{"messages": [{"content": "OQ = 12. Point E is obtained by rotating point Q by angle 5pi/6 about point O. Point I is the midpoint of segment QO. Let X be the line through point E perpendicular to segment EO. Point J is the reflection of point Q across line X. What is the circumradius of triangle IEJ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "47.15", "dataset": "MATH"}
{"messages": [{"content": "GC = 6. Let Q be the midpoint of segment GC. Let B be the rotation of point Q by angle pi/4 around point G. Line X is the perpendicular bisector of segment BG. Let D be the midpoint of segment BQ. Line R passes through point G parallel to line X. Point E is the midpoint of segment DB. Point W is the reflection of point Q across line R. Find 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": "5.15", "dataset": "MATH"}
{"messages": [{"content": "ID = 11. Point R is the reflection of point D across point I. Let F be the rotation of point D by angle pi/4 around point I. Point Z is the reflection of point I across point F. Let J be the centroid of triangle RFZ. Point H is the reflection of point R across segment FZ. Find the length of segment JH. Express your 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": "IS = 3. Point F is obtained by rotating point I by angle 5pi/12 about point S. B is constructed so that FB = 3. Point V is the reflection of point S across segment IF. Let Q be the midpoint of segment IS. Let Y be the line through points B and V. Point K is the reflection of point Q across line Y. 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": "3.69", "dataset": "MATH"}
{"messages": [{"content": "IU = 6. Let O be the rotation of point U by angle pi/6 around point I. Point H is the midpoint of segment OU. Point M is the centroid of triangle OIH. Point S is the incenter of triangle OIH. What is the distance between points M 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": "1.27", "dataset": "MATH"}
{"messages": [{"content": "Let TC = 3. Point E is obtained by rotating point T by angle pi/2 about point C. Point J is the reflection of point E across point C. Let H be the circumcenter of triangle TJE. Let L be the incenter of triangle TJE. Point A is the reflection of point L across point H. Find the distance between points T 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": "HE = 1. Let P be the rotation of point H by angle 5pi/12 around point E. Point I is the reflection of point E across point P. Point B is the midpoint of segment PH. Point X is the reflection of point I across point B. What is the inradius of triangle PBX? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.12", "dataset": "MATH"}
{"messages": [{"content": "AR = 5. Point C is obtained by rotating point A by angle 5pi/12 about point R. Let H be the perpendicular bisector of segment CR. Point G is the midpoint of segment AC. Point J is the reflection of point A across line H. Let V be the midpoint of segment JG. Point Y is the reflection of point V across segment AC. Find the circumradius of triangle RYC. Express your 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": "AL = 3. Let R be the rotation of point L by angle pi/4 around point A. Let Q be the circumcenter of triangle ALR. Line Z passes through point Q parallel to segment AL. Segment LR and line Z intersect at point Y. Let F be the midpoint of segment LR. Find 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": "0.38", "dataset": "MATH"}
{"messages": [{"content": "BQ = 7. Point H is the midpoint of segment BQ. Point V is obtained by rotating point B by angle pi/6 about point Q. Line E passes through point H perpendicular to segment HB. Point D is the reflection of point V across segment HB. Point X is the reflection of point D across line E. Find the inradius of triangle BXV. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.27", "dataset": "MATH"}
{"messages": [{"content": "Let DR = 6. Let J be the rotation of point R by angle pi/3 around point D. Let N be the incenter of triangle DRJ. Let Z be the midpoint of segment NR. Let M be the midpoint of segment RJ. What is the measure of angle ZMN, 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": "BR = 8. Let X be the reflection of point R across point B. Let Y be the rotation of point B by angle 7pi/12 around point X. Let F be the reflection of point B across point X. Circle A is the circumcircle of triangle RBY. Point D is the reflection of point Y across segment RB. Let V be the inversion of point F in circle A. Point P is the reflection of point V across point X. Point H is the circumcenter of triangle RBY. Find the measure of angle DPH, 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": "2.05", "dataset": "MATH"}
{"messages": [{"content": "Let UB = 10. Point D is obtained by rotating point U by angle 11pi/12 about point B. Let K be the angle bisector of angle BUD. Line W passes through point U perpendicular to line K. Point E is the reflection of point B across line W. Find the inradius of triangle EDB. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.65", "dataset": "MATH"}
{"messages": [{"content": "UH = 9. Point J is obtained by rotating point H by angle 3pi/4 about point U. Let F be the angle bisector of angle UHJ. Point M is the reflection of point J across line F. Point N is the midpoint of segment MH. Let P be the reflection of point H across point U. Point K is the centroid of triangle JUN. 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": "7.42", "dataset": "MATH"}
{"messages": [{"content": "CW = 12. Point O is obtained by rotating point W by angle 5pi/6 about point C. Point T is the midpoint of segment CO. Point K is the midpoint of segment CW. Point X is the reflection of point C across point W. Find the circumradius of triangle TXK. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "56.7", "dataset": "MATH"}
{"messages": [{"content": "OD = 1. Let S be the midpoint of segment OD. Point K is obtained by rotating point D by angle pi/6 about point S. Let I be the midpoint of segment KO. Line F passes through points O and S. Let V be the reflection of point D across point I. Point B is the reflection of point V across line F. Find the inradius of triangle OBV. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.03", "dataset": "MATH"}
{"messages": [{"content": "Let QH = 4. Let O be the rotation of point H by angle 5pi/6 around point Q. Let J be the midpoint of segment OQ. Line V is the angle bisector of angle OJH. Point L is the midpoint of segment QH. Point E is the reflection of point L across line V. Find the measure of angle LEJ, 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": "1.28", "dataset": "MATH"}
{"messages": [{"content": "Let SF = 7. Line P passes through point S perpendicular to segment SF. Let X be the rotation of point S by angle 5pi/6 around point F. Let D be the midpoint of segment XS. Point O is the reflection of point D across line P. Let L be the centroid of triangle SDO. What is the distance between points L 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": "6.56", "dataset": "MATH"}
{"messages": [{"content": "IV = 11. Point M is obtained by rotating point I by angle 3pi/4 about point V. Let S be the reflection of point M across point V. Point F is the midpoint of segment MV. Line Y is the angle bisector of angle SIV. Let H be the rotation of point F by angle pi around point I. Line W is the perpendicular bisector of segment MV. Let Q be the reflection of line Y through point H. Line U passes through point V parallel to line W. Lines Q and U intersect at point K. Find the measure of angle HKI, 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": "1012.45", "dataset": "MATH"}
{"messages": [{"content": "Let RO = 5. Let Z be the perpendicular bisector of segment RO. Point T lies on line Z at distance 8 from point R. Point E is obtained by rotating point R by angle 7pi/12 about point T. Let V be the orthocenter of triangle OER. Point U is the midpoint of segment ER. Let F be the midpoint of segment VO. Find 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": "8.38", "dataset": "MATH"}
{"messages": [{"content": "HE = 8. Point V is obtained by rotating point E by angle pi/3 about point H. Point I is the midpoint of segment EV. Point P is the centroid of triangle VHE. Let B be the midpoint of segment EP. Find the length of segment BI. Express your 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": "PA = 2. Point S is obtained by rotating point P by angle pi/6 about point A. Line L is the perpendicular bisector of segment SA. Let C be the midpoint of segment AS. Point W is the reflection of point P across line L. Let I be the reflection of point W across point S. What is the circumradius of triangle CSI? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.91", "dataset": "MATH"}
{"messages": [{"content": "Let ID = 6. Let N be the rotation of point I by angle 11pi/12 around point D. Line K is the perpendicular bisector of segment ID. Point A is obtained by rotating point N by angle pi about point I. Let U be the line through point A parallel to line K. Let Q be the rotation of point N by angle 11pi/12 around point A. Line J is the reflection of line K across point I. Point Y is the intersection of segment QD and line J. Point V is the reflection of point Y across line U. Find the measure of angle AVI, 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": "1.77", "dataset": "MATH"}
{"messages": [{"content": "Let AB = 11. Let D be the rotation of point B by angle pi/12 around point A. Point S is the reflection of point A across point D. Let F be the orthocenter of triangle BAS. Circle J is the circumcircle of triangle BAS. Point Q is the inversion of point F with respect to circle J. 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": "54.01", "dataset": "MATH"}
{"messages": [{"content": "ZK = 10. Point H is obtained by rotating point Z by angle 5pi/12 about point K. Let W be the midpoint of segment ZH. Let F be the midpoint of segment ZK. Point C is the reflection of point W across point Z. Line G is the perpendicular bisector of segment KH. Point Q is the reflection of point W across line G. Point X is the midpoint of segment FC. 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": "6.03", "dataset": "MATH"}
{"messages": [{"content": "XD = 7. Let Q be the midpoint of segment DX. Point O is obtained by rotating point Q by angle 2pi/3 about point X. Point Z is the midpoint of segment DQ. Point A is the reflection of point Q across point O. Let G be the midpoint of segment QZ. Point Y is the circumcenter of triangle AZQ. 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": "4.3", "dataset": "MATH"}
{"messages": [{"content": "GF = 2. Let R be the rotation of point G by angle 11pi/12 around point F. Circle E is the incircle of triangle GRF. Let U be the midpoint of segment RF. Point P is the inversion of point U with respect to circle E. 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": "1.97", "dataset": "MATH"}
{"messages": [{"content": "HS = 8. Let Q be the rotation of point H by angle pi/6 around point S. Point B is the midpoint of segment HS. Point X is the midpoint of segment QS. Point P is the midpoint of segment XQ. Find the measure of angle BPH, 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": "6.0", "dataset": "MATH"}
{"messages": [{"content": "KX = 7. Let N be the reflection of point X across point K. Point L is obtained by rotating point K by angle 3pi/4 about point N. Let H be the midpoint of segment KL. Let R be the angle bisector of angle HLX. Point V is the reflection of point K across line R. What is the inradius of triangle HVN? Express your 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": "YD = 11. Let V be the rotation of point Y by angle pi/12 around point D. Point N is the midpoint of segment VD. Let P be the orthocenter of triangle VNY. Let H be the centroid of triangle VNY. What is the measure of angle NHP, 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": "1.24", "dataset": "MATH"}
{"messages": [{"content": "Let UI = 3. Point V is obtained by rotating point U by angle 7pi/12 about point I. Point A is obtained by rotating point I by angle pi/2 about point U. Let X be the circumcircle of triangle IAV. Let F be the incenter of triangle IAV. Let Q be the inversion of point F in circle X. What is 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": "4.44", "dataset": "MATH"}
{"messages": [{"content": "Let AE = 6. Point X is obtained by rotating point E by angle pi/2 about point A. Let V be the rotation of point X by angle 3pi/4 around point A. Point W is the reflection of point V across point E. Point C is the reflection of point V across point A. Let R be the midpoint of segment XV. Find the circumradius of triangle WCR. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19.97", "dataset": "MATH"}
{"messages": [{"content": "Let WD = 3. Line G passes through points D and W. Point T is obtained by rotating point D by angle pi/4 about point W. Let N be the rotation of point D by angle 2pi/3 around point T. Let M be the perpendicular bisector of segment TN. Let K be the reflection of line M across line G. Let B be the reflection of line K across line M. Point S is the reflection of point D across line B. Find the circumradius of triangle NST. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.45", "dataset": "MATH"}
{"messages": [{"content": "Let ZQ = 3. Point E is obtained by rotating point Q by angle 5pi/12 about point Z. Point I is obtained by rotating point Q by angle 3pi/4 about point E. Let P be the reflection of point E across point Q. Let G be the centroid of triangle EZI. What is the circumradius of triangle PIG? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.02", "dataset": "MATH"}
