the reflection of point D across line N. Find the length of segment ML. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21.56", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center G and circumradius 3. Let M be the rotation of point G by angle 5pi/12 around point D. Let Q be the rotation of point E by angle 7pi/12 around point A. Point Z is the reflection of point A across point D. Line P is the perpendicular bisector of segment QD. Let L be the angle bisector of angle MGD. Line Y is the reflection of line L across line P. Let J be the perpendicular bisector of segment GE. Let H be the reflection of line P across line J. Point R is the intersection of lines H and Y. What is the length of segment RZ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.76", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEGHI be a regular octagon with center F and circumradius 6. Line Y passes through points H and B. Point N is the reflection of point F across point A. Line O is the reflection of line Y across point N. Point K is the reflection of point D across line O. What is the length of segment NK? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34.29", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFGHI be a regular octagon with center C and circumradius 4. Let J be the reflection of point D across point B. Let W be the reflection of point E across point F. Let R be the midpoint of segment JH. Let S be the reflection of point R across point W. What is the length of segment RS? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.34", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGHIJKLM is a regular dodecagon with center F and circumradius 8. Point Y is the reflection of point A across point I. Let P be the circumcircle of polygon ABCDEGHIJKLM. Point R is the midpoint of segment GY. Let Z be the inversion of point R in circle P. What is the distance between points Z and J? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.33", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJK is a regular 11-gon with center S and circumradius 11. Point V is the reflection of point H across point G. Line U passes through points I and C. Line O is the perpendicular bisector of segment HI. Line M is the reflection of line O across point G. Point X is the intersection of lines U and M. What is the distance between points V and X? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "130.53", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center Q and circumradius 5. Let R be the angle bisector of angle HAG. Let X be the midpoint of segment IE. Point S is the reflection of point X across point E. Point U is the reflection of point H across line R. What is the length of segment SU? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.17", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center Y and circumradius 3. Let T be the perpendicular bisector of segment CB. Let W be the reflection of line T through point A. Point V is the midpoint of segment YC. Let F be the midpoint of segment DA. Point Z is the reflection of point F across line W. What is the length of segment ZV? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.06", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center Y and circumradius 6. Line Q is the angle bisector of angle CJY. Let W be the reflection of point H across point F. Point X is the reflection of point E across line Q. Point T is the reflection of point X across point W. Point M is the reflection of point W across segment BT. What is the length of segment IM? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.41", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center H and circumradius 6. Let O be the reflection of point G across point B. Point R is the midpoint of segment GD. Let Y be the perpendicular bisector of segment RO. Point X is the reflection of point G across line Y. Find the length of segment EX. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22.79", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center M and circumradius 11. Let S be the angle bisector of angle DMA. Let V be the perpendicular bisector of segment EA. Let I be the reflection of line V across line S. Line Z passes through point D parallel to line S. Lines I and Z intersect at point L. What is the measure of angle ELM, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.63", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center X and circumradius 6. Point N is the reflection of point B across point J. Let M be the line through point X perpendicular to segment XN. Let V be the reflection of line M through point C. Let L be the reflection of line V across line M. Line T is the reflection of line L across point B. Point O is the reflection of point E across line T. Find the length of segment OG. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18.56", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center L and circumradius 6. Let W be the angle bisector of angle JEI. Let O be the perpendicular bisector of segment JC. Let M be the reflection of line O through point H. Point Q is the reflection of point D across line M. Point X is the reflection of point Q across line W. What is the length of segment XI? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21.28", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center G and circumradius 5. Line H passes through points B and A. Point W is on line H. Let U be the midpoint of segment BE. Find the measure of angle WBU, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.63", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center T and circumradius 6. Line I passes through points T and E. Line V is the reflection of line I across point D. Point U is obtained by rotating point D by angle pi about point E. Let X be the midpoint of segment TD. Let G be the line through points E and X. Lines V and G intersect at point Y. Find the length of segment UY. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26.15", "dataset": "MATH"}
{"messages": [{"content": "BCDEFGHIJ is a regular 9-gon with center A and circumradius 2. Let Y be the reflection of point G across point J. Let P be the perpendicular bisector of segment DY. Let Z be the reflection of line P through point B. Point R is the intersection of segment DY and line Z. What is the distance between points R and F? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.81", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center O and circumradius 7. Let I be the circumcircle of polygon ABCDEFG. Let T be the midpoint of segment ED. Point J is the midpoint of segment CB. Circle H is the reflection of circle I across point C. Point X is the inversion of point T with respect to circle H. Find the distance between points J and X. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.71", "dataset": "MATH"}
{"messages": [{"content": "ABCDEG is a regular hexagon with center F and circumradius 11. Let S be the reflection of point A across point G. Line P passes through points B and E. Let R be the midpoint of segment FS. Line M passes through point E perpendicular to line P. Let U be the reflection of line M through point F. Line X passes through points E and R. Lines X and U intersect at point J. What is the distance between points J and S? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39.66", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGIJK is a regular 10-gon with center H and circumradius 5. Point W is the reflection of point H across segment FB. Let Y be the reflection of point W across point B. Let R be the midpoint of segment FB. Point V is the midpoint of segment RY. Find the distance between points G and V. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.79", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFHIJK be a regular 10-gon with center G and circumradius 3. Let P be the line through points J and E. Point L is the reflection of point E across point K. Line T passes through point L parallel to line P. Let W be the reflection of point A across point F. Segment AB and line T intersect at point X. Find the measure of angle WLX, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.63", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center P and circumradius 1. Let R be the midpoint of segment CA. Let W be the line through points D and F. Line S is the reflection of line W across point B. Point M is the reflection of point R across line S. Find the distance between points E and M. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.77", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center Q and circumradius 12. Let P be the angle bisector of angle GEQ. Let T be the line through point Q perpendicular to line P. Let V be the midpoint of segment QD. Point M is the reflection of point V across line T. What is the distance between points M and B? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.42", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center Q and circumradius 5. Point H is the midpoint of segment DF. Let S be the perpendicular bisector of segment HQ. Let W be the line through point B parallel to line S. Point M is the reflection of point A across line W. What is the distance between points M and E? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.23", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center Y and circumradius 2. Point W is the reflection of point Y across point A. Let Z be the rotation of point A by angle 5pi/12 around point B. Let J be the circumcircle of polygon ABCDEFGH. Circle T is the reflection of circle J across point B. Point M is the inversion of point G with respect to circle T. Point X is the intersection of segments WE and ZH. Find the length of segment XM. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.24", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center M and circumradius 11. Point S is obtained by rotating point E by angle pi about point B. Let Q be the midpoint of segment EM. Point W is the reflection of point B across segment AQ. Point G is the reflection of point S across segment QC. What is the distance between points G and W? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54.53", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center L and circumradius 12. Let Y be the rotation of point A by angle 2pi/3 around point L. Let V be the rotation of point L by angle pi/3 around point Y. Let Z be the line through point V parallel to segment YB. Let N be the line through point L perpendicular to line Z. Point P is the reflection of point A across line N. Find the distance between points P and V. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.18", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center S and circumradius 4. Let M be the midpoint of segment SA. Let L be the midpoint of segment SF. Point Z is the reflection of point M across point L. Point H is the reflection of point M across segment CA. What is the distance between points Z and H? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.0", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center Z and circumradius 3. Let R be the circumcircle of polygon ABCD. Point J is obtained by rotating point Z by angle pi/6 about point B. Line H is the angle bisector of angle ACJ. Let Q be the reflection of circle R through point A. Point T is the inversion of point C with respect to circle Q. Line W passes through point T perpendicular to line H. Point N is the reflection of point D across line W. Find the distance between points Z and N. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.56", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center W and circumradius 8. Point M is the reflection of point D across segment HG. Point X is the midpoint of segment HG. Let Y be the midpoint of segment XC. What is the length of segment MY? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.2", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center Q and circumradius 8. Let R be the circumcircle of polygon ABCDEFGHI. Point K is obtained by rotating point E by angle 7pi/12 about point A. Point X is the midpoint of segment KQ. Let Y be the reflection of circle R through point A. Point Z is the inversion of point X with respect to circle R. Point J is the inversion of point Z with respect to circle Y. Find the length of segment IJ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.49", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center K and circumradius 11. Let Q be the line through points H and B. Point Z is on line Q. Let O be the angle bisector of angle ZHC. Point P is the reflection of point G across line O. Line V passes through point D perpendicular to line O. Point X is the reflection of point P across segment FD. Point W is the reflection of point P across line V. What is the length of segment XW? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56.06", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJK be a regular 11-gon with center W and circumradius 12. Point Y is the midpoint of segment CJ. Let M be the perpendicular bisector of segment KI. Point N is the reflection of point Y across line M. Let R be the midpoint of segment YC. Point P is the midpoint of segment NC. What is the distance between points P and R? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.53", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center Q and circumradius 8. Let T be the rotation of point D by angle pi/12 around point B. Point J is the midpoint of segment QD. Let Y be the perpendicular bisector of segment CT. Let Z be the reflection of point A across point C. Line X passes through point J perpendicular to line Y. Let M be the reflection of point Z across point J. Point V is the reflection of point M across line X. Find the length of segment CV. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "32.55", "dataset": "MATH"}