{"messages": [{"content": "UW = 12. Point D is the midpoint of segment WU. Point Y is the reflection of point D across point U. Point S is obtained by rotating point D by angle pi/6 about point Y. Let G be the reflection of point W across point S. Point T is obtained by rotating point U by angle pi/4 about point W. Let X be the circumcenter of triangle GTY. Let V be the incenter of triangle DGX. Let M be the circumcenter of triangle DGX. What is the distance between points M 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": "1.05", "dataset": "MATH"}
{"messages": [{"content": "Let SL = 4. Let Y be the rotation of point S by angle 7pi/12 around point L. Point W is obtained by rotating point Y by angle 11pi/12 about point S. Let Z be the midpoint of segment YS. Let X be the reflection of point W across point Z. Point P is the reflection of point W across point Y. 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": "6.35", "dataset": "MATH"}
{"messages": [{"content": "Let ZQ = 6. Let V be the rotation of point Z by angle 7pi/12 around point Q. Point Y is the centroid of triangle QZV. Point C is the midpoint of segment ZY. Point F is the reflection of point Y across segment QZ. Let A be the midpoint of segment VF. Find the distance between points A 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": "3.9", "dataset": "MATH"}
{"messages": [{"content": "Let EA = 5. Let Z be the midpoint of segment EA. Let P be the reflection of point Z across point E. Let H be the rotation of point A by angle 3pi/4 around point P. Let S be the reflection of point H across point P. Point N is the reflection of point Z across segment HS. Line Q is the perpendicular bisector of segment ZH. Point I is the centroid of triangle HSZ. Let W be the midpoint of segment IN. Point R is the reflection of point N across line Q. Find the measure of angle PWR, 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": "8.78", "dataset": "MATH"}
{"messages": [{"content": "DL = 4. Point M is obtained by rotating point L by angle 2pi/3 about point D. Let G be the midpoint of segment DL. Point J is the midpoint of segment DM. Point Z is the centroid of triangle LJG. 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": "1.76", "dataset": "MATH"}
{"messages": [{"content": "Let LD = 6. Let O be the line through points L and D. Let R be the rotation of point D by angle 3pi/4 around point L. Line E is the perpendicular bisector of segment RL. Let X be the reflection of line E through point L. Lines X and O intersect at point C. Let K be the midpoint of segment CD. Let V be the midpoint of segment KR. Point W lies on line E. Point Q is the reflection of point C across line E. Let A be the rotation of point C by angle pi/2 around point Q. Point H is the centroid of triangle AWD. Point T is the reflection of point V across segment DQ. What is the length of segment CT? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.25", "dataset": "MATH"}
{"messages": [{"content": "Let QS = 7. Point D is obtained by rotating point Q by angle pi/2 about point S. Circle T is the circumcircle of triangle DSQ. Let I be the angle bisector of angle SDQ. Point R is the reflection of point Q across line I. Let F be the incenter of triangle DSQ. Let Z be the inversion of point F in circle T. Find 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": "5.36", "dataset": "MATH"}
{"messages": [{"content": "HC = 3. Point B is obtained by rotating point C by angle 5pi/12 about point H. Point U is obtained by rotating point H by angle pi/12 about point B. Line F passes through point H parallel to segment UC. Point O is the reflection of point C across line F. Point P is the reflection of point B across line F. Find the measure of angle POH, 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": "4.35", "dataset": "MATH"}
{"messages": [{"content": "IK = 5. Let W be the midpoint of segment IK. Point R is the midpoint of segment WI. Point E is obtained by rotating point R by angle 5pi/6 about point I. Point Z is the reflection of point E across point W. Circle D is the incircle of triangle ZKE. Let G be the perpendicular bisector of segment KE. Let J be the line through point R parallel to line G. Point Y is the reflection of point W across line G. Point H is the intersection of line J and segment KE. Let O be the inversion of point H in circle D. Find the measure of angle YOK, 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": "0.38", "dataset": "MATH"}
{"messages": [{"content": "LX = 4. Let N be the rotation of point X by angle 5pi/12 around point L. Point T is the reflection of point N across point L. Point R is obtained by rotating point T by angle 5pi/6 about point L. Point Y is the midpoint of segment TX. Let K be the angle bisector of angle NTY. Point D is the reflection of point X across line K. Find the circumradius of triangle XRD. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.06", "dataset": "MATH"}
{"messages": [{"content": "Let GJ = 7. Point C is the reflection of point J across point G. Let N be the rotation of point G by angle pi/3 around point J. Let W be the reflection of point N across point J. Let F be the angle bisector of angle JWC. Line P is the perpendicular bisector of segment JG. Let L be the incircle of triangle NJG. Point V is the inversion of point J with respect to circle L. Line I is the reflection of line P across line F. Segment JG and line I intersect at point U. Find 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": "1.62", "dataset": "MATH"}
{"messages": [{"content": "Let VF = 5. Line G passes through points V and F. Line D passes through point V perpendicular to line G. Point S lies on line D at distance 2 from point F. Point A is the midpoint of segment SF. Line R is the angle bisector of angle SVA. Point U is the midpoint of segment SV. Point I is the reflection of point A across line R. Line W is the perpendicular bisector of segment FU. Point N is the reflection of point I across line W. Find the measure of angle INF, 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": "4.78", "dataset": "MATH"}
{"messages": [{"content": "Let WE = 3. Let F be the rotation of point W by angle 2pi/3 around point E. Let A be the reflection of point F across point W. Let M be the reflection of point F across point A. Point I is the reflection of point F across segment WE. Find 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": "18.73", "dataset": "MATH"}
{"messages": [{"content": "Let QT = 12. Point O is the reflection of point Q across point T. Line S is the angle bisector of angle OTQ. Point C is the reflection of point T across point O. Point U lies on line S at distance 6 from point C. Line N is the angle bisector of angle OQU. Line I is the reflection of line S across line N. Point P is the reflection of point C across line I. Find the inradius of triangle TQP. Express your 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 RC = 8. Let RW = 12. Point J is the reflection of point W across point C. Let Z be the midpoint of segment JR. Let S be the centroid of triangle RCJ. What is 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": "2.0", "dataset": "MATH"}
{"messages": [{"content": "LQ = 4. Point U is the midpoint of segment LQ. Point X is obtained by rotating point Q by angle 2pi/3 about point L. Point W is obtained by rotating point U by angle pi/4 about point L. Let O be the reflection of point W across point X. Point Z is the reflection of point L across point O. Point R is the reflection of point X across point L. Find the inradius of triangle ZOR. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.2", "dataset": "MATH"}
{"messages": [{"content": "FE = 10. Point A is obtained by rotating point E by angle 11pi/12 about point F. Let U be the midpoint of segment EF. Let L be the reflection of point A across point F. Let T be the incenter of triangle EAF. What is the measure of angle TUL, 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": "4.85", "dataset": "MATH"}
{"messages": [{"content": "Let LY = 4. Point J is obtained by rotating point L by angle pi about point Y. Let M be the line through points J and L. Let O be the rotation of point J by angle 5pi/12 around point L. Let H be the rotation of point J by angle pi/12 around point Y. Line C passes through point O perpendicular to line M. Point Q is the reflection of point J across line C. Line G is the perpendicular bisector of segment OY. Point D is the reflection of point J across line G. What is the measure of angle QHD, 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": "51.42", "dataset": "MATH"}
{"messages": [{"content": "CE = 6. Line U is the perpendicular bisector of segment EC. Let Q be the reflection of line U through point E. Point N is on line Q with |NE| = 12. Point H is the reflection of point E across line Q. Let V be the midpoint of segment HN. Point P is the circumcenter of triangle CNV. What is the length of segment PC? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.85", "dataset": "MATH"}
{"messages": [{"content": "Let CO = 7. Let U be the reflection of point O across point C. Let G be the rotation of point C by angle 3pi/4 around point O. Let B be the perpendicular bisector of segment OC. Line D is the reflection of line B across point C. Let S be the circumcenter of triangle UCG. Line D and segment GU intersect at point L. Point I is the midpoint of segment GL. 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": "2.88", "dataset": "MATH"}
{"messages": [{"content": "LT = 9. Let Y be the rotation of point L by angle 3pi/4 around point T. Let F be the midpoint of segment YT. Let D be the midpoint of segment FL. Let H be the orthocenter of triangle LYF. Let P be the reflection of point H across point D. What is the measure of angle PHL, 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": "177.04", "dataset": "MATH"}
{"messages": [{"content": "PZ = 7. Point I is obtained by rotating point Z by angle pi/2 about point P. Let O be the line through point I perpendicular to segment PI. Let B be the incircle of triangle PIZ. Point T is the reflection of point Z across line O. Let W be the inversion of point T in circle B. Let U be the midpoint of segment IZ. What is the distance between points U 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": "1.75", "dataset": "MATH"}
{"messages": [{"content": "SB = 10. Let Q be the reflection of point B across point S. Let X be the rotation of point B by angle 7pi/12 around point Q. Let J be the midpoint of segment SQ. Point R is obtained by rotating point S by angle 5pi/12 about point Q. Let D be the midpoint of segment JX. Let H be the circumcenter of triangle RQX. Point P is the reflection of point D across point Q. 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": "19.34", "dataset": "MATH"}
{"messages": [{"content": "PC = 9. Let Y be the midpoint of segment PC. Let Z be the rotation of point Y by angle 7pi/12 around point P. Let H be the midpoint of segment YZ. Let M be the midpoint of segment CZ. Let G be the line through point P perpendicular to segment PC. Point D is the reflection of point P across point M. Point B is the reflection of point D across line G. What is the measure of angle YHB, 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": "7.25", "dataset": "MATH"}
{"messages": [{"content": "Let MY = 4. Point W is obtained by rotating point Y by angle pi/3 about point M. Line U is the perpendicular bisector of segment WY. Line X is the reflection of line U across point W. Point R is the reflection of point W across line X. What is the inradius of triangle RMW? Express your 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": "YX = 1. Let D be the rotation of point X by angle 5pi/12 around point Y. Point W is the midpoint of segment YD. Point E is obtained by rotating point X by angle 11pi/12 about point Y. Point I is the midpoint of segment XW. Line F passes through points E and I. Point K is the reflection of point W across line F. Let R be the midpoint of segment XD. What is the circumradius of triangle WKR? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.35", "dataset": "MATH"}
{"messages": [{"content": "YR = 7. Let V be the rotation of point Y by angle 3pi/4 around point R. Let W be the rotation of point R by angle 2pi/3 around point Y. Point H is the circumcenter of triangle RVW. Point J is the reflection of point Y across segment VW. Let F be the centroid of triangle HJV. What is the distance between points F 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": "5.82", "dataset": "MATH"}
{"messages": [{"content": "NA = 4. Point X is the midpoint of segment AN. Let G be the rotation of point A by angle 7pi/12 around point X. Point C is the orthocenter of triangle GXN. Point H is the centroid of triangle GXN. Find the circumradius of triangle NHC. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.84", "dataset": "MATH"}
{"messages": [{"content": "Let VM = 5. Let L be the midpoint of segment VM. Let I be the rotation of point M by angle 5pi/6 around point L. Let D be the circumcenter of triangle LIV. Line G passes through point M parallel to segment IV. Point E is the reflection of point I across line G. Let Y be the midpoint of segment VD. 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": "9.31", "dataset": "MATH"}
{"messages": [{"content": "AI = 5. Let J be the rotation of point I by angle 7pi/12 around point A. Point R is the reflection of point J across segment IA. Point X is the orthocenter of triangle RAJ. Let E be the midpoint of segment AJ. What is the measure of angle EAX, 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 KH = 10. Point P is the reflection of point K across point H. Line O is the perpendicular bisector of segment PK. Let C be the rotation of point P by angle pi/12 around point K. Let G be the midpoint of segment CK. Let J be the midpoint of segment HC. Point U is the midpoint of segment CG. Let Y be the orthocenter of triangle PCU. Point T is the reflection of point Y across line O. Find the measure of angle YTJ, 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": "16.41", "dataset": "MATH"}
{"messages": [{"content": "Let XV = 2. Point Y is obtained by rotating point V by angle pi/2 about point X. Line A is the angle bisector of angle YVX. Point M is the reflection of point X across line A. Let G be the midpoint of segment VX. Point Z is the reflection of point M across segment XY. 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": "2.12", "dataset": "MATH"}
{"messages": [{"content": "Let UY = 10. Point N is obtained by rotating point U by angle pi/3 about point Y. Let A be the reflection of point Y across point U. Circle Q is the incircle of triangle ANY. Point S is the inversion of point A with respect to circle Q. 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": "5.71", "dataset": "MATH"}
{"messages": [{"content": "JV = 6. Point F is obtained by rotating point V by angle pi/6 about point J. Point S is the midpoint of segment FJ. Let N be the line through point J perpendicular to segment JF. Let Y be the line through point F parallel to line N. Point X is the reflection of point J across line Y. Let H be the rotation of point J by angle 2pi/3 around point S. Let B be the reflection of point X across point H. Point K is the reflection of point B across point V. Point P is the reflection of point K across segment JF. Point W is the midpoint of segment PJ. What is the measure of angle WVX, 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": "9.32", "dataset": "MATH"}
{"messages": [{"content": "EZ = 11. Let D be the reflection of point E across point Z. Let B be the rotation of point D by angle 2pi/3 around point E. Let R be the perpendicular bisector of segment BZ. Point N is the reflection of point D across line R. Let H be the centroid of triangle EBN. 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": "4.16", "dataset": "MATH"}
{"messages": [{"content": "NH = 1. Let M be the reflection of point H across point N. Point E is obtained by rotating point N by angle pi/6 about point M. Point S is obtained by rotating point E by angle 11pi/12 about point N. Point L is the reflection of point N across point E. Let F be the perpendicular bisector of segment MS. Line U is the perpendicular bisector of segment LN. Let Z be the reflection of line F across line U. Point Q is the reflection of point H across line U. Line T is the reflection of line F across point L. Lines Z and T intersect at point V. What is the measure of angle MQV, 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": "0.53", "dataset": "MATH"}
{"messages": [{"content": "ED = 12. Let O be the midpoint of segment ED. Let B be the rotation of point O by angle pi/4 around point E. Let F be the reflection of point D across point B. Point I is the reflection of point F across point O. Let J be the centroid of triangle DEB. Find the circumradius of triangle BJI. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "21.69", "dataset": "MATH"}
{"messages": [{"content": "OK = 8. Line B is the perpendicular bisector of segment OK. Let T be the midpoint of segment OK. Line W is the reflection of line B across point O. Point P is on line W with |PK| = 4. Point I is the orthocenter of triangle TPO. Point Q is the reflection of point I across line B. Let N be the incircle of triangle TPO. Let R be the inversion of point K in circle N. Point L is the midpoint of segment OT. Point S is the reflection of point L across point R. Find the measure of angle SQR, 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": "1.95", "dataset": "MATH"}
{"messages": [{"content": "Let LE = 1. Let N be the rotation of point E by angle pi/12 around point L. Point T is the reflection of point E across point N. Let W be the circumcenter of triangle NLE. Point Z is the reflection of point W across segment NL. Line I passes through points Z and W. Segment LE and line I intersect at point C. 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": "0.58", "dataset": "MATH"}
{"messages": [{"content": "Let DK = 10. Let I be the reflection of point K across point D. Line B passes through point I perpendicular to segment KD. Let V be the rotation of point K by angle 7pi/12 around point D. Point M is the reflection of point D across line B. Let F be the line through point V parallel to segment KD. Point Z is the reflection of point K across line F. Let A be the reflection of point V across point M. What is the circumradius of triangle ZVA? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "188.39", "dataset": "MATH"}
{"messages": [{"content": "DF = 8. Point H is obtained by rotating point F by angle 3pi/4 about point D. Let M be the reflection of point H across point D. Point N is the midpoint of segment DH. Point X is the reflection of point H across segment FD. Point B is the reflection of point X across point M. Point I is the reflection of point B across segment NF. Find the measure of angle BHI, 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": "79.36", "dataset": "MATH"}
{"messages": [{"content": "ZW = 1. Let M be the rotation of point W by angle 2pi/3 around point Z. Point S is the orthocenter of triangle ZWM. Point F is obtained by rotating point S by angle pi/6 about point Z. Let J be the rotation of point F by angle 7pi/12 around point W. Line T is the perpendicular bisector of segment MZ. Point C is the midpoint of segment MS. Let V be the line through point C perpendicular to line T. Let X be the line through point F perpendicular to line V. Point H is the midpoint of segment WF. Point D is the reflection of point J across line X. Find the measure of angle HCD, 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": "0.86", "dataset": "MATH"}
{"messages": [{"content": "Let GA = 6. Point P is the midpoint of segment GA. Point U is obtained by rotating point P by angle pi/4 about point G. Let J be the midpoint of segment UA. Let T be the midpoint of segment PJ. Point L is the reflection of point U across point P. What is the inradius of triangle LJT? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.24", "dataset": "MATH"}
{"messages": [{"content": "Let SE = 8. Point V is obtained by rotating point E by angle 5pi/6 about point S. Let G be the rotation of point V by angle 5pi/12 around point E. Let Y be the midpoint of segment SG. Point U is the reflection of point E across point S. Let W be the orthocenter of triangle UGY. 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": "17.85", "dataset": "MATH"}
{"messages": [{"content": "Let VH = 8. Let Y be the rotation of point V by angle pi/4 around point H. Let C be the reflection of point Y across point V. Point B is obtained by rotating point Y by angle 11pi/12 about point C. Point K is the reflection of point V across point B. Let W be the midpoint of segment HV. Point N is the reflection of point B across point C. Find the circumradius of triangle NWK. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26.16", "dataset": "MATH"}
{"messages": [{"content": "Let HJ = 6. Point I is obtained by rotating point H by angle 5pi/12 about point J. Let K be the orthocenter of triangle IHJ. Point Z is the reflection of point J across segment IH. Point X is the reflection of point Z across segment HJ. What is the distance between points K 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": "9.21", "dataset": "MATH"}
{"messages": [{"content": "Let QU = 1. Point R is obtained by rotating point Q by angle 5pi/6 about point U. Point D is the reflection of point R across point Q. Point I is the midpoint of segment RQ. Point W is the midpoint of segment UD. What is the circumradius of triangle WIU? Express your 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": "MF = 9. Let J be the vector from point F to point M. Let A be point M translated by vector J. Let X be the rotation of point M by angle 5pi/6 around point A. Line T is the perpendicular bisector of segment AF. Point N is the midpoint of segment AX. Point G is the reflection of point X across line T. Let C be the reflection of point N across point G. Point Q is the circumcenter of triangle FCG. 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": "22.59", "dataset": "MATH"}
{"messages": [{"content": "Let UD = 12. Let K be the midpoint of segment DU. Let B be the rotation of point D by angle pi/3 around point K. Point J is the midpoint of segment DK. Let N be the rotation of point B by angle 7pi/12 around point D. Let L be the rotation of point D by angle 11pi/12 around point J. Let R be the perpendicular bisector of segment JL. Point P is the reflection of point N across line R. Let S be the line through point U parallel to segment ND. Point Z is the reflection of point D across line S. Find the circumradius of triangle PZL. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.15", "dataset": "MATH"}
{"messages": [{"content": "Let KY = 2. Let Q be the midpoint of segment KY. Point C is obtained by rotating point Q by angle 3pi/4 about point Y. Let L be the midpoint of segment YC. Let N be the orthocenter of triangle YCK. Let G be the circumcenter of triangle YCK. Let H be the line through points K and L. Point W is the incenter of triangle YCK. Let I be the reflection of line H through point W. Let Z be the reflection of point N across point G. Point J is the reflection of point Y across line I. What is the measure of angle JCZ, 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": "1.27", "dataset": "MATH"}
{"messages": [{"content": "XP = 11. Let L be the vector from point X to point P. Let J be point P translated by vector L. Let U be the rotation of point J by angle 3pi/4 around point P. Let I be the reflection of point X across point U. Point A is the midpoint of segment IU. Let S be the reflection of point A across point J. 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": "42.57", "dataset": "MATH"}
{"messages": [{"content": "BU = 2. Point M is obtained by rotating point U by angle pi/6 about point B. Let P be the rotation of point B by angle 2pi/3 around point M. Let S be the perpendicular bisector of segment UB. Let O be the rotation of point U by angle pi/4 around point P. Line G passes through point O parallel to line S. Point R is the reflection of point P across line S. Line C passes through points M and R. Line Q is the reflection of line G across point R. Let H be the midpoint of segment BU. Let V be the reflection of line C across line Q. Point W is the reflection of point H across line V. Find the measure of angle PWO, 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": "0.8", "dataset": "MATH"}
{"messages": [{"content": "Let DH = 7. Let X be the rotation of point H by angle 7pi/12 around point D. Point W is the midpoint of segment HD. Point C is the midpoint of segment WH. Point R is the circumcenter of triangle XCD. What is the measure of angle RXH, 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": "9.86", "dataset": "MATH"}
{"messages": [{"content": "CD = 2. Point W is the reflection of point C across point D. Point N is obtained by rotating point W by angle pi/4 about point D. Let H be the centroid of triangle DCN. Line S is the perpendicular bisector of segment HC. Point Q is the reflection of point D across line S. Point U is the reflection of point Q across segment ND. What is the measure of angle WUQ, 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": "3.19", "dataset": "MATH"}
{"messages": [{"content": "CZ = 2. Let O be the perpendicular bisector of segment ZC. Let W be the midpoint of segment CZ. Point M is obtained by rotating point Z by angle pi/4 about point C. Let X be the rotation of point C by angle pi/4 around point W. Point Y is obtained by rotating point X by angle pi/3 about point Z. Point F is the reflection of point X across point Y. Let V be the incircle of triangle CFX. Circle L is the reflection of circle V across line O. Let I be the inversion of point Z in circle L. Let H be the circumcenter of triangle CFX. Find the measure of angle MIH, 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": "0.48", "dataset": "MATH"}
{"messages": [{"content": "ZN = 11. Let I be the rotation of point Z by angle pi/12 around point N. Point Y is the reflection of point N across segment ZI. Let V be the reflection of point Y across point Z. Line A passes through point V perpendicular to segment ZI. Line S passes through point V perpendicular to line A. Point L is the reflection of point Z across line S. Point U is the reflection of point Z across point L. Find the measure of angle UZN, 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": "31.32", "dataset": "MATH"}
{"messages": [{"content": "Let AK = 3. Line I is the perpendicular bisector of segment KA. Let G be the rotation of point A by angle 11pi/12 around point K. Let V be the reflection of point G across point K. Point W is the reflection of point G across line I. Let P be the circumcircle of triangle AVW. Point T is the inversion of point G with respect to circle P. Let L be the midpoint of segment VW. What is the inradius of triangle LTK? Express your 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": "HD = 6. Let Z be the rotation of point H by angle 2pi/3 around point D. Let S be the orthocenter of triangle DZH. Let J be the reflection of point Z across point S. Let T be the midpoint of segment DJ. 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": "13.08", "dataset": "MATH"}