{"messages": [{"content": "BCDEFG is a regular hexagon with center A and circumradius 5. Let I be the perpendicular bisector of segment EG. Let N be the line through point C perpendicular to line I. Let S be the circumcircle of polygon BCDEFG. Let H be the reflection of circle S across line N. Point O is the inversion of point A with respect to circle H. Find the distance between points E and O. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.9", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center X and circumradius 7. Let Y be the vector from point D to point C. Let J be the midpoint of segment CB. Let P be the line through points E and A. Let I be point J translated by vector Y. Line T passes through points C and I. Let K be the reflection of line T across line P. Let N be the reflection of point E across point J. Let S be the reflection of line K through point I. Point O is the reflection of point A across line S. Let Q be the reflection of point N across point X. Find the distance between points Q and O. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "64.56", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGI is a regular octagon with center H and circumradius 9. Line L is the perpendicular bisector of segment FA. Let W be the line through point B perpendicular to line L. Point P is the reflection of point I across line W. Point Q is the reflection of point P across line L. What is the distance between points B and Q? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.63", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center X and circumradius 1. Let N be the line through points H and D. Let J be the line through point E perpendicular to line N. Line Q passes through point G parallel to line J. Point Y is the reflection of point C across line Q. What is the distance between points B and Y? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.14", "dataset": "MATH"}
{"messages": [{"content": "ACDEFGHIJKLM is a regular dodecagon with center B and circumradius 11. Let U be the angle bisector of angle KMH. Line R passes through point H parallel to line U. Point S is the reflection of point K across line R. Point Y is the midpoint of segment JS. Find the distance between points Y and M. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.77", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center W and circumradius 9. Let T be the line through points G and A. Line L passes through point W parallel to line T. Line L and segment FE intersect at point P. What is the distance between points P and A? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.67", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center J and circumradius 11. Point L is the reflection of point A across point B. Point G is the reflection of point C across point L. Point X is the midpoint of segment AC. Point W is the reflection of point X across point G. What is the length of segment LW? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34.35", "dataset": "MATH"}
{"messages": [{"content": "Let BCDEF be a regular 5-gon with center A and circumradius 4. Point M is the reflection of point A across segment FE. Let O be the midpoint of segment FE. Let P be the angle bisector of angle OBM. Point X is the reflection of point A across line P. Find the length of segment XM. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.6", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center N and circumradius 4. Point V is the reflection of point B across point A. Let R be the line through points C and V. Point G is the midpoint of segment VF. Point Q is the midpoint of segment VG. Point U is the reflection of point D across line R. Let L be the midpoint of segment QU. What is the length of segment LF? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.82", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center H and circumradius 9. Let T be the angle bisector of angle DCE. Point N is the reflection of point D across line T. Let L be the midpoint of segment CE. Find the measure of angle DNL, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.31", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFGHI be a regular octagon with center C and circumradius 5. Let M be the line through points G and I. Let U be the line through point F parallel to line M. Line K is the reflection of line U across point E. Line O passes through point C perpendicular to line U. Let V be the reflection of line O through point I. Lines V and K intersect at point Z. What is the length of segment ZB? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center P and circumradius 12. Line O is the angle bisector of angle CBE. Line Z is the reflection of line O across point A. Point G lies on line Z at distance 9 from point F. Line L is the reflection of line O across point F. Let W be the reflection of line L across line O. Let S be the reflection of line O across line W. Point N is the reflection of point G across line S. What is the distance between points N and C? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "162.25", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center H and circumradius 1. Line X is the perpendicular bisector of segment BA. Let S be the reflection of line X through point A. Line L is the reflection of line S across point C. Point Z is the midpoint of segment BA. Point G is the reflection of point E across point Z. Point M is the reflection of point A across line L. Find the distance between points G and M. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.9", "dataset": "MATH"}
{"messages": [{"content": "Let EA = 3. BCDFGH is a regular hexagon with center A and circumradius 12. Let L be the line through point H parallel to segment GA. Point Z is the reflection of point D across line L. Point R is the reflection of point Z across segment GA. Find the length of segment GR. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEGHIJKL be a regular 11-gon with center F and circumradius 2. Line Z is the perpendicular bisector of segment CE. Let M be the line through point I parallel to line Z. Line P passes through point H parallel to segment CE. Point T is the intersection of lines P and M. What is the distance between points E and T? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.01", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center Q and circumradius 8. Let V be the line through points A and B. Point R is the reflection of point G across line V. Line P is the perpendicular bisector of segment HQ. Point N is the reflection of point F across line P. What is the length of segment NR? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22.38", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEFGHIJKLM be a regular dodecagon with center B and circumradius 8. Line O passes through points H and E. Let U be the angle bisector of angle BAK. Line X is the reflection of line O across line U. Point N is the reflection of point H across line X. Find the length of segment NE. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31.66", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center Z and circumradius 3. Point K is obtained by rotating point E by angle 3pi/4 about point C. Let M be the line through points B and C. Let N be the perpendicular bisector of segment DK. Let I be the reflection of line N through point E. Lines M and I intersect at point J. Find the distance between points J and D. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.1", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDFGHI be a regular octagon with center E and circumradius 7. Let O be the midpoint of segment HE. Line L is the angle bisector of angle OIB. Let J be the circumcircle of polygon ABCDFGHI. Circle T is the reflection of circle J across line L. Let Z be the inversion of point F in circle T. Find the distance between points Z and F. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.02", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center S and circumradius 3. Point F is the reflection of point B across point D. Let G be the reflection of point S across point D. Let U be the circumcircle of polygon ABCD. Let N be the inversion of point F in circle U. What is the distance between points N and G? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center M and circumradius 9. Let Q be the perpendicular bisector of segment CE. Let Y be the reflection of point C across point G. Let N be the reflection of line Q through point K. Line T is the reflection of line N across line Q. Point U is the reflection of point D across line T. Let Z be the angle bisector of angle KUY. Point V is the reflection of point H across line Z. What is the distance between points C and V? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27.16", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center Q and circumradius 8. Let H be the midpoint of segment ED. Line K passes through points A and D. Let N be the line through point E perpendicular to line K. Let S be the reflection of point H across point E. Let W be the perpendicular bisector of segment HG. Let Y be the circumcircle of polygon ABCDEFG. Point P is the reflection of point S across line W. Circle I is the reflection of circle Y across line N. Line L passes through points F and P. Let U be the reflection of circle I across line L. Let R be the inversion of point Q in circle U. Find the distance between points R and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.91", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center R and circumradius 3. Let T be the midpoint of segment DA. Let N be the angle bisector of angle FET. Point I is the reflection of point D across line N. Point P is the reflection of point I across segment DA. What is the distance between points P and H? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.97", "dataset": "MATH"}
{"messages": [{"content": "ABCE is a square with center D and circumradius 7. Point P lies on segment CA. Line H is the perpendicular bisector of segment BP. Point S is the reflection of point D across line H. Line O passes through point S parallel to line H. Point Q is the reflection of point P across line O. What is the distance between points Q and S? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center P and circumradius 4. Let K be the rotation of point C by angle pi around point G. Point V is the midpoint of segment KG. Point N is the reflection of point P across point V. Line O passes through point N perpendicular to segment HK. Let I be the line through point P perpendicular to line O. Point T is the reflection of point F across line I. Let R be the midpoint of segment HK. Find the distance between points R and T. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.55", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center V and circumradius 3. Point U is on segment BA. Let G be the line through point F perpendicular to segment BA. Point Y is the reflection of point U across point V. Point M is the reflection of point U across line G. What is the distance between points Y and M? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.94", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center T and circumradius 8. Point X is obtained by rotating point C by angle pi/6 about point F. Line W is the perpendicular bisector of segment AF. Line P is the reflection of line W across point C. Line K passes through point D perpendicular to line P. Point Y is the reflection of point E across line K. What is the length of segment YX? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.84", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJK be a regular 11-gon with center Q and circumradius 9. Let V be the line through points B and J. Let L be the rotation of point D by angle 5pi/6 around point C. Line N passes through points L and K. Let O be the line through points A and H. Lines V and N intersect at point P. Point M is the intersection of lines N and O. What is the distance between points M and P? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.03", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center K and circumradius 7. Let Z be the line through points C and F. Let O be the circumcircle of polygon ABCDEF. Line W passes through point D parallel to line Z. Circle P is the reflection of circle O across line W. Point Y is the inversion of point F with respect to circle P. Find the distance between points Y and C. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.62", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center M and circumradius 1. Let X be the midpoint of segment MH. Let J be the midpoint of segment XA. Point Y is the reflection of point D across point E. Let Q be the midpoint of segment EY. Find the distance between points Q and J. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.83", "dataset": "MATH"}