{"messages": [{"content": "Let TW = 6. Point P is obtained by rotating point W by angle pi/4 about point T. Point I is the midpoint of segment PT. Point C is the midpoint of segment TW. Point A is the reflection of point P across segment TW. What is the circumradius of triangle ICA? Express your 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": "LI = 3. Point W is obtained by rotating point I by angle 5pi/6 about point L. Let X be the reflection of point W across point I. Point Y is the midpoint of segment XI. Point E is the reflection of point Y across point W. Find the circumradius of triangle ELW. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "22.44", "dataset": "MATH"}
{"messages": [{"content": "RQ = 5. Let J be the rotation of point R by angle pi/2 around point Q. Let D be the centroid of triangle RJQ. Point E is the reflection of point D across segment RJ. Let L be the midpoint of segment RQ. 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": "3.44", "dataset": "MATH"}
{"messages": [{"content": "Let CP = 5. Point U is obtained by rotating point C by angle pi/4 about point P. Point V is the circumcenter of triangle PUC. Point H is the orthocenter of triangle PUC. Point D is the midpoint of segment HC. 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": "1.49", "dataset": "MATH"}
{"messages": [{"content": "Let CW = 6. Let Y be the rotation of point W by angle pi/12 around point C. Let A be the circumcenter of triangle CWY. Let Z be the reflection of point C across point W. Let Q be the reflection of point A across point Z. What is 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": "14.42", "dataset": "MATH"}
{"messages": [{"content": "Let AB = 7. Point Y is obtained by rotating point B by angle pi/2 about point A. Let L be the reflection of point A across point B. Let D be the midpoint of segment YB. Let C be the midpoint of segment LB. 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": "12.62", "dataset": "MATH"}
{"messages": [{"content": "CQ = 2. Let N be the midpoint of segment CQ. Let O be the vector from point Q to point N. Let Z be point C translated by vector O. Point K is obtained by rotating point Q by angle 11pi/12 about point Z. Point F is the reflection of point Q across segment NK. Point W is the reflection of point K across point Z. Circle M is the incircle of triangle FZK. Point Y is the inversion of point Z with respect to circle M. Point P is the reflection of point F across segment CQ. Find the measure of angle YWP, 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": "1.59", "dataset": "MATH"}
{"messages": [{"content": "CB = 3. Point Q is the midpoint of segment CB. Point F is the midpoint of segment QB. Let P be the rotation of point Q by angle pi/6 around point F. Point D is the orthocenter of triangle CPQ. Point X is the reflection of point P across point D. Let S be the midpoint of segment XF. Let U be the midpoint of segment DS. What is the measure of angle QUD, 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": "0.03", "dataset": "MATH"}
{"messages": [{"content": "Let CS = 4. Let O be the midpoint of segment SC. Point Z is obtained by rotating point O by angle 5pi/12 about point C. Line H is the angle bisector of angle SCZ. Let F be the reflection of point O across point S. Point X is the midpoint of segment FC. Point B is the reflection of point X across line H. What is the inradius of triangle XBZ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.37", "dataset": "MATH"}
{"messages": [{"content": "CY = 12. Point X is the reflection of point C across point Y. Let O be the perpendicular bisector of segment YX. Point G is obtained by rotating point Y by angle pi/2 about point X. Point A is the reflection of point X across point G. Let U be the rotation of point G by angle 7pi/12 around point C. Point T is the circumcenter of triangle ACU. Let L be the midpoint of segment CU. Point S is the reflection of point T across line O. What is the distance between points L 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": "26.58", "dataset": "MATH"}
{"messages": [{"content": "GC = 1. Let T be the reflection of point G across point C. Let D be the rotation of point T by angle pi/4 around point G. Line A passes through point T perpendicular to segment DC. Let E be the midpoint of segment DC. Point F is the reflection of point E across line A. Point U is the reflection of point D across point T. Let K be the midpoint of segment FC. Let P be the reflection of point K across point E. What is the measure of angle GPU, 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": "2.95", "dataset": "MATH"}
{"messages": [{"content": "Let JN = 3. Let G be the perpendicular bisector of segment JN. Let S be the rotation of point N by angle 5pi/12 around point J. Line C is the reflection of line G across point J. Line E is the perpendicular bisector of segment SJ. Let K be the reflection of line E across line C. Point X is the reflection of point N across line K. Point T is the reflection of point X across segment JN. Point I is the reflection of point S across segment JN. What is the measure of angle IXT, 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": "3.79", "dataset": "MATH"}
{"messages": [{"content": "Let HJ = 8. Point U is obtained by rotating point J by angle pi/12 about point H. Let S be the rotation of point H by angle pi/6 around point U. Point F is the midpoint of segment JU. Line X is the perpendicular bisector of segment SU. Point W is the reflection of point F across point S. Point G is the reflection of point J across line X. 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": "8.99", "dataset": "MATH"}
{"messages": [{"content": "VK = 2. Let G be the reflection of point V across point K. Point T is obtained by rotating point V by angle 7pi/12 about point G. Let O be the reflection of point T across point K. Point Q is the reflection of point T across point V. Point I is the reflection of point K across segment OQ. Point W is the reflection of point G across segment TO. What is the circumradius of triangle WQI? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.14", "dataset": "MATH"}
{"messages": [{"content": "Let TS = 11. Point B is obtained by rotating point T by angle pi/12 about point S. Point X is the midpoint of segment TB. Let Z be the centroid of triangle TBS. Point O is the reflection of point Z across segment TB. Let H be the reflection of point S across point O. 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": "18.18", "dataset": "MATH"}
{"messages": [{"content": "AV = 6. Let K be the midpoint of segment AV. Point Q is obtained by rotating point V by angle 5pi/12 about point A. Let P be the angle bisector of angle AVQ. Let Z be the orthocenter of triangle QVK. Point B is the reflection of point Z across line P. Let F be the angle bisector of angle VQA. Let T be the circumcenter of triangle QVK. Point N is the reflection of point T across point B. Point L is the reflection of point N across line F. What is the distance between points L 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": "6.69", "dataset": "MATH"}
{"messages": [{"content": "Let OH = 2. Point C is obtained by rotating point O by angle 11pi/12 about point H. Point Y is the midpoint of segment HC. Line X passes through points O and C. Point I is the reflection of point H across line X. What is the circumradius of triangle IYC? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.06", "dataset": "MATH"}
{"messages": [{"content": "Let SL = 3. Point R is obtained by rotating point L by angle pi about point S. Line T passes through point L perpendicular to segment LS. Point F is obtained by rotating point R by angle 3pi/4 about point L. Let I be the reflection of line T through point S. Let C be the perpendicular bisector of segment LS. Let D be the reflection of line C through point L. Let W be the rotation of point F by angle 7pi/12 around point S. Point P is the reflection of point S across point F. Line X is the reflection of line I across line D. Point H is the reflection of point W across line X. Find the circumradius of triangle PLH. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.48", "dataset": "MATH"}
{"messages": [{"content": "FE = 5. Point B is obtained by rotating point F by angle pi/4 about point E. Let I be the orthocenter of triangle BFE. Point V is the centroid of triangle BFE. Point N is the midpoint of segment BI. Find 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": "1.49", "dataset": "MATH"}
{"messages": [{"content": "AX = 2. Let P be the rotation of point A by angle pi/2 around point X. Point N is obtained by rotating point A by angle pi/12 about point X. Point C is the orthocenter of triangle ANX. Let Z be the midpoint of segment PC. Let Y be the perpendicular bisector of segment AN. Point B is the reflection of point Z across line Y. Find the length of segment BZ. Express your 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": "Let KM = 11. Point C is obtained by rotating point K by angle 7pi/12 about point M. Point X is the circumcenter of triangle KCM. Let Z be the line through point X parallel to segment KC. Segment MK and line Z intersect at point L. Point H is the orthocenter of triangle KCM. Find the length of segment LH. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.84", "dataset": "MATH"}
{"messages": [{"content": "OR = 7. Let A be the midpoint of segment RO. Point I is obtained by rotating point R by angle 7pi/12 about point A. Let N be the angle bisector of angle IOA. Point D is the reflection of point I across line N. Let J be the line through point R perpendicular to line N. Point F is the reflection of point D across line J. Point T is the midpoint of segment AR. Find the inradius of triangle TAF. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.33", "dataset": "MATH"}
{"messages": [{"content": "DY = 9. Let N be the rotation of point Y by angle 5pi/6 around point D. Line O passes through point N parallel to segment YD. Point M is the midpoint of segment DY. Let A be the angle bisector of angle NDM. Let U be the perpendicular bisector of segment YD. Let I be the reflection of line O across line A. Line K is the reflection of line U across point D. Lines I and K intersect at point R. Find the inradius of triangle NMR. Express your 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": "MK = 8. Point J is the midpoint of segment KM. Point R is obtained by rotating point K by angle pi/4 about point M. Let C be the rotation of point R by angle pi/6 around point J. Let Y be the rotation of point K by angle pi/3 around point R. Line T is the angle bisector of angle YRC. Let L be the midpoint of segment CY. Point I is the reflection of point L across line T. Find the inradius of triangle KIL. Express your 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 PJ = 8. Point C is obtained by rotating point P by angle pi/2 about point J. Circle Y is the circumcircle of triangle CPJ. Let I be the reflection of point P across point J. Let B be the reflection of point C across point J. Let S be the reflection of point J across point I. Let A be the midpoint of segment SC. Let X be the inversion of point A in circle Y. Let E be the angle bisector of angle PCX. Let M be the line through point S parallel to line E. Let T be the line through point B perpendicular to line M. Point W is the reflection of point A 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": "25.51", "dataset": "MATH"}
{"messages": [{"content": "Let UX = 10. Let Y be the midpoint of segment UX. Point Q is the midpoint of segment YU. Let T be the perpendicular bisector of segment XU. Point L is obtained by rotating point Q by angle 11pi/12 about point U. Let V be the reflection of line T through point X. Point R is the reflection of point Q across line V. Let H be the midpoint of segment RL. Find the inradius of triangle HLQ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.27", "dataset": "MATH"}
{"messages": [{"content": "Let UK = 3. Let G be the rotation of point K by angle 5pi/6 around point U. Point E is the midpoint of segment GK. Let N be the angle bisector of angle KEG. Point F is the midpoint of segment GU. Point Z is the reflection of point E across point F. Point J is the reflection of point Z across line N. What is the measure of angle FJK, 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": "1.69", "dataset": "MATH"}
{"messages": [{"content": "TZ = 2. Let Y be the rotation of point T by angle pi/3 around point Z. Point G is the reflection of point Y across point T. Let S be the circumcenter of triangle YZG. Let M be the incircle of triangle YZG. Let V be the inversion of point S in circle M. 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": "0.83", "dataset": "MATH"}
{"messages": [{"content": "Let JD = 7. Let Z be the midpoint of segment DJ. Let I be the rotation of point J by angle pi/6 around point Z. Let O be the midpoint of segment JI. Let R be the circumcenter of triangle ODI. Let N be the midpoint of segment RD. Find the distance between points O 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": "4.57", "dataset": "MATH"}
{"messages": [{"content": "Let EG = 2. Let R be the rotation of point E by angle 5pi/12 around point G. Let M be the rotation of point E by angle 3pi/4 around point G. Let A be the reflection of point R across point G. Let S be the reflection of point M across point G. Point P is the centroid of triangle SRE. What is the measure of angle GPA, 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": "1.22", "dataset": "MATH"}
{"messages": [{"content": "Let PJ = 5. Point K is the reflection of point P across point J. Let Q be the rotation of point J by angle pi/2 around point P. Point X is the reflection of point K across segment JQ. Let E be the midpoint of segment JQ. Point F is the reflection of point Q across point X. Find the measure of angle EKF, 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": "56.25", "dataset": "MATH"}
{"messages": [{"content": "PB = 8. Let Z be the rotation of point B by angle 5pi/6 around point P. Let G be the incircle of triangle BPZ. Let Y be the orthocenter of triangle BPZ. Point K is the inversion of point Z with respect to circle G. Let D be the midpoint of segment KY. 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": "12.85", "dataset": "MATH"}
{"messages": [{"content": "OB = 9. Let C be the rotation of point B by angle 7pi/12 around point O. Let M be the rotation of point C by angle pi/3 around point O. Let A be the rotation of point C by angle pi/12 around point B. Let H be the line through point B perpendicular to segment OB. Let S be the line through point C perpendicular to line H. Point I is the reflection of point M across line S. 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": "12.73", "dataset": "MATH"}
{"messages": [{"content": "Let TU = 5. Let N be the rotation of point U by angle 11pi/12 around point T. Let B be the perpendicular bisector of segment UT. Point M is the reflection of point N across line B. Let X be the reflection of line B through point N. Point H is the reflection of point T across line X. Point Y is the reflection of point M across segment HU. Find the distance between points Y 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": "34.17", "dataset": "MATH"}
{"messages": [{"content": "NB = 9. Point K is the midpoint of segment NB. Let J be the midpoint of segment KN. Point M is obtained by rotating point J by angle 5pi/12 about point N. Point V is the reflection of point N across point M. Point L is the circumcenter of triangle VJK. Point R is the reflection of point B across point L. What is the distance between points R 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": "8.72", "dataset": "MATH"}
{"messages": [{"content": "AU = 5. Point F is the midpoint of segment UA. Point N is obtained by rotating point F by angle pi/2 about point A. Point B is the centroid of triangle AFN. Line O is the perpendicular bisector of segment UA. Point K is the reflection of point B across line O. Point J is the orthocenter of triangle NKU. What is 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": "16.01", "dataset": "MATH"}
{"messages": [{"content": "Let DR = 2. Point Z is obtained by rotating point D by angle pi/2 about point R. Let C be the circumcircle of triangle ZDR. Circle P is the reflection of circle C across point Z. Point U is the incenter of triangle ZDR. Let O be the inversion of point U in circle P. What is the measure of angle UOD, 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": "GL = 2. Line N is the perpendicular bisector of segment GL. Point R is on line N with |RL| = 5. Let U be the perpendicular bisector of segment RL. Let X be the incenter of triangle LRG. Let Y be the line through point L parallel to line U. Line A passes through point G parallel to line Y. Point H is the midpoint of segment RX. Line A and segment LR intersect at point M. Find the length of segment MH. Express your 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 YK = 10. Let R be the rotation of point K by angle 3pi/4 around point Y. Point S is the midpoint of segment KY. Let Z be the incenter of triangle SKR. Let Q be the centroid of triangle SKR. 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": "2.97", "dataset": "MATH"}
{"messages": [{"content": "TS = 9. Line K passes through points T and S. Point Q is obtained by rotating point S by angle pi/6 about point T. Let F be the rotation of point S by angle 5pi/12 around point T. Point C is the midpoint of segment TF. Line V passes through point C parallel to line K. Point P is obtained by rotating point Q by angle pi/6 about point T. Point E is the reflection of point P across line V. Point I is the reflection of point Q across segment SP. Point W is the reflection of point I across segment PT. Find the measure of angle WEQ, 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": "17.47", "dataset": "MATH"}
{"messages": [{"content": "Let AH = 9. Let X be the midpoint of segment HA. Point G is obtained by rotating point X by angle 7pi/12 about point H. Point C is obtained by rotating point X by angle pi/2 about point H. Let W be the reflection of point C across point G. Point E is the midpoint of segment WA. Point R is the centroid of triangle GWA. 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": "1.68", "dataset": "MATH"}
{"messages": [{"content": "RZ = 4. Point D is the midpoint of segment ZR. Let C be the reflection of point D across point Z. Point T is obtained by rotating point C by angle pi/6 about point D. Let Y be the orthocenter of triangle ZCT. Point V is the circumcenter of triangle ZCT. What is the inradius of triangle YVT? Express your 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": "TR = 6. Point S is obtained by rotating point R by angle 2pi/3 about point T. Point D is the reflection of point S across point R. Point Z is the midpoint of segment DT. Let U be the centroid of triangle TRS. What is the measure of angle ZTU, 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": "7.79", "dataset": "MATH"}
{"messages": [{"content": "Point U is the centroid of triangle M. Let W be the line through point U parallel to segment MB. Let E be the reflection of line W through point X. Point R is the reflection of point M across line E. What is the length of segment RB? Express your 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": "DN = 9. Point U is obtained by rotating point D by angle pi/3 about point N. Let S be the perpendicular bisector of segment UN. Point P is the reflection of point U across point D. Line E is the reflection of line S across point U. Line B is the reflection of line E across point D. Point C is the reflection of point D across line B. Let Z be the circumcenter of triangle NUP. Find the measure of angle DZC, 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": "23.38", "dataset": "MATH"}
{"messages": [{"content": "ZM = 4. Let X be the rotation of point M by angle pi/12 around point Z. Let C be the centroid of triangle MZX. Point P is the reflection of point C across segment MZ. Let D be the circumcenter of triangle PZC. Find the length of segment DC. Express your 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": "SK = 12. Point H is obtained by rotating point K by angle 11pi/12 about point S. Point V is the midpoint of segment SK. Point B is the midpoint of segment KH. Point F is the reflection of point S across segment KH. Let Z be the midpoint of segment VS. 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": "4.04", "dataset": "MATH"}
{"messages": [{"content": "HK = 2. Point A is obtained by rotating point H by angle pi/6 about point K. Let V be the circumcircle of triangle AKH. Let R be the incenter of triangle AKH. Let D be the inversion of point R in circle V. Point L is the midpoint of segment AK. 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": "2.23", "dataset": "MATH"}
{"messages": [{"content": "BI = 10. Point S is obtained by rotating point I by angle 3pi/4 about point B. Let Z be the midpoint of segment SI. Point U is the reflection of point Z across segment BS. Point K is the reflection of point I across segment BS. Find the measure of angle KUZ, 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": "UL = 9. Let Z be the rotation of point U by angle pi/2 around point L. Point H is the midpoint of segment UZ. Line K passes through point Z perpendicular to segment LZ. Let B be the rotation of point L by angle 5pi/6 around point U. Line T passes through point U parallel to segment LZ. Point O is the reflection of point B across point H. Let P be the reflection of line T through point B. Point G is the reflection of point O across line P. Point X lies on line K. Let Y be the reflection of point H across point Z. Find the measure of angle YGX, 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": "138.31", "dataset": "MATH"}
{"messages": [{"content": "SU = 6. Point F is obtained by rotating point S by angle pi/3 about point U. Let C be the rotation of point F by angle pi/12 around point U. Let G be the angle bisector of angle CFS. Point E is the reflection of point U across line G. Point D is the reflection of point E across segment US. Find the circumradius of triangle DEF. Express your 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 IX = 7. Point B is obtained by rotating point I by angle 7pi/12 about point X. Let N be the midpoint of segment XI. Point W is the reflection of point X across segment BI. Let U be the perpendicular bisector of segment BW. Let Z be the angle bisector of angle BNX. Line H is the reflection of line Z across line U. Let K be the circumcenter of triangle WNX. Segment WN and line H intersect at point D. What is 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": "1.13", "dataset": "MATH"}
{"messages": [{"content": "JT = 4. Let L be the midpoint of segment TJ. Line X is the angle bisector of angle JLT. Point Z is the midpoint of segment TL. Point R is on line X with |RL| = 11. Line F passes through points Z and R. Line G is the reflection of line F across point T. Point N is the reflection of point R across line G. Let M be the midpoint of segment LN. Let A be the reflection of point M across point N. What is the inradius of triangle TAN? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.32", "dataset": "MATH"}
{"messages": [{"content": "ZO = 12. Point T is the reflection of point Z across point O. Point L is obtained by rotating point T by angle 5pi/6 about point O. Let G be the incircle of triangle TLO. Point U is the orthocenter of triangle TLO. Point V is the inversion of point L with respect to circle G. Find the measure of angle UZV, 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": "238.36", "dataset": "MATH"}
{"messages": [{"content": "GT = 6. Point V is obtained by rotating point T by angle 5pi/12 about point G. Let B be the rotation of point V by angle 5pi/6 around point G. Let H be the midpoint of segment TV. Point Z is the midpoint of segment VB. Let P be the centroid of triangle ZHG. 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": "2.2", "dataset": "MATH"}
{"messages": [{"content": "Let FS = 10. Let E be the vector from point S to point F. Let Q be the midpoint of segment FS. Let U be the rotation of point Q by angle pi/2 around point F. Point A is obtained by rotating point S by angle 11pi/12 about point F. Line X is the angle bisector of angle AQF. Let K be the reflection of line X through point U. Point O is the translation of point A by vector E. Line R passes through point O perpendicular to line X. Point V is the midpoint of segment QS. Let T be the reflection of line R through point Q. Point Z is the intersection of lines T and K. Point C is the orthocenter of triangle OZV. What is the length of segment FC? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "124.64", "dataset": "MATH"}
{"messages": [{"content": "Let LN = 10. Point O is the reflection of point N across point L. Let G be the perpendicular bisector of segment NO. Let V be the vector from point O to point N. Point D is the translation of point L by vector V. Point K is on line G with |KL| = 11. Let B be the line through points K and N. Point Q is the reflection of point L across line B. Let E be the midpoint of segment DK. Let C be the centroid of triangle QNE. Find the measure of angle CLE, 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": "2.61", "dataset": "MATH"}
{"messages": [{"content": "BC = 7. Point G is the reflection of point B across point C. Point U is obtained by rotating point C by angle 11pi/12 about point B. Let F be the rotation of point B by angle pi/4 around point C. Let Y be the midpoint of segment FG. Let P be the orthocenter of triangle UBY. Let X be the incenter of triangle UBY. Find 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": "412.21", "dataset": "MATH"}
{"messages": [{"content": "Let EA = 3. Let K be the rotation of point A by angle 11pi/12 around point E. Let W be the midpoint of segment AK. Point Z is obtained by rotating point K by angle 11pi/12 about point W. Point T is obtained by rotating point E by angle 5pi/6 about point W. Let S be the rotation of point Z by angle pi/2 around point E. Point O is the reflection of point T across point A. Point J is the reflection of point S across segment EK. Point V is the midpoint of segment OE. What is the measure of angle JTV, 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": "5.03", "dataset": "MATH"}
{"messages": [{"content": "CU = 11. Point D is obtained by rotating point U by angle 5pi/6 about point C. Let H be the centroid of triangle CDU. Point F is the reflection of point C across point D. Let P be the line through points H and F. Line P and segment UC intersect at point V. Point O is the reflection of point V across segment CD. What is the length of segment FO? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "25.9", "dataset": "MATH"}