{"messages": [{"content": "Let ABCEFGHIJKLM be a regular dodecagon with center D and circumradius 5. Line Y is the angle bisector of angle MBH. Point X is the midpoint of segment JH. Point S is the reflection of point K across line Y. Point T is the midpoint of segment SX. Find the length of segment TL. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.46", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center F and circumradius 12. Point U is the midpoint of segment CD. Point L is the reflection of point U across point B. Let I be the rotation of point L by angle pi around point U. Point E is the midpoint of segment LF. Point K is the reflection of point I across point A. Let P be the circumcircle of polygon ABCD. Let T be the inversion of point K in circle P. Point G is the midpoint of segment TE. What is the length of segment GI? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54.32", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center H and circumradius 4. Let G be the reflection of point E across point D. Let J be the perpendicular bisector of segment GF. Line I is the angle bisector of angle HCF. Let W be the reflection of line J across line I. Let X be the reflection of line W across line J. Point P is the reflection of point H across line X. Find the distance between points P and D. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.28", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center Q and circumradius 1. Let K be the circumcircle of polygon ABCDEFGH. Let W be the midpoint of segment QC. Let N be the perpendicular bisector of segment DW. Circle Y is the reflection of circle K across line N. Let V be the inversion of point G in circle Y. What is the distance between points W and V? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.93", "dataset": "MATH"}
{"messages": [{"content": "ABCDFG is a regular hexagon with center E and circumradius 10. Line P passes through point C parallel to segment AB. Let Q be the midpoint of segment AB. Point J is the reflection of point Q across line P. What is the distance between points D and J? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJLM be a regular dodecagon with center K and circumradius 2. Let V be the rotation of point L by angle pi/6 around point E. Point T is the reflection of point I across point C. Let W be the perpendicular bisector of segment ML. Let S be the reflection of line W through point V. Point Q is the reflection of point T across line S. Find the length of segment TQ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.38", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFH be a regular 7-gon with center G and circumradius 8. Point K is the midpoint of segment AD. Point L is the reflection of point F across point H. Let V be the line through points L and K. Let Q be the reflection of line V through point E. Point N is the reflection of point D across line Q. What is the length of segment CN? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35.18", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center R and circumradius 3. Let T be the vector from point E to point J. Point Z is the translation of point A by vector T. Let V be the midpoint of segment IE. Line S passes through points J and A. Let Q be the circumcircle of polygon ABCDEFGHIJ. Point Y is the reflection of point V across line S. Point W is the inversion of point Y with respect to circle Q. Find the distance between points W and Z. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.18", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center U and circumradius 4. Point Y is the midpoint of segment IU. Segments CG and EI intersect at point X. What is the distance between points X and Y? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.06", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHI be a regular 9-gon with center T and circumradius 5. Let V be the line through points B and T. Let J be the rotation of point E by angle pi/3 around point G. Let O be the midpoint of segment DJ. Line L passes through point E perpendicular to line V. Point U is the reflection of point O across line L. Point Q is the reflection of point I across segment UO. Find the distance between points B and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.7", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center U and circumradius 5. Let S be the line through points K and L. Point O is the reflection of point E across line S. Point Z is the reflection of point F across segment OB. What is the length of segment HZ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18.03", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center Y and circumradius 1. Let X be the angle bisector of angle LEH. Point V is the reflection of point L across line X. Line U passes through points A and B. Line M is the reflection of line U across point H. Point Q is the reflection of point K across line M. What is the distance between points Q and V? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.84", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHI be a regular 9-gon with center X and circumradius 7. Line W is the angle bisector of angle DHE. Point K is the reflection of point E across line W. Let S be the midpoint of segment KG. Point R is the reflection of point H across point S. What is the length of segment HR? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.1", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center V and circumradius 2. Point Z is the midpoint of segment GD. Let W be the reflection of point C across point G. Line O is the angle bisector of angle ZWH. Let J be the midpoint of segment CB. Point Q is the midpoint of segment HJ. Point N is the reflection of point B across line O. Point I is the midpoint of segment QG. Point X is the midpoint of segment NI. Find the length of segment HX. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.1", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center W and circumradius 9. Line V is the angle bisector of angle IGE. Let T be the line through point A perpendicular to line V. Point X is the reflection of point F across line T. Point U is the reflection of point X across line V. Find the length of segment DU. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.73", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center Q and circumradius 1. Let P be the midpoint of segment AC. Let U be the midpoint of segment DB. Let T be the midpoint of segment UP. What is the length of segment ET? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.39", "dataset": "MATH"}
{"messages": [{"content": "ACDEF is a regular 5-gon with center B and circumradius 12. Let Y be the reflection of point F across point C. Let V be the perpendicular bisector of segment BA. Line G is the reflection of line V across point Y. Point Q is the reflection of point C across line G. Find the length of segment QF. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.28", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center W and circumradius 8. Point Q is the midpoint of segment HC. Point S is the reflection of point D across point F. Let K be the circumcircle of polygon ABCDEFGH. Point N is the inversion of point S with respect to circle K. Point Z is the midpoint of segment BN. Find the distance between points Z and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.57", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center L and circumradius 6. Let O be the circumcircle of polygon ABCDEFGH. Let S be the angle bisector of angle FAB. Circle P is the reflection of circle O across line S. Let Z be the inversion of point B in circle P. What is the distance between points E and Z? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22.64", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGIJKLM be a regular dodecagon with center H and circumradius 2. Let R be the circumcircle of polygon ABCDEFGIJKLM. Circle Q is the reflection of circle R across point B. Let Y be the line through point M perpendicular to segment BI. Let P be the reflection of circle Q across line Y. Let T be the inversion of point G in circle P. Find the measure of angle TIM, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.83", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFHIJ be a regular 9-gon with center G and circumradius 1. Line Y passes through point A parallel to segment EF. Point L is the reflection of point D across line Y. Let Z be the reflection of point G across point L. Line R is the angle bisector of angle GFI. Point P is the reflection of point B across line R. Find the distance between points P and Z. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.55", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center Y and circumradius 9. Let E be the perpendicular bisector of segment CY. Let Z be the reflection of line E through point C. Let T be the line through points C and B. Point S is the intersection of lines Z and T. Find the distance between points D and S. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.23", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center U and circumradius 6. Let J be the circumcircle of polygon ABCDEFG. Line R is the angle bisector of angle CGE. Line L passes through point E parallel to line R. Line O is the reflection of line R across line L. Let Q be the reflection of circle J across line O. Let I be the inversion of point A in circle Q. What is the distance between points I and B? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.43", "dataset": "MATH"}
{"messages": [{"content": "BCDEFGH is a regular 7-gon with center A and circumradius 12. Let Y be the midpoint of segment GC. Line L passes through point Y parallel to segment ED. Point I is the reflection of point B across line L. Point X is the reflection of point I across segment ED. Point U is the reflection of point X across point Y. What is the length of segment UD? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "36.69", "dataset": "MATH"}
{"messages": [{"content": "BCDEFG is a regular hexagon with center A and circumradius 4. Let Q be the line through point G parallel to segment CE. Point V is the reflection of point B across segment CE. Line K passes through points E and V. Lines Q and K intersect at point R. Find the distance between points R and E. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIKLM be a regular dodecagon with center J and circumradius 6. Let P be the reflection of point I across point H. Let Q be the midpoint of segment MP. Point Z is the midpoint of segment IB. Find the distance between points Q and Z. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.34", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJK be a regular 11-gon with center T and circumradius 4. Line O is the angle bisector of angle DIE. Point V is the reflection of point G across line O. Let N be the midpoint of segment GB. What is the distance between points N and V? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.17", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center L and circumradius 11. Let P be the line through points A and B. Line Y passes through point F parallel to line P. Let W be the reflection of line Y across line P. Point K is the reflection of point A across line W. What is the distance between points L and K? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43.71", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center Q and circumradius 6. Let M be the reflection of point D across point E. Let Z be the midpoint of segment QC. Let G be the angle bisector of angle ZAQ. Point U is the reflection of point Z across line G. What is the distance between points U and M? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.57", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center G and circumradius 6. Let X be the circumcircle of polygon ABCDEF. Point I is the midpoint of segment FE. Let H be the inversion of point I in circle X. What is the length of segment CH? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.49", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center M and circumradius 4. Point W is the reflection of point C across point M. Let T be the line through point M perpendicular to segment AF. Point V is the reflection of point W across line T. Find the measure of angle EVB, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.35", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center L and circumradius 1. Let O be the circumcircle of polygon ABCDEFGHI. Point V is the reflection of point E across segment HB. Point U is the inversion of point V with respect to circle O. Find the length of segment AU. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.56", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center Q and circumradius 5. Point X is the reflection of point Q across point D. Line V passes through points X and Q. Point L is the midpoint of segment AX. Point N is the reflection of point L across line V. Find the length of segment XN. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.17", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center R and circumradius 8. Line W is the angle bisector of angle ECF. Point O is the midpoint of segment RD. Let J be the perpendicular bisector of segment RD. Point V lies on line J. Let M be the line through point O perpendicular to line W. Point P is the reflection of point C across line M. Find the measure of angle VCP, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.26", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center U and circumradius 1. Line T passes through point C perpendicular to segment CA. Point L is the reflection of point D across line T. Line H is the perpendicular bisector of segment LC. Point R is the reflection of point B across line H. What is the distance between points R and D? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.16", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDF be a regular 5-gon with center E and circumradius 10. Let I be the midpoint of segment BC. Point J is the midpoint of segment IF. Let M be the circumcircle of polygon ABCDF. Point T is the inversion of point J with respect to circle M. Find the distance between points T and F. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "94.72", "dataset": "MATH"}
{"messages": [{"content": "Let BCDEF be a regular 5-gon with center A and circumradius 9. Let R be the perpendicular bisector of segment FA. Point Q is obtained by rotating point D by angle 11pi/12 about point E. Let N be the line through points D and B. Line J is the reflection of line R across point Q. Line T is the reflection of line J across point F. Point V is the intersection of lines T and N. Let S be the midpoint of segment AD. Find the length of segment VS. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.87", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center J and circumradius 12. Line P is the angle bisector of angle JFE. Let U be the line through point B parallel to line P. Let T be the angle bisector of angle CAE. Line Y is the reflection of line T across line U. Point I is on line P. Line V is the reflection of line Y across point A. Point S is the reflection of point I across line V. Find the measure of angle ABS, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.52", "dataset": "MATH"}
{"messages": [{"content": "ABCEFGHIJKLM is a regular dodecagon with center D and circumradius 3. Let S be the midpoint of segment LB. Let U be the line through point L perpendicular to segment LB. V is constructed so that KV = 6. Point O is on line U with |OV| = 3. Let W be the circumcircle of polygon ABCEFGHIJKLM. Let P be the reflection of line U through point S. Point Y is the inversion of point S with respect to circle W. Line N passes through point Y perpendicular to line P. Let Z be the line through point O parallel to line N. Let R be the reflection of line P across line U. Let T be the line through point H parallel to line Z. Point X is the intersection of lines T and R. Find the distance between points X and K. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.05", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center H and circumradius 10. Let Q be the vector from point C to point A. Point O is the translation of point D by vector Q. Let T be the perpendicular bisector of segment HB. Let V be the angle bisector of angle DCO. Let I be the reflection of line V through point D. Let Y be the reflection of line T across line I. Point U is the midpoint of segment BA. Point P is the midpoint of segment HB. Point J is the reflection of point P across segment BA. Point M is the reflection of point B across line Y. Find the measure of angle MJU, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.14", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center Z and circumradius 7. Line S passes through points I and H. Let X be the perpendicular bisector of segment DJ. Line P is the reflection of line S across line X. Line O passes through points B and J. Lines P and O intersect at point M. Find the distance between points H and M. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "45.79", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center X and circumradius 3. Point V is the reflection of point G across point E. Line Z passes through points D and V. Line Z and segment AX intersect at point M. What is the distance between points J and M? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.38", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEFGHI be a regular octagon with center B and circumradius 11. Let L be the reflection of point A across point C. Point V is the midpoint of segment DI. Point T is the midpoint of segment VD. What is the length of segment LT? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.61", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center W and circumradius 6. Point N is the midpoint of segment FE. Let J be the midpoint of segment BN. What is the length of segment FJ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.41", "dataset": "MATH"}
{"messages": [{"content": "Let BCDEFGHIJKLM be a regular dodecagon with center A and circumradius 9. Let R be the midpoint of segment LI. Line U is the perpendicular bisector of segment IM. Let Q be the line through point J parallel to line U. Point V is the reflection of point D across line Q. What is the distance between points V and R? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18.3", "dataset": "MATH"}
{"messages": [{"content": "ABCEFGHI is a regular octagon with center D and circumradius 3. Let T be the perpendicular bisector of segment DG. Let W be the line through point I perpendicular to line T. Point J is the reflection of point E across point H. Point U is the reflection of point H across line W. Let L be the midpoint of segment CJ. Let O be the angle bisector of angle LUF. Line R passes through point F parallel to line O. Point K is the intersection of line R and segment IG. What is the distance between points C and K? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.49", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center R and circumradius 12. Let M be the reflection of point E across point R. Let W be the perpendicular bisector of segment MF. Line J passes through point D parallel to line W. Point P is the reflection of point F across line J. Let I be the midpoint of segment DM. Find the distance between points I and P. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.57", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center I and circumradius 2. Let W be the midpoint of segment GA. Point V is the midpoint of segment HA. Let T be the perpendicular bisector of segment VD. Point X is the reflection of point G across line T. Point S is the reflection of point I across segment BW. What is the distance between points X and S? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.27", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFHI is a regular octagon with center G and circumradius 6. Let R be the circumcircle of polygon ABCDEFHI. Let T be the reflection of circle R through point E. Let V be the inversion of point A in circle T. Point P is the midpoint of segment VE. What is the measure of angle PAB, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.18", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center X and circumradius 6. Line Z is the angle bisector of angle ABD. Let V be the line through point D parallel to line Z. Let K be the line through point A parallel to line V. Point R is the reflection of point C across line K. What is the distance between points C and R? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21.71", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFGHIJKL be a regular 11-gon with center C and circumradius 9. Let Q be the midpoint of segment FA. Point W is the reflection of point Q across segment IH. Point R is the reflection of point C across point F. Find the distance between points R and W. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.84", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFGHI be a regular octagon with center C and circumradius 6. Line S is the perpendicular bisector of segment DF. Point W is the midpoint of segment EB. Point R is the reflection of point W across line S. Find the length of segment RA. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.24", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center Y and circumradius 10. Let O be the circumcircle of polygon ABCDEFGH. Line U passes through points A and C. Point J is the reflection of point D across line U. Point T is the inversion of point J with respect to circle O. Line R passes through point D parallel to segment EC. Point M is the reflection of point C across line R. Find the length of segment TM. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.88", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center R and circumradius 12. Let M be the circumcircle of polygon ABCDEFGHIJKL. Circle O is the reflection of circle M across point A. Line P passes through points H and K. Point X is the reflection of point F across line P. Let Y be the inversion of point X in circle O. What is the length of segment YR? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.98", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHJKLM be a regular dodecagon with center I and circumradius 3. Let Z be the circumcircle of polygon ABCDEFGHJKLM. Let R be the reflection of circle Z through point H. Line Q passes through points F and K. Circle U is the reflection of circle R across line Q. Point P is the reflection of point G across point H. Line W is the perpendicular bisector of segment KM. Point Y is the reflection of point P across line W. Let S be the inversion of point Y in circle U. Find the distance between points H and S. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.36", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDFGH be a regular 7-gon with center E and circumradius 11. Line Q passes through point F parallel to segment AH. Point I is the reflection of point B across line Q. Line O is the perpendicular bisector of segment AH. Point Z is the reflection of point I across line O. Find the measure of angle HZC, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.9", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center Z and circumradius 3. Let Y be the circumcircle of polygon ABCDEFG. Line R is the perpendicular bisector of segment ZG. Circle L is the reflection of circle Y across line R. Point P is the inversion of point D with respect to circle L. Let Q be the midpoint of segment FZ. Find the length of segment QP. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center J and circumradius 10. Let Q be the midpoint of segment EF. Let Y be the midpoint of segment QB. Let K be the line through points J and C. Let O be the angle bisector of angle JDE. Let I be the reflection of line O through point C. Point S is the intersection of lines I and K. What is the distance between points S and Y? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31.32", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center W and circumradius 2. Let Z be the reflection of point A across point E. Let L be the line through points F and D. Point X is the reflection of point Z across line L. Let Q be the reflection of point X across point A. Find the measure of angle EZQ, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.52", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center Y and circumradius 8. Let Z be the midpoint of segment BC. Line S passes through points Z and B. Let T be the reflection of point Z across point C. Let E be the reflection of line S through point Y. Let J be the circumcircle of polygon ABCD. Point P is the midpoint of segment BT. Let I be the reflection of circle J across line S. Point O is the reflection of point Y across point B. Let M be the reflection of line E through point P. Circle V is the reflection of circle I across line M. Let G be the inversion of point O in circle V. Find the length of segment GC. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.49", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center F and circumradius 1. Let N be the angle bisector of angle AFE. Let Q be the line through point A parallel to line N. Let V be the reflection of line N across line Q. Let Y be the midpoint of segment EF. Point W is the reflection of point Y across point F. Point R is the reflection of point W across line V. What is the distance between points R and Y? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.49", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center I and circumradius 10. Point R is obtained by rotating point I by angle 5pi/12 about point D. Let W be the circumcircle of polygon ABCDEF. Line G passes through points I and F. Point V is the reflection of point R across line G. Point N is the inversion of point V with respect to circle W. What is the distance between points V and N? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.96", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEF be a regular 5-gon with center B and circumradius 5. Point Q is the midpoint of segment AC. Let H be the perpendicular bisector of segment DQ. Line M is the reflection of line H across point E. Point J is the reflection of point F across line M. What is the length of segment JC? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19.59", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center N and circumradius 3. Let K be the reflection of point F across point A. Line L passes through point A perpendicular to segment KC. Point M is the reflection of point G across line L. Find the distance between points M and D. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.74", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEFGHI be a regular octagon with center B and circumradius 7. Let M be the rotation of point H by angle 5pi/6 around point C. Let K be the midpoint of segment CH. Line P is the perpendicular bisector of segment IM. Point L is the reflection of point D across line P. Find the length of segment KL. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.16", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJK is a regular 11-gon with center Q and circumradius 3. Let W be the angle bisector of angle EAK. Let V be the reflection of line W through point C. Line O passes through point B parallel to line V. Let X be the reflection of line V across line O. Point M is the reflection of point Q across line X. Let R be the reflection of line X through point M. Point P is the reflection of point A across line V. Point Z is the reflection of point D across line R. Find the distance between points Z and P. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40.45", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGIJKLM be a regular dodecagon with center H and circumradius 4. Let N be the circumcircle of polygon ABCDEFGIJKLM. Let P be the midpoint of segment ME. Let Q be the angle bisector of angle JEI. Let R be the reflection of line Q through point P. Circle T is the reflection of circle N across point D. Circle Z is the reflection of circle T across line R. Let Y be the inversion of point K in circle Z. What is the length of segment YF? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.74", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFHIJK be a regular 10-gon with center G and circumradius 2. Let U be the midpoint of segment FH. Let X be the angle bisector of angle BDU. Point L is the reflection of point K across line X. Let R be the circumcircle of polygon ABCDEFHIJK. Point O is the inversion of point L with respect to circle R. What is the distance between points O and C? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.11", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center S and circumradius 3. Point J is the intersection of segments EB and DG. Let H be the reflection of point F across point G. Let M be the midpoint of segment JH. What is the distance between points M and H? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.88", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center O and circumradius 8. Let N be the circumcircle of polygon ABCDEFGHI. Let K be the reflection of point D across point G. Point M is the reflection of point B across point K. Point X is the inversion of point M with respect to circle N. What is the distance between points X and I? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.58", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center R and circumradius 11. Line Y passes through points B and L. Let N be the circumcircle of polygon ABCDEFGHIJKL. Line S passes through point H parallel to line Y. Let P be the reflection of circle N across line S. Let U be the inversion of point J in circle P. What is the distance between points C and U? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.29", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center P and circumradius 4. Point N is obtained by rotating point P by angle pi/12 about point A. Point G is the midpoint of segment DN. Point T is the reflection of point E across point G. Point I is the reflection of point C across point B. Point M is the reflection of point I across point P. Find the length of segment MT. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.84", "dataset": "MATH"}
{"messages": [{"content": "Let ABCD be a square with center Z and circumradius 11. Let M be the reflection of point A across point C. Point I is the midpoint of segment DM. Point Q is obtained by rotating point M by angle pi/4 about point I. Let N be the rotation of point M by angle 2pi/3 around point Q. Let L be the line through point C perpendicular to segment MI. Let F be the line through point N parallel to segment MI. Line T passes through point Q parallel to line L. Lines F and T intersect at point R. Find the distance between points Z and R. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42.37", "dataset": "MATH"}
{"messages": [{"content": "ACDEFG is a regular hexagon with center B and circumradius 3. Point V is the reflection of point A across point C. Let O be the reflection of point E across point F. Point X is the midpoint of segment GO. Point N is the midpoint of segment DV. Find the distance between points N and X. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.94", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center V and circumradius 5. Let P be the line through points A and H. Let Y be the line through point B perpendicular to line P. Point I is the reflection of point B across line P. Let R be the line through point H parallel to line Y. Point L is the reflection of point I across line R. Point N is the reflection of point L across line P. Find the distance between points H and N. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.07", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGH be a regular octagon with center M and circumradius 1. Line Q is the angle bisector of angle AFH. Point O is the reflection of point C across line Q. Point Y is the reflection of point E across segment BO. Find the distance between points M and Y. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.36", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center L and circumradius 4. Let Y be the circumcircle of polygon ABCDEF. Let R be the midpoint of segment LF. Line Q is the angle bisector of angle LRD. Let J be the reflection of point L across point B. Circle W is the reflection of circle Y across line Q. Point X is the inversion of point L with respect to circle W. Point T is the reflection of point X across point J. Find the distance between points T and X. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.08", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center L and circumradius 5. Let Z be the circumcircle of polygon ABCDEFGH. Let V be the midpoint of segment DG. Point X is the inversion of point V with respect to circle Z. Line J is the angle bisector of angle LGE. Point O is the reflection of point F across line J. What is the length of segment OX? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.99", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGIJKLM is a regular dodecagon with center H and circumradius 6. Point S is the midpoint of segment MK. Point Y is the reflection of point I across point B. Point P is the reflection of point J across segment SA. Find the length of segment PY. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22.61", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJ be a regular 10-gon with center K and circumradius 4. Let Y be the midpoint of segment FI. Point N is the reflection of point H across point D. Let M be the line through points Y and K. Let U be the midpoint of segment GH. Point Z is the reflection of point N across line M. Find the length of segment UZ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.52", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFHIJKLM be a regular dodecagon with center G and circumradius 11. Line O is the angle bisector of angle FGM. Line S passes through point L perpendicular to line O. Point Z is the reflection of point B across line S. Let T be the midpoint of segment CZ. Find the length of segment TK. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.5", "dataset": "MATH"}
{"messages": [{"content": "ABCEFGHIJKLM is a regular dodecagon with center D and circumradius 10. Let P be the perpendicular bisector of segment BA. Line Z passes through point G parallel to line P. Point S is the reflection of point D across line Z. What is the length of segment LS? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.94", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center M and circumradius 4. Line V passes through point A parallel to segment BH. Point L is the reflection of point H across line V. Point Q is the midpoint of segment AG. What is the distance between points Q and L? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.28", "dataset": "MATH"}
{"messages": [{"content": "ABCDFGHI is a regular octagon with center E and circumradius 6. Let R be the circumcircle of polygon ABCDFGHI. Let M be the reflection of circle R through point I. Let N be the reflection of point H across point A. Point W is the inversion of point N with respect to circle M. Find the length of segment WH. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.06", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGIJKLM be a regular dodecagon with center H and circumradius 4. Line W is the perpendicular bisector of segment MB. Line O passes through point C perpendicular to line W. Let Y be the circumcircle of polygon ABCDEFGIJKLM. Let N be the reflection of circle Y across line O. Circle S is the reflection of circle N across point G. Point P is the midpoint of segment GK. Point T is the inversion of point P with respect to circle S. Find the length of segment TD. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.31", "dataset": "MATH"}
{"messages": [{"content": "HS = 6. Let ABCDEFGI be a regular octagon with center S and circumradius 1. Let K be the reflection of point D across point E. Let W be the circumcircle of polygon ABCDEFGI. Let Z be the inversion of point K in circle W. Point P is the reflection of point F across segment ZA. Find the distance between points E and P. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.15", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center W and circumradius 1. Let M be the midpoint of segment EI. Line Y is the perpendicular bisector of segment IJ. Let V be the line through point I parallel to line Y. Point X is the reflection of point E across line V. What is the measure of angle MIX, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.52", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center W and circumradius 6. Let H be the perpendicular bisector of segment ED. Let T be the reflection of line H through point C. Point P is the reflection of point G across line T. Point J is the midpoint of segment PB. Find the length of segment JD. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.55", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center T and circumradius 5. Let W be the line through point F parallel to segment BT. Line S passes through point G parallel to segment FJ. Lines S and W intersect at point Q. Find the measure of angle IQE, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.26", "dataset": "MATH"}