{"messages": [{"content": "KE = 5. Let C be the rotation of point K by angle 5pi/6 around point E. Let M be the centroid of triangle CEK. Point G is the orthocenter of triangle CEK. Let N be the reflection of point M across point C. Let Q be the midpoint of segment MG. What is the measure of angle NKQ, 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": "67.89", "dataset": "MATH"}
{"messages": [{"content": "Let AM = 9. Let Z be the rotation of point M by angle pi/3 around point A. Let F be the circumcenter of triangle MAZ. Point I is the reflection of point F across segment ZM. Point O is the reflection of point I across segment MA. What is 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": "13.75", "dataset": "MATH"}
{"messages": [{"content": "Let LR = 3. Let Z be the rotation of point R by angle pi/12 around point L. Let Y be the midpoint of segment LR. Point W is the orthocenter of triangle LZY. Let T be the midpoint of segment WR. What is the circumradius of triangle ZTW? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "86.39", "dataset": "MATH"}
{"messages": [{"content": "BP = 1. Point S is obtained by rotating point P by angle pi about point B. Let K be the rotation of point S by angle pi/12 around point P. Let G be the midpoint of segment SK. Let X be the circumcenter of triangle BKG. What is the circumradius of triangle KXP? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.9", "dataset": "MATH"}
{"messages": [{"content": "Let GX = 2. Let H be the line through point G perpendicular to segment GX. Let E be the perpendicular bisector of segment GX. Point W is on line E with |WX| = 5. Point C is the midpoint of segment WX. Point L is the reflection of point G across point C. Let Q be the reflection of line E across line H. Point A is the midpoint of segment GX. Point I is the reflection of point C across line Q. What is the measure of angle LIA, 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": "13.75", "dataset": "MATH"}
{"messages": [{"content": "Let LI = 4. Let Q be the rotation of point L by angle 3pi/4 around point I. Line F is the perpendicular bisector of segment IL. Point H is the midpoint of segment IQ. Point D is the reflection of point H across line F. What is the circumradius of triangle DIH? Express your 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": "IX = 6. Point D is obtained by rotating point I by angle pi/6 about point X. Let T be the reflection of point I across point D. Point H is the midpoint of segment XT. Let A be the midpoint of segment HT. Find the inradius of triangle AXD. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.96", "dataset": "MATH"}
{"messages": [{"content": "Let WD = 8. Point G is the midpoint of segment WD. Point N is obtained by rotating point G by angle pi/2 about point D. Let A be the reflection of point G across point N. Let J be the midpoint of segment WN. Point U is the incenter of triangle AWG. What is the measure of angle UJN, 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": "DU = 4. Point W is obtained by rotating point D by angle 11pi/12 about point U. Let J be the midpoint of segment DW. Let C be the midpoint of segment UW. Let G be the midpoint of segment CJ. What is the inradius of triangle GUW? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.13", "dataset": "MATH"}
{"messages": [{"content": "GT = 1. Let R be the rotation of point T by angle 11pi/12 around point G. Point K is the circumcenter of triangle RGT. Point A is the incenter of triangle RGT. Point S is the midpoint of segment RT. Point J is the midpoint of segment AS. Line F is the perpendicular bisector of segment GT. Point Z is the reflection of point S across point K. Point D is the reflection of point Z across line F. Find 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": "0.98", "dataset": "MATH"}
{"messages": [{"content": "CN = 1. Let F be the rotation of point N by angle 7pi/12 around point C. Point J is the orthocenter of triangle CNF. Let H be the circumcenter of triangle CNF. Let E be the reflection of point N across point C. Let V be the reflection of point J across point N. Find the measure of angle VHE, 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 GD = 8. Let Z be the rotation of point D by angle 7pi/12 around point G. Point I is the midpoint of segment DZ. Let N be the line through point I parallel to segment GD. Let W be the centroid of triangle ZGD. Point V is the reflection of point W across line N. What is the distance between points V 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": "1.62", "dataset": "MATH"}
{"messages": [{"content": "Line X is the perpendicular bisector of segment O17D. Let Z be the line through point T parallel to line X. Point F is the reflection of point T across segment O17D. Point C is the reflection of point T across line Z. Let M be the circumcenter of triangle D. Find the measure of angle MCF, 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": "HS = 2. Let Z be the perpendicular bisector of segment HS. Point K lies on line Z at distance 4 from point S. Point G is the midpoint of segment HK. Point N is the centroid of triangle HKS. Line C is the perpendicular bisector of segment NK. Point F is the reflection of point K across point H. Point D is the reflection of point G across line C. Point Y is the reflection of point F across segment KS. Find 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": "8.13", "dataset": "MATH"}
{"messages": [{"content": "OJ = 9. Let T be the rotation of point O by angle pi/4 around point J. Let H be the reflection of point O across point J. Line F is the angle bisector of angle HOJ. Point Y is the reflection of point T across line F. Find the circumradius of triangle OYT. Express your 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 WZ = 10. Let K be the midpoint of segment ZW. Point O is obtained by rotating point K by angle pi/3 about point Z. Point L is the reflection of point W across point O. Let C be the reflection of point O across point K. Let G be the midpoint of segment LC. Let D be the reflection of point W across point L. Find the inradius of triangle WDG. Express your 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": "EC = 10. Point A is obtained by rotating point C by angle 7pi/12 about point E. Circle W is the incircle of triangle AEC. Let O be the centroid of triangle AEC. Let M be the inversion of point O in circle W. 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": "11.43", "dataset": "MATH"}
{"messages": [{"content": "Let LF = 12. LN = 12. Point S is obtained by rotating point L by angle 2pi/3 about point F. Point Q is the midpoint of segment FL. Let U be the circumcenter of triangle FLN. Point J is the reflection of point S across point U. Point G is the midpoint of segment QJ. What is 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": "7.94", "dataset": "MATH"}
{"messages": [{"content": "HU = 7. Point G is obtained by rotating point H by angle 7pi/12 about point U. Let P be the midpoint of segment HG. Point J is the reflection of point P across segment GU. Point E is the orthocenter of triangle UHG. What is the measure of angle EUJ, 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": "AF = 9. Line S passes through points A and F. Point D is obtained by rotating point A by angle 11pi/12 about point F. Point O is the reflection of point D across line S. Let E be the line through point F perpendicular to line S. Point V is on line E. Point I is the reflection of point D across point A. Point X is the reflection of point F across point D. Point N is the midpoint of segment XO. Find the measure of angle AIN, 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": "15.37", "dataset": "MATH"}
{"messages": [{"content": "Let TB = 9. Point V is obtained by rotating point T by angle pi/12 about point B. Point L is the centroid of triangle BTV. Point Z is the midpoint of segment LB. Point W is the reflection of point L across segment TV. 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": "8.92", "dataset": "MATH"}
{"messages": [{"content": "WX = 9. Point P is obtained by rotating point X by angle 5pi/6 about point W. Line L passes through point W perpendicular to segment WX. Let A be the reflection of point X across point P. Line T is the perpendicular bisector of segment WP. Let E be the reflection of line L across line T. Point F is the reflection of point A across line E. Point Q is the reflection of point F across segment AW. Find the measure of angle XQA, 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": "243.94", "dataset": "MATH"}
{"messages": [{"content": "Let GV = 3. Point H is obtained by rotating point V by angle pi/2 about point G. Point Q is the midpoint of segment VH. Let E be the midpoint of segment VG. Point U is the midpoint of segment HE. What is the length of segment QU? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.75", "dataset": "MATH"}
{"messages": [{"content": "Let ZK = 11. Point M is the reflection of point Z across point K. Let V be the rotation of point M by angle 5pi/6 around point K. Point O is the reflection of point V across point K. Let I be the midpoint of segment VZ. Point F is the reflection of point I across segment VM. What is the distance between points F 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": "2.85", "dataset": "MATH"}
{"messages": [{"content": "TK = 7. Point C is the reflection of point T across point K. Let G be the rotation of point T by angle pi/3 around point C. Let B be the angle bisector of angle GKT. Point W is the reflection of point G across line B. Let H be the line through points T and K. Line N passes through point G parallel to line H. Point E is the reflection of point T across point G. Let M be the reflection of line N through point E. Point D is the reflection of point W across line M. Find the measure of angle TGD, 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": "285.68", "dataset": "MATH"}
{"messages": [{"content": "Let OH = 8. Point Z is the reflection of point H across point O. Let I be the midpoint of segment HO. Let M be the reflection of point O across point Z. Let U be the rotation of point I by angle 7pi/12 around point H. Point R is obtained by rotating point U by angle pi/12 about point H. Let W be the midpoint of segment ZM. Let L be the line through point Z parallel to segment RW. Point A is the intersection of segment OR and line L. Find 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": "3.53", "dataset": "MATH"}
{"messages": [{"content": "VM = 2. Line R passes through points V and M. Let K be the reflection of point M across point V. Point I is obtained by rotating point V by angle pi/6 about point K. Let Z be the midpoint of segment MV. Let G be the reflection of point V across point I. Let O be the reflection of point I across point Z. Let P be the midpoint of segment IK. Point J is the reflection of point O across line R. Point L is the reflection of point J across segment GZ. What is the circumradius of triangle OLJ? Express your 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": "Let FT = 1. Point G is obtained by rotating point F by angle 3pi/4 about point T. Let L be the midpoint of segment TF. Let U be the incenter of triangle FTG. What is the circumradius of triangle ULG? Express your 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 IX = 5. Let N be the rotation of point I by angle pi/3 around point X. Let G be the circumcenter of triangle XIN. Let E be the midpoint of segment XI. Let V be the line through point E perpendicular to segment IN. Point B is the reflection of point I across line V. Point P is the reflection of point G across line V. Find the length of segment BP. Express your 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": "QP = 4. Let O be the rotation of point P by angle 5pi/6 around point Q. Point V is the centroid of triangle OPQ. Point S is the reflection of point V across point O. Let Z be the reflection of point S across point V. 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": "7.73", "dataset": "MATH"}
{"messages": [{"content": "Let WL = 7. Point B is the midpoint of segment WL. Let V be the midpoint of segment WB. Let Q be the angle bisector of angle WVL. Point M is the reflection of point L across point W. Let O be the rotation of point B by angle 7pi/12 around point M. Point P is the reflection of point M across line Q. Point H is the reflection of point P across segment OV. Point F lies on segment OV. Find the measure of angle MHF, 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": "37.99", "dataset": "MATH"}
{"messages": [{"content": "YV = 9. Let K be the vector from point Y to point V. Let X be point V translated by vector K. Let P be the midpoint of segment VX. Let E be the rotation of point P by angle pi/2 around point Y. Let D be the reflection of point V across point E. Let M be the incenter of triangle DPE. Let S be the circumcenter of triangle DPE. Let J be the midpoint of segment ED. What is the measure of angle MJS, 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": "1.15", "dataset": "MATH"}
{"messages": [{"content": "Let GF = 11. Let J be the rotation of point F by angle pi/4 around point G. Point B is the reflection of point G across point J. Point C is the orthocenter of triangle GBF. Point H is the midpoint of segment FG. What is the measure of angle CHB, 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": "101.13", "dataset": "MATH"}
{"messages": [{"content": "Let ZG = 1. Let R be the rotation of point Z by angle pi/6 around point G. Let N be the midpoint of segment ZR. Point P is the midpoint of segment NZ. Point J is the reflection of point G across point Z. Let U be the circumcenter of triangle RPG. What is the length of segment JU? Express your 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 QL = 5. Let K be the rotation of point Q by angle pi/3 around point L. Let Y be the rotation of point K by angle 7pi/12 around point L. Point W is the reflection of point L across segment KY. Let A be the midpoint of segment KW. Let C be the circumcircle of triangle YQL. Point X is the inversion of point K with respect to circle C. What is the length of segment XA? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.27", "dataset": "MATH"}
{"messages": [{"content": "IR = 2. Let Y be the line through points R and I. Let V be the line through point R perpendicular to line Y. Point Z is on line V. Point X is the midpoint of segment IZ. Let H be the midpoint of segment IR. Point N is the centroid of triangle HXZ. 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": "0.33", "dataset": "MATH"}
{"messages": [{"content": "Let RN = 9. Let H be the rotation of point R by angle pi/2 around point N. Point U is the midpoint of segment HN. Let Y be the orthocenter of triangle RUH. Let M be the circumcircle of triangle RUH. Line B passes through point R perpendicular to segment HR. Point S is the reflection of point N across line B. Let L be the inversion of point N in circle M. What is the measure of angle SYL, 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": "54.0", "dataset": "MATH"}
{"messages": [{"content": "DV = 6. Let M be the reflection of point V across point D. Let A be the perpendicular bisector of segment DM. Let H be the vector from point M to point V. Point P is on line A with |PD| = 10. Let W be point P translated by vector H. Point U is the midpoint of segment VP. Let O be the line through point P perpendicular to segment VP. Point F is the reflection of point U across line O. Let S be the reflection of point D across point W. Find the measure of angle SDF, 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": "210.0", "dataset": "MATH"}
{"messages": [{"content": "Let MP = 9. Let K be the rotation of point P by angle 3pi/4 around point M. Point N is obtained by rotating point P by angle 2pi/3 about point K. Point Q is obtained by rotating point N by angle 5pi/12 about point P. Let O be the rotation of point M by angle pi/6 around point N. Point Y is the midpoint of segment OQ. Let X be the circumcenter of triangle OQP. Find the circumradius of triangle YNX. Express your 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": "Let NQ = 3. Point H is obtained by rotating point N by angle pi/12 about point Q. Point E is the reflection of point Q across segment NH. Point F is the incenter of triangle NHQ. Let P be the reflection of point E across point H. Point W is the reflection of point P across point F. Let Z be the orthocenter of triangle NEW. Find the measure of angle FWZ, 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": "2.31", "dataset": "MATH"}
{"messages": [{"content": "Let IY = 12. Let O be the rotation of point Y by angle pi/3 around point I. Point B is the midpoint of segment OI. Point J is the centroid of triangle BOY. Point M is the midpoint of segment OY. Find the measure of angle IMJ, 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": "LO = 1. Point F is the reflection of point O across point L. Point P is obtained by rotating point F by angle 2pi/3 about point L. Let W be the rotation of point L by angle pi/2 around point F. Point T is the midpoint of segment FL. Point U is the reflection of point W across point O. Point M is the reflection of point W across point T. Point J is the reflection of point U across point M. Point Q is the incenter of triangle PJF. Find the measure of angle QOU, 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": "1.5", "dataset": "MATH"}
{"messages": [{"content": "XQ = 4. Point H is the reflection of point X across point Q. Let Y be the rotation of point H by angle pi/12 around point X. Let M be the midpoint of segment XQ. Point C is the reflection of point H across segment YX. What is the inradius of triangle MYC? Express your 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": "Let IV = 1. Let C be the rotation of point I by angle 5pi/6 around point V. Let H be the orthocenter of triangle IVC. Point B is the reflection of point C across point V. Let R be the midpoint of segment CI. Point Z is the reflection of point B across point R. 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": "4.23", "dataset": "MATH"}
{"messages": [{"content": "QF = 7. Let Y be the vector from point Q to point F. Point E is the translation of point F by vector Y. Let M be the rotation of point F by angle 5pi/12 around point Q. Line L is the perpendicular bisector of segment MF. Point U is the reflection of point E across line L. Find the circumradius of triangle UEM. Express your 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": "NB = 9. Let J be the rotation of point B by angle pi/12 around point N. Point H is the incenter of triangle NBJ. Point E is on segment BJ. Point I is the midpoint of segment HE. Point W is the midpoint of segment EN. What is the distance between points I 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": "3.95", "dataset": "MATH"}
{"messages": [{"content": "FQ = 8. Point E is obtained by rotating point F by angle 5pi/6 about point Q. Let H be the reflection of point Q across point F. Point N is the reflection of point H across segment EQ. Let R be the midpoint of segment EN. Let B be the midpoint of segment EQ. Line O passes through point N perpendicular to segment EQ. Point J is the reflection of point E across line O. Let A be the incircle of triangle BRN. Let P be the inversion of point R in circle A. 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": "33.03", "dataset": "MATH"}
{"messages": [{"content": "QH = 1. Let N be the rotation of point Q by angle pi/2 around point H. Point V is the midpoint of segment HN. Let G be the circumcenter of triangle HVQ. Point F is the incenter of triangle HVQ. Find the measure of angle QFG, 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": "0.0", "dataset": "MATH"}
{"messages": [{"content": "Let ZP = 9. Let B be the rotation of point Z by angle 5pi/6 around point P. Point W is the orthocenter of triangle BPZ. Point S is the centroid of triangle BPZ. Point Y is the incenter of triangle BPZ. Point U is the reflection of point S across segment ZB. Point J is the midpoint of segment YU. Find the measure of angle JWZ, 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 WM = 7. Point K is obtained by rotating point W by angle 2pi/3 about point M. Point A is the reflection of point M across point K. Point L is the reflection of point A across segment MW. Let N be the incenter of triangle ALW. Find the distance between points L 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": "13.33", "dataset": "MATH"}
{"messages": [{"content": "Let EX = 1. Point G is obtained by rotating point E by angle 3pi/4 about point X. Let K be the reflection of point E across point G. Point M is the orthocenter of triangle XKG. Point B is the reflection of point E across point M. What is the distance between points B 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": "13.51", "dataset": "MATH"}
{"messages": [{"content": "Let ME = 11. Let Y be the rotation of point M by angle 11pi/12 around point E. Point T is the circumcenter of triangle MYE. Let I be the orthocenter of triangle MYE. Point J is the reflection of point T across segment EM. Point O is the midpoint of segment YE. Find the measure of angle OJI, 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": "229.59", "dataset": "MATH"}
{"messages": [{"content": "WG = 10. Point Y is obtained by rotating point G by angle 7pi/12 about point W. Let I be the rotation of point W by angle 3pi/4 around point G. Point A is the reflection of point G across point I. Point O is the reflection of point A across segment GW. Let H be the midpoint of segment YI. Let X be the circumcenter of triangle OYI. Point Q is the midpoint of segment XY. What is the measure of angle QHA, 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": "37.25", "dataset": "MATH"}
{"messages": [{"content": "Let AF = 12. Let L be the rotation of point A by angle pi/4 around point F. Let J be the line through points L and F. Point C is the midpoint of segment AF. Line H is the reflection of line J across point A. Point R is the reflection of point L across line H. Point Y is the circumcenter of triangle LAC. Let U be the circumcircle of triangle LAC. Let W be the inversion of point Y in circle U. What is the measure of angle WRL, 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": "401.88", "dataset": "MATH"}
{"messages": [{"content": "Let PR = 8. Let C be the rotation of point R by angle 5pi/6 around point P. Point W is the midpoint of segment CR. Point T is the midpoint of segment PC. Point M is the reflection of point T across segment PW. What is the length of segment MP? Express your 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 DU = 12. Point B is the reflection of point U across point D. Let Z be the rotation of point D by angle pi/12 around point B. Point S is the centroid of triangle ZUB. Let X be the incenter of triangle ZUB. Point W is the midpoint of segment XU. Find the measure of angle WSX, 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": "1.43", "dataset": "MATH"}
{"messages": [{"content": "Let HE = 9. Let L be the rotation of point H by angle pi/4 around point E. Let V be the reflection of point E across point L. Point D is obtained by rotating point L by angle pi about point H. Let G be the midpoint of segment DV. Let P be the incenter of triangle LED. Let C be the reflection of point G across point H. Find the measure of angle HCP, 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": "7.17", "dataset": "MATH"}
{"messages": [{"content": "Let AJ = 4. Line Z is the perpendicular bisector of segment JA. Point W lies on line Z at distance 8 from point J. Point F is the midpoint of segment AW. Point E is the reflection of point W across point J. Point S is the orthocenter of triangle FWE. 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": "20.55", "dataset": "MATH"}
{"messages": [{"content": "Let GP = 5. Line L is the perpendicular bisector of segment GP. Let A be the line through points G and P. Let H be the line through point G perpendicular to line A. Point T is the reflection of point P across point G. Let O be the rotation of point G by angle 5pi/6 around point P. Line E is the reflection of line H across line L. Let F be the rotation of point P by angle 3pi/4 around point G. Point C is the reflection of point G across line E. Point V is the reflection of point O across segment GF. Let N be the midpoint of segment CV. 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": "12.18", "dataset": "MATH"}
{"messages": [{"content": "Let FR = 4. Let P be the vector from point R to point F. Point H is obtained by rotating point R by angle pi/4 about point F. Let Q be the angle bisector of angle HRF. Let Y be point H translated by vector P. Let K be the line through points H and Y. Line D is the reflection of line Q across point F. Let S be point F translated by vector P. Point Z is the reflection of point R across line K. Let E be the line through point Z parallel to line D. Point G is the midpoint of segment SF. Circle L is the circumcircle of triangle SFH. Let T be the reflection of circle L across line E. Point B is the inversion of point G with respect to circle T. What is the measure of angle YBF, 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": "7.9", "dataset": "MATH"}
{"messages": [{"content": "OE = 11. L is placed 3 units away from E. Point Q is obtained by rotating point E by angle pi/3 about point L. Point U is obtained by rotating point L by angle 2pi/3 about point Q. Line G is the perpendicular bisector of segment EL. Point Z is the reflection of point U across line G. Let H be the reflection of point Z across point E. Let J be the midpoint of segment ZQ. Point F is the reflection of point J across segment EL. 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": "7.5", "dataset": "MATH"}
{"messages": [{"content": "Let X be the orthocenter of triangle B. Point F is the reflection of point Y across point B. Point D is the midpoint of segment XF. Point N is the reflection of point B across point D. What is the distance between points Y 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": "1.15", "dataset": "MATH"}
{"messages": [{"content": "Let LU = 2. Let J be the midpoint of segment UL. Point G is obtained by rotating point J by angle 11pi/12 about point L. Let H be the perpendicular bisector of segment GU. Line N is the reflection of line H across point G. Point Z is the reflection of point G across line N. Point M is the reflection of point Z across line H. Point V is the reflection of point L across point M. Let A be the midpoint of segment GL. Find the measure of angle UVA, 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": "0.13", "dataset": "MATH"}
{"messages": [{"content": "Let LC = 5. Point Q is the midpoint of segment LC. Let D be the rotation of point C by angle pi/3 around point Q. Let S be the reflection of point Q across point D. Let W be the circumcenter of triangle DCS. Let Y be the circumcircle of triangle DCS. Let T be the reflection of circle Y through point L. Let F be the inversion of point S in circle T. Find the measure of angle WFQ, 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": "7.02", "dataset": "MATH"}