{"messages": [{"content": "Let UI = 6. Let ABCDE be a regular 5-gon with center I and circumradius 4. Let N be the line through points I and D. Let T be the line through point B perpendicular to line N. Let R be the line through point E perpendicular to line T. Point M is the reflection of point B across line R. Let Z be the circumcircle of polygon ABCDE. Point F lies on line T. Let L be the inversion of point M in circle Z. Find the measure of angle FML, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.31", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEFGHIJKLM be a regular dodecagon with center B and circumradius 2. Point R is the reflection of point K across point C. Let Q be the angle bisector of angle IFM. Point Z is obtained by rotating point B by angle 2pi/3 about point K. Let X be the reflection of line Q through point J. Line Y is the reflection of line X across line Q. Line T is the reflection of line Y across point R. Let S be the reflection of line T across line Y. Point P is the reflection of point Z across line S. What is the length of segment HP? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.24", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDFGHI be a regular octagon with center E and circumradius 2. Let J be the line through points E and A. Let Y be the reflection of line J through point I. Point Z is the midpoint of segment GC. Line K passes through point I perpendicular to segment GC. Line P is the reflection of line K across line Y. Point O is the reflection of point B across line P. What is the distance between points O and Z? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.5", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEFG be a regular hexagon with center B and circumradius 11. Point P is obtained by rotating point E by angle 5pi/12 about point G. Let I be the circumcircle of polygon ACDEFG. Let S be the inversion of point P in circle I. Circle O is the reflection of circle I across point E. Point V is the midpoint of segment CG. Let N be the inversion of point V in circle O. Find the distance between points N and S. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.65", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center S and circumradius 3. Let U be the rotation of point S by angle pi/6 around point H. Point Y is the midpoint of segment BU. Let V be the midpoint of segment US. Let Z be the midpoint of segment AY. Find the length of segment ZV. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.39", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center X and circumradius 5. Point L is the reflection of point F across point D. Let W be the midpoint of segment LC. Line T is the angle bisector of angle GFL. Point Y is the reflection of point X across line T. Let Q be the midpoint of segment YW. Find the distance between points H and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.68", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center Q and circumradius 7. Let U be the midpoint of segment QG. Let T be the midpoint of segment BQ. Let P be the reflection of point J across point D. Point N is the reflection of point P across segment UT. What is the distance between points N and Q? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "20.09", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGIJKLM be a regular dodecagon with center H and circumradius 7. Let X be the reflection of point I across point J. Line P is the perpendicular bisector of segment FD. Line Q is the reflection of line P across point J. Let Y be the reflection of line Q through point X. Point U is the reflection of point L across line Y. What is the distance between points L and U? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.75", "dataset": "MATH"}
{"messages": [{"content": "DA = 7. Let BCEFGHIJKLMN be a regular dodecagon with center A and circumradius 8. Point U is the reflection of point N across segment EJ. Point V is the reflection of point U across segment JI. What is the length of segment VN? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDFG is a regular hexagon with center E and circumradius 12. Let L be the midpoint of segment CD. Let O be the reflection of point D across point A. Let V be the perpendicular bisector of segment LA. Line M passes through point O perpendicular to line V. Point I is the reflection of point G across line M. Find the distance between points B and I. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.25", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center W and circumradius 10. Let T be the perpendicular bisector of segment EK. Line P passes through point W perpendicular to line T. Line O is the reflection of line P across point F. Point Z is the reflection of point W across line O. What is the length of segment ZJ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.32", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFG be a regular 7-gon with center T and circumradius 12. Let X be the midpoint of segment CF. Let W be the rotation of point F by angle 3pi/4 around point X. Line I is the angle bisector of angle TGA. Let U be the reflection of point C across point W. Point J is the reflection of point U across line I. Line V passes through point B parallel to segment DA. Point L is the reflection of point F across line V. Find the length of segment LJ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31.16", "dataset": "MATH"}
{"messages": [{"content": "SZ = 5. ABCDEFGH is a regular octagon with center S and circumradius 10. Point V is the reflection of point Z across point A. Point T is the reflection of point S across point V. Let K be the perpendicular bisector of segment GT. Point I is the reflection of point A across line K. What is the distance between points T and I? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "14.14", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center W and circumradius 4. Let I be the perpendicular bisector of segment CW. Point Q is the reflection of point G across line I. Point U is the reflection of point W across point E. Point Z is the reflection of point U across line I. Let V be the reflection of point B across point Z. What is the distance between points V and Q? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19.04", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center J and circumradius 1. Line U is the angle bisector of angle JBE. Line M passes through points A and B. Line X is the reflection of line U across line M. Point Z is the reflection of point D across line X. What is the distance between points D and Z? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.39", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center G and circumradius 4. Point X is obtained by rotating point E by angle pi/3 about point G. Line V is the angle bisector of angle GXA. Let O be the circumcircle of polygon ABCDE. Circle N is the reflection of circle O across line V. Let M be the inversion of point G in circle N. Find the distance between points M and C. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.24", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHJKL is a regular 11-gon with center I and circumradius 8. Line R is the perpendicular bisector of segment AL. Let N be the rotation of point D by angle pi/2 around point G. Line X passes through points H and N. Let U be the reflection of line R across line X. Point T is the reflection of point K across segment FB. Line P passes through point T perpendicular to line U. Point Q is the reflection of point J across line P. Find the distance between points Q and H. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34.99", "dataset": "MATH"}
{"messages": [{"content": "Let ACDE be a square with center B and circumradius 9. Let Z be the reflection of point C across point D. Point T is the reflection of point C across point E. Point Q is the midpoint of segment ZE. Find the measure of angle TZQ, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.79", "dataset": "MATH"}
{"messages": [{"content": "ABCDEGHIJKLM is a regular dodecagon with center F and circumradius 11. Line Z is the angle bisector of angle AGI. Point O is the reflection of point I across line Z. Point T is the reflection of point O across segment JM. What is the distance between points C and T? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26.31", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJK is a regular 11-gon with center O and circumradius 1. Point S is the midpoint of segment DI. Point M is the reflection of point H across point S. Point T is the midpoint of segment KH. Find the distance between points T and M. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.43", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center V and circumradius 3. Let Z be the circumcircle of polygon ABCDEF. Line H is the angle bisector of angle AFD. Circle G is the reflection of circle Z across line H. Point M is the inversion of point E with respect to circle G. What is the length of segment MD? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.19", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center Q and circumradius 9. Let J be the midpoint of segment CA. Let T be the reflection of point D across point A. Point R is the midpoint of segment JF. What is the distance between points T and R? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.71", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center M and circumradius 2. Let R be the angle bisector of angle IAM. Line Z passes through point I perpendicular to line R. Point W is the reflection of point G across line Z. Let P be the midpoint of segment IW. What is the distance between points C and P? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.17", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center T and circumradius 3. Let J be the rotation of point B by angle pi/12 around point D. Line Q passes through points E and B. Let W be the line through point F parallel to line Q. Let N be the reflection of line W through point B. Line Y passes through points J and T. Point R is the intersection of lines Y and N. Find the length of segment RF. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.81", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center S and circumradius 1. Let U be the circumcircle of polygon ABCDEF. Let I be the reflection of circle U through point E. Let Y be the reflection of circle I through point A. Point K is the inversion of point S with respect to circle Y. Line J is the angle bisector of angle KSC. Point X is the reflection of point K across line J. What is the length of segment AX? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.78", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center I and circumradius 11. Let Q be the circumcircle of polygon ABCDEF. Line M is the perpendicular bisector of segment FC. Circle J is the reflection of circle Q across point A. Circle N is the reflection of circle J across line M. Let W be the inversion of point I in circle N. Find the measure of angle CIW, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.05", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center Y and circumradius 5. Let V be the reflection of point Y across point C. Let S be the perpendicular bisector of segment CA. Line K is the reflection of line S across point V. Point O is the reflection of point A across line K. Find the distance between points D and O. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40.25", "dataset": "MATH"}
{"messages": [{"content": "Let JU = 11. Let ABCDEF be a regular hexagon with center U and circumradius 5. Point P is the midpoint of segment DC. Let G be the line through points U and D. Let O be the line through point P parallel to line G. Let I be the line through points D and B. Lines O and I intersect at point S. What is the distance between points E and S? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.61", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center U and circumradius 4. Line S is the perpendicular bisector of segment AF. Let O be the reflection of point A across point B. Point X is the reflection of point O across line S. Let I be the perpendicular bisector of segment XF. Point Q is the reflection of point A across line I. Find the distance between points E and Q. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.93", "dataset": "MATH"}