{"messages": [{"content": "KI = 11. Point A is obtained by rotating point I by angle 11pi/12 about point K. Let V be the reflection of point I across point A. Point F is the reflection of point A across point I. Point X is the orthocenter of triangle AKI. Let O be the perpendicular bisector of segment AX. Point D is the reflection of point V across line O. What is the distance between points D 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": "103.55", "dataset": "MATH"}
{"messages": [{"content": "Let HI = 2. Let V be the reflection of point I across point H. Let E be the reflection of point H across point I. Point J is obtained by rotating point V by angle 5pi/12 about point H. Point B is obtained by rotating point J by angle pi/3 about point V. Point C is the reflection of point J across point I. Let S be the incenter of triangle BCE. Circle M is the circumcircle of triangle BCE. Let Y be the inversion of point S in circle M. 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": "2.6", "dataset": "MATH"}
{"messages": [{"content": "WZ = 3. Point Q is obtained by rotating point Z by angle pi/2 about point W. Point L is the incenter of triangle QWZ. Circle E is the circumcircle of triangle QWZ. Let N be the reflection of point L across point Q. Let H be the inversion of point N in circle E. Find 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": "3.29", "dataset": "MATH"}
{"messages": [{"content": "Let BP = 11. Let D be the rotation of point P by angle 11pi/12 around point B. Let A be the midpoint of segment PD. Circle G is the incircle of triangle DAB. Let C be the circumcenter of triangle DAB. Point E is the inversion of point P with respect to circle G. Line M passes through points E and A. Point F is the intersection of line M and segment BD. Let L be the midpoint of segment CA. 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": "1.2", "dataset": "MATH"}
{"messages": [{"content": "Let RK = 10. Let S be the rotation of point R by angle 7pi/12 around point K. Let J be the incenter of triangle SKR. Let A be the centroid of triangle SKR. Point H is the midpoint of segment JR. Find the measure of angle RAH, 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": "1.32", "dataset": "MATH"}
{"messages": [{"content": "Let EO = 5. Let L be the rotation of point O by angle pi/3 around point E. Let F be the reflection of point O across point L. Point Q is obtained by rotating point E by angle 7pi/12 about point O. Line A is the perpendicular bisector of segment OF. Point I is the reflection of point Q across line A. Point Z is the midpoint of segment IQ. What is the inradius of triangle ZFQ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.43", "dataset": "MATH"}
{"messages": [{"content": "QN = 4. Point M is the midpoint of segment QN. Point G is obtained by rotating point Q by angle 2pi/3 about point M. Let H be the rotation of point N by angle 3pi/4 around point M. Point Z is the midpoint of segment MQ. Point J is the reflection of point G across segment MH. What is the inradius of triangle ZMJ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.17", "dataset": "MATH"}
{"messages": [{"content": "Let VO = 1. Line U passes through points O and V. Let X be the rotation of point V by angle pi/6 around point O. Line H passes through point X perpendicular to line U. Let F be the rotation of point X by angle 2pi/3 around point V. Let J be the line through point F perpendicular to line H. Point W is the reflection of point X across line J. Let M be the orthocenter of triangle OFW. What is the measure of angle FXM, 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": "0.43", "dataset": "MATH"}
{"messages": [{"content": "XV = 10. Point W is obtained by rotating point X by angle pi/6 about point V. Point R is the midpoint of segment WX. Let C be the orthocenter of triangle WXV. Point F is the reflection of point R across segment XV. Find the distance between points F 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": "4.87", "dataset": "MATH"}
{"messages": [{"content": "YE = 2. Line N is the perpendicular bisector of segment EY. Let P be the reflection of point E across point Y. Point S lies on line N at distance 1 from point E. Point D is the midpoint of segment ES. Let L be the reflection of point E across point P. Point R is the midpoint of segment LD. Let J be the midpoint of segment DR. Find 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": "0.25", "dataset": "MATH"}
{"messages": [{"content": "Let BV = 7. Let S be the rotation of point V by angle 5pi/12 around point B. Let M be the rotation of point B by angle 2pi/3 around point V. Point C is the centroid of triangle MBS. Point W is the orthocenter of triangle MBS. Point Q is the incenter of triangle MBS. Line H is the perpendicular bisector of segment CQ. Point J is the reflection of point M across line H. Find 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": "7.34", "dataset": "MATH"}
{"messages": [{"content": "PB = 11. Line D is the perpendicular bisector of segment BP. Let Y be the midpoint of segment PB. Point V lies on line D at distance 5 from point Y. Let C be the reflection of line D through point B. Point U is the reflection of point P across line C. Let T be the midpoint of segment PU. Let J be the centroid of triangle PUV. 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": "4.03", "dataset": "MATH"}
{"messages": [{"content": "Let QY = 7. Point E is the reflection of point Q across point Y. Point R is the reflection of point E across point Q. Line I is the perpendicular bisector of segment RQ. Point Z lies on line I at distance 9 from point Q. Let T be the reflection of point Z across point E. Point A is the midpoint of segment YR. Let S be the midpoint of segment QT. Point F is the midpoint of segment QA. Find the measure of angle ZFS, 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": "74.81", "dataset": "MATH"}
{"messages": [{"content": "FI = 10. Let V be the reflection of point I across point F. Let Y be the reflection of point F across point V. Let W be the rotation of point Y by angle 3pi/4 around point V. Point T is the reflection of point F across point Y. Let X be the midpoint of segment TW. Let S be the reflection of point F across point T. What is the inradius of triangle WSX? Express your 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": "VR = 10. Let W be the line through point R perpendicular to segment RV. Point Z is obtained by rotating point R by angle pi/4 about point V. Let P be the midpoint of segment ZR. Line A passes through point P parallel to line W. Point D is the midpoint of segment RV. Point H is the midpoint of segment ZD. Point K is on segment RV. Point U is the reflection of point K across line A. What is the measure of angle UHP, 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": "4.42", "dataset": "MATH"}
{"messages": [{"content": "LS = 9. Let W be the rotation of point L by angle pi/2 around point S. Let M be the reflection of point W across point L. Let T be the centroid of triangle WLS. Point A is the reflection of point T across segment LS. What is the distance between points A 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": "16.16", "dataset": "MATH"}
{"messages": [{"content": "GP = 8. Point D is obtained by rotating point P by angle pi/12 about point G. Line J passes through points D and G. Point C is the reflection of point D across segment GP. Point W is the reflection of point P across line J. Let N be the midpoint of segment DP. What is the inradius of triangle CNG? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.28", "dataset": "MATH"}
{"messages": [{"content": "Point H is the midpoint of segment TI24. Let N be the midpoint of segment GH. Point I is the reflection of point G across point L. Point R is the midpoint of segment HI. 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": "1.0", "dataset": "MATH"}
{"messages": [{"content": "Let HW = 5. Let R be the rotation of point W by angle pi/3 around point H. Let X be the rotation of point R by angle 11pi/12 around point H. Point B is the midpoint of segment WX. Let M be the midpoint of segment XR. Point G is the reflection of point B across segment HX. Let P be the incenter of triangle HXM. Let L be the midpoint of segment GR. What is the measure of angle XPL, 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": "2.49", "dataset": "MATH"}
{"messages": [{"content": "LS = 4. Let G be the midpoint of segment LS. Line X passes through point G perpendicular to segment LS. Let Z be the reflection of point S across point L. Let O be the reflection of point G across point S. Let H be the line through point S parallel to line X. Line T is the reflection of line X across line H. Point A is obtained by rotating point S by angle 5pi/12 about point O. Let B be the angle bisector of angle OAZ. Line Y is the reflection of line T across point G. Let J be the rotation of point G by angle pi/4 around point L. Point D is the reflection of point A across segment LS. Point K is the reflection of point L across line B. Let E be the line through point J perpendicular to line Y. Line W is the angle bisector of angle DSO. Point Q is the reflection of point K across line E. Point F is the reflection of point D across line W. Let I be the reflection of point F across point K. Find the measure of angle FIQ, 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": "1.13", "dataset": "MATH"}
{"messages": [{"content": "Let OZ = 11. Line S is the perpendicular bisector of segment OZ. Point W is on line S with |WO| = 11. Point K is the circumcenter of triangle ZOW. Point J is the incenter of triangle ZOW. Point P is the reflection of point J across segment WZ. Find 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": "6.68", "dataset": "MATH"}
{"messages": [{"content": "Let NI = 2. Let A be the rotation of point I by angle pi/2 around point N. Point H is the midpoint of segment NA. Point K is the reflection of point H across point I. Let F be the perpendicular bisector of segment KH. Point R is the reflection of point N across line F. Point X is the centroid of triangle IHR. What is 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": "1.74", "dataset": "MATH"}
{"messages": [{"content": "CT = 9. Let V be the reflection of point C across point T. Let K be the midpoint of segment VT. Let S be the rotation of point V by angle 5pi/12 around point T. Point G is the orthocenter of triangle SKV. Let H be the midpoint of segment KV. Point L is the reflection of point G across point C. Let M be the reflection of point T across point L. 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": "47.53", "dataset": "MATH"}
{"messages": [{"content": "Let SD = 8. Line Z is the perpendicular bisector of segment DS. Point E is on line Z with |ES| = 3. Line B passes through point S parallel to segment DE. Point C is the reflection of point D across line B. Point Q is the midpoint of segment ES. 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": "9.18", "dataset": "MATH"}
{"messages": [{"content": "Let DP = 11. Point K is obtained by rotating point P by angle pi/6 about point D. Point Y is the reflection of point D across point K. Circle F is the incircle of triangle PKY. Let N be the inversion of point P in circle F. 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": "12.26", "dataset": "MATH"}
{"messages": [{"content": "Let HV = 3. Point R is obtained by rotating point V by angle pi/2 about point H. Point Z is obtained by rotating point V by angle pi/6 about point R. Let U be the midpoint of segment VZ. Point I is the reflection of point H across segment ZR. Let L be the midpoint of segment IZ. Find the measure of angle RLU, 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": "4.2", "dataset": "MATH"}
{"messages": [{"content": "IO = 9. Point P is obtained by rotating point O by angle 2pi/3 about point I. Point X is the circumcenter of triangle PIO. Point U is the incenter of triangle PIO. Point V is the reflection of point P across point U. Circle D is the circumcircle of triangle PIO. Let C be the inversion of point V in circle D. Find 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": "4.72", "dataset": "MATH"}
{"messages": [{"content": "Point Y is the midpoint of segment IA. Let L be the angle bisector of angle UIY. Point M is the circumcenter of triangle I. Point W is the midpoint of segment MU. Point O is the reflection of point W across point Y. Point X is the reflection of point O across line L. 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": "1.23", "dataset": "MATH"}
{"messages": [{"content": "EX = 4. Let B be the midpoint of segment EX. Line S is the perpendicular bisector of segment BX. Line M is the reflection of line S across point X. Point N is on line M with |NE| = 7. Point R is the reflection of point X across point N. Point D is the reflection of point E across segment NB. Find the inradius of triangle XRD. Express your 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": "FQ = 6. Point K is the midpoint of segment QF. Let U be the rotation of point K by angle pi/12 around point F. Point Z is obtained by rotating point Q by angle 3pi/4 about point F. Let A be the reflection of point Q across point Z. Point B is the reflection of point A across segment UZ. Find the circumradius of triangle FAB. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.5", "