{"messages": [{"content": "Let ABCEF be a regular 5-gon with center D and circumradius 11. Let H be the midpoint of segment BE. Point G is the midpoint of segment CE. Let R be the midpoint of segment CH. What is the length of segment RG? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.23", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center G and circumradius 7. Line S is the perpendicular bisector of segment DA. Let Q be the reflection of line S through point F. Let Z be the reflection of point G across point D. Point X is the reflection of point Z across line Q. What is the distance between points X and Z? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center Q and circumradius 1. Point U is the reflection of point G across point A. Line V passes through points U and A. Let R be the reflection of point K across point G. Line X passes through point R parallel to line V. Line W passes through points K and A. Line O is the perpendicular bisector of segment RD. Let M be the reflection of line O across line W. Lines X and M intersect at point Z. Find the distance between points Z and B. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.7", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJK is a regular 11-gon with center S and circumradius 10. Point P is obtained by rotating point S by angle pi/2 about point G. Point T is the reflection of point A across point S. Point L is the reflection of point T across point I. Point R is the reflection of point P across segment DI. Line Y passes through point R perpendicular to segment DI. Point U is the reflection of point L across line Y. Point Z is the reflection of point U across segment DI. What is the distance between points Z and R? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center X and circumradius 12. Point M is the reflection of point C across point H. Let T be the reflection of point E across point A. Point R is the reflection of point T across segment EM. Find the length of segment FR. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "39.67", "dataset": "MATH"}
{"messages": [{"content": "CR = 7. ABDEF is a regular 5-gon with center R and circumradius 2. Point U is the midpoint of segment FD. Point P is the reflection of point B across segment ED. Let Z be the midpoint of segment UF. What is the distance between points P and Z? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.95", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center N and circumradius 12. Let Q be the perpendicular bisector of segment EB. Let T be the reflection of point N across point D. Line R is the perpendicular bisector of segment FT. Point S is on line Q. Line L passes through point S parallel to line R. Line K passes through point C parallel to segment AN. Lines K and L intersect at point P. What is the measure of angle PBN, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.05", "dataset": "MATH"}
{"messages": [{"content": "ABCEF is a regular 5-gon with center D and circumradius 4. Line V is the angle bisector of angle ADF. Point T is the reflection of point C across point F. Let J be the line through point T perpendicular to line V. Let R be the line through point E parallel to line J. Let K be the perpendicular bisector of segment BT. Point Z is the reflection of point C across line R. Line L is the reflection of line V across line K. Point U is the reflection of point B across line L. What is the distance between points U and Z? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.12", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGIJKLM be a regular dodecagon with center H and circumradius 12. Line S is the angle bisector of angle LIA. Let U be the perpendicular bisector of segment KE. Let W be the reflection of line U across line S. Point P is the reflection of point J across line S. Point T is the reflection of point P across line W. Let O be the midpoint of segment LB. What is the distance between points O and T? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "24.4", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center F and circumradius 1. Let Z be the line through points D and A. Line V passes through point E parallel to line Z. Point Y is the reflection of point B across point C. Point N is the reflection of point F across line V. Let S be the reflection of point Y across point N. Find the distance between points Y and S. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.63", "dataset": "MATH"}
{"messages": [{"content": "Let BCDEFGHI be a regular octagon with center A and circumradius 10. Point O is the midpoint of segment AI. Point K is the midpoint of segment BO. Point L is the reflection of point A across segment IB. What is the distance between points K and L? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.59", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center M and circumradius 5. Point V is obtained by rotating point E by angle 2pi/3 about point A. Line U is the angle bisector of angle EFD. Point J is the reflection of point M across line U. Point H is the reflection of point J across point C. Line N is the perpendicular bisector of segment HD. Let T be the circumcircle of polygon ABCDEF. Point S is the reflection of point V across line N. Point X is the inversion of point S with respect to circle T. What is the distance between points M and X? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.16", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGH is a regular octagon with center I and circumradius 11. Let L be the reflection of point C across point A. Point Z is the midpoint of segment AG. Let P be the line through point G perpendicular to segment LD. Point O is the reflection of point Z across line P. Find the distance between points D and O. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.44", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center Q and circumradius 9. Let K be the midpoint of segment BC. Point Y is the midpoint of segment KD. Let P be the midpoint of segment CQ. What is the length of segment YP? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.15", "dataset": "MATH"}
{"messages": [{"content": "Let ABCEFG be a regular hexagon with center D and circumradius 8. Point W is the midpoint of segment GF. Let J be the perpendicular bisector of segment FC. Point X is the reflection of point W across line J. Point O is the midpoint of segment AX. What is the distance between points C and O? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.72", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFGHIJKLM be a regular dodecagon with center C and circumradius 4. Point Z is the reflection of point E across segment IL. Let P be the midpoint of segment HM. What is the length of segment ZP? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.55", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center K and circumradius 1. Let X be the midpoint of segment CA. Let J be the line through point D perpendicular to segment EC. Let Y be the line through point X perpendicular to line J. Point N is the reflection of point A across line Y. Find the distance between points N and E. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.87", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center P and circumradius 9. Point W is obtained by rotating point P by angle pi/3 about point F. Point L is the reflection of point G across segment DW. Point R is the midpoint of segment DW. Point T is the midpoint of segment LR. Find the distance between points T and D. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.6", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center U and circumradius 10. Let P be the line through points I and U. Line N passes through point J parallel to line P. Point R is the intersection of segment AU and line N. Find the measure of angle RIC, in radians. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.31", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center I and circumradius 3. Point L is the reflection of point E across point C. Let Z be the line through points A and F. Let V be the circumcircle of polygon ABCDEF. Let N be the reflection of circle V through point A. Circle Y is the reflection of circle N across line Z. Point M is the inversion of point L with respect to circle Y. What is the distance between points I and M? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.97", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center J and circumradius 2. Line G passes through points J and C. Line Q passes through points A and E. Point K is the intersection of lines G and Q. Let R be the midpoint of segment KB. What is the length of segment RK? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.47", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJK be a regular 11-gon with center O and circumradius 6. Let Y be the circumcircle of polygon ABCDEFGHIJK. Point V is the reflection of point O across point A. Line S is the perpendicular bisector of segment OV. Circle U is the reflection of circle Y across line S. Let L be the inversion of point C in circle U. Find the distance between points L and I. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.55", "dataset": "MATH"}
{"messages": [{"content": "Let ABCE be a square with center D and circumradius 9. Point U is the reflection of point C across point E. Let Q be the midpoint of segment UD. Point J is the reflection of point A across segment UQ. Find the length of segment UJ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABCEFG be a regular hexagon with center D and circumradius 7. Let J be the reflection of point C across point G. Let Y be the midpoint of segment FE. Let H be the circumcircle of polygon ABCEFG. Let U be the reflection of circle H through point Y. Let V be the inversion of point B in circle U. What is the length of segment JV? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.52", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center S and circumradius 1. Let Y be the line through points E and A. Point Z is the midpoint of segment SD. Line P is the perpendicular bisector of segment BD. Let G be the reflection of line P through point Z. Point I is the intersection of lines Y and G. What is the distance between points I and Z? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.07", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center S and circumradius 2. Let V be the rotation of point L by angle pi/12 around point J. Let P be the line through points V and C. Let U be the line through point S parallel to line P. Let R be the reflection of line U through point A. Point W is the reflection of point C across line R. Find the length of segment WD. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.25", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHI is a regular 9-gon with center L and circumradius 2. Line X passes through points B and A. Line R is the reflection of line X across point E. Point Z is the reflection of point H across line R. Point J is the reflection of point F across point Z. What is the distance between points G and J? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.77", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center Z and circumradius 11. Line L is the angle bisector of angle FAC. Line T is the reflection of line L across point F. Point P is the reflection of point B across line T. Point W is the midpoint of segment PE. Point J is the midpoint of segment AW. Find the length of segment AJ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.82", "dataset": "MATH"}
{"messages": [{"content": "BCDEF is a regular 5-gon with center A and circumradius 4. Point V is the reflection of point B across point E. Let I be the midpoint of segment BF. Line M is the perpendicular bisector of segment IA. Point O is the reflection of point B across line M. Point Q is the reflection of point V across point O. Point K is the midpoint of segment BQ. Find the length of segment OK. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.61", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFG be a regular hexagon with center C and circumradius 10. Let W be the rotation of point B by angle 5pi/12 around point D. Point J is the reflection of point A across point F. Line H passes through points E and W. Point N is the reflection of point J across line H. Point Z is the midpoint of segment WA. Find the length of segment ZN. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "34.22", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center L and circumradius 3. Let W be the midpoint of segment AB. Point H is the reflection of point A across segment WE. Find the distance between points H and E. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.2", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHI be a regular 9-gon with center Z and circumradius 10. Line X passes through point B perpendicular to segment AD. Point J is the reflection of point F across line X. Point O is the midpoint of segment AD. Find the length of segment JO. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18.34", "dataset": "MATH"}
{"messages": [{"content": "ABDEFG is a regular hexagon with center C and circumradius 4. Let I be the midpoint of segment FD. Let X be the midpoint of segment BI. Line T passes through points A and X. Point V is the intersection of line T and segment FD. Let H be the circumcircle of polygon ABDEFG. Let Q be the inversion of point V in circle H. Find the length of segment DQ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.06", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEFGHIJK be a regular 10-gon with center B and circumradius 3. Point X is the reflection of point K across segment FG. Let O be the midpoint of segment FG. Let Z be the angle bisector of angle CXD. Let S be the circumcircle of polygon ACDEFGHIJK. Let W be the reflection of circle S across line Z. Let Y be the inversion of point D in circle W. Find the distance between points O and Y. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.07", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center Z and circumradius 2. Line O passes through points E and L. Let R be the line through point C parallel to segment GK. Point P is the intersection of lines R and O. Point W is the reflection of point P across segment GK. What is the length of segment WE? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.24", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center K and circumradius 6. Let T be the line through point C perpendicular to segment BC. Let I be the reflection of point C across point D. Point W is the reflection of point K across line T. What is the distance between points W and I? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.6", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFG is a regular 7-gon with center V and circumradius 3. Let N be the midpoint of segment DB. Let W be the line through points A and D. Point K is the reflection of point N across line W. Let M be the reflection of point V across point F. Find the distance between points M and K. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.71", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center P and circumradius 12. Let V be the midpoint of segment ED. Let J be the circumcircle of polygon ABCDEF. Point N is the inversion of point V with respect to circle J. Point G is the reflection of point C across segment NB. What is the length of segment NG? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18.33", "dataset": "MATH"}
{"messages": [{"content": "ACDEFGHI is a regular octagon with center B and circumradius 6. Point Z is the reflection of point F across point H. Let T be the line through points H and E. Let Q be the line through point D parallel to line T. Let N be the reflection of line Q across line T. Let J be the reflection of point E across point I. Point R is the reflection of point J across line N. What is the distance between points R and Z? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21.86", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center Y and circumradius 1. Point Z is the midpoint of segment LK. Let Q be the circumcircle of polygon ABCDEFGHIJKL. Point R is the reflection of point B across point Z. Let U be the inversion of point R in circle Q. What is the distance between points L and U? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.73", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJKL is a regular dodecagon with center U and circumradius 1. Point V is the reflection of point K across segment JC. Let P be the reflection of point V across point J. Point N is the reflection of point P across segment JC. What is the length of segment NC? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.24", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center U and circumradius 6. Let F be the perpendicular bisector of segment DU. Let R be the reflection of line F through point D. Let G be the reflection of line F across line R. Point H is the reflection of point E across line G. Point Q is the midpoint of segment DU. Find the distance between points Q and H. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25.79", "dataset": "MATH"}
{"messages": [{"content": "Let ABCEFGHIJKLM be a regular dodecagon with center D and circumradius 10. Line O is the perpendicular bisector of segment EJ. Let V be the reflection of line O through point L. Let Y be the reflection of line O across line V. Point P is the reflection of point D across line Y. Let S be the line through points H and P. Lines S and Y intersect at point U. Find the distance between points G and U. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31.21", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center S and circumradius 7. Let K be the midpoint of segment DS. Let O be the midpoint of segment DC. Find the distance between points K and O. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.5", "dataset": "MATH"}
{"messages": [{"content": "ACDEFGHI is a regular octagon with center B and circumradius 2. Let O be the reflection of point C across point G. Let P be the reflection of point G across point E. Point J is the reflection of point P across segment OD. Find the distance between points J and B. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.01", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center L and circumradius 2. Let K be the midpoint of segment BA. Line O is the angle bisector of angle FBK. Line X passes through point A perpendicular to line O. Point J is the reflection of point F across line X. Find the distance between points E and J. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.01", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEF be a regular 5-gon with center C and circumradius 2. Point X is obtained by rotating point B by angle pi/3 about point A. Let G be the circumcircle of polygon ABDEF. Let T be the inversion of point X in circle G. Point Y is the reflection of point B across point T. Let K be the midpoint of segment AC. Find the length of segment YK. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21.57", "dataset": "MATH"}
{"messages": [{"content": "ABCDEFGHIJ is a regular 10-gon with center X and circumradius 11. Let S be the midpoint of segment JX. Let W be the rotation of point E by angle 2pi/3 around point S. Point L is obtained by rotating point I by angle 5pi/6 about point C. Let O be the angle bisector of angle WXB. Let Y be the reflection of line O through point L. Let M be the reflection of line O across line Y. Point N is the reflection of point W across line M. Find the distance between points N and W. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "287.83", "dataset": "MATH"}
{"messages": [{"content": "Let ACDEFGHIJKLM be a regular dodecagon with center B and circumradius 4. Line S passes through points F and I. Let X be the line through points M and F. Line O is the reflection of line S across line X. Line R is the reflection of line O across point C. Point U is the intersection of line R and segment LA. What is the length of segment CU? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.93", "dataset": "MATH"}
{"messages": [{"content": "Let ACDE be a square with center B and circumradius 1. Let O be the midpoint of segment BC. Point M is the reflection of point D across point E. Point I is the reflection of point M across point O. Point T is the reflection of point E across segment MA. What is the distance between points T and I? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.0", "dataset": "MATH"}
{"messages": [{"content": "Let ABDEFGHI be a regular octagon with center C and circumradius 12. Point S is the midpoint of segment HC. Let M be the reflection of point F across point S. Point U is the reflection of point M across segment HG. Find the length of segment IU. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.0", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center K and circumradius 8. Let J be the circumcircle of polygon ABCDEF. Line N passes through point C parallel to segment DA. Let W be the line through point D perpendicular to line N. Circle L is the reflection of circle J across line W. Let X be the inversion of point E in circle L. Find the distance between points F and X. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.65", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJK be a regular 11-gon with center N and circumradius 12. Let L be the angle bisector of angle JIN. Let M be the angle bisector of angle GDF. Line V is the reflection of line M across line L. Let Q be the line through points C and I. Lines V and Q intersect at point R. Find the distance between points N and R. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "40.69", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center M and circumradius 11. Let Y be the line through points B and M. Point N lies on line Y at distance 2 from point B. Let Q be the midpoint of segment MN. Line W passes through points Q and A. Let L be the reflection of point M across point D. Point S lies on line W. Let X be the midpoint of segment BS. Point R is the reflection of point L across segment EX. Find the length of segment ER. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21.34", "dataset": "MATH"}
{"messages": [{"content": "ABCDEF is a regular hexagon with center O and circumradius 5. Line Z is the perpendicular bisector of segment EB. Let Q be the reflection of line Z through point E. Let U be the line through point B parallel to line Q. Point N is the reflection of point C across line U. What is the distance between points N and F? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.23", "dataset": "MATH"}
{"messages": [{"content": "ABCDE is a regular 5-gon with center T and circumradius 10. Let V be the reflection of point D across point B. Let Q be the circumcircle of polygon ABCDE. Let S be the reflection of circle Q through point A. Point I is the inversion of point E with respect to circle S. What is the length of segment IV? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27.45", "dataset": "MATH"}
{"messages": [{"content": "ABDEF is a regular 5-gon with center C and circumradius 11. Point T is the midpoint of segment DC. Let S be the angle bisector of angle DEA. Point P is the reflection of point T across line S. Let X be the reflection of point E across point B. What is the length of segment PX? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31.46", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEFGHIJKL be a regular dodecagon with center P and circumradius 10. Let Q be the circumcircle of polygon ABCDEFGHIJKL. Point W is the midpoint of segment CF. Circle V is the reflection of circle Q across point B. Point O is the inversion of point L with respect to circle V. What is the length of segment OW? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "16.37", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDE be a regular 5-gon with center O and circumradius 1. Let G be the midpoint of segment AB. Let Z be the midpoint of segment AD. What is the measure of angle AGZ, in radians? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.26", "dataset": "MATH"}
{"messages": [{"content": "Let ABCDEF be a regular hexagon with center K and circumradius 10. Point V is the reflection of point F across point D. Point L is the reflection of point V across segment KC. Let Q be the perpendicular bisector of segment KC. Point X is the reflection of point L across line Q. Let U be the midpoint of segment KC. Find the distance between points X and U. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22.91", "dataset": "MATH"}
{"messages": [{"content": "ABCD is a square with center T and circumradius 8. Let U be the rotation of point A by angle pi/12 around point C. Let K be the midpoint of segment DU. Line E passes through points C and K. Line R is the reflection of line E across point A. Point S is the reflection of point K across line R. Find the length of segment SD. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.2", "dataset": "MATH"}
{"messages": [{"content": "ABCEFGHIJ is a regular 9-gon with center D and circumradius 6. Line K is the angle bisector of angle EAI. Line T passes through point H perpendicular to line K. Point M is the reflection of point H across point D. Point O is the reflection of point J across point M. Point P is the reflection of point H across point A. Point R is the reflection of point P across line T. Find the distance between points R and O. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "29.49", "dataset": "MATH"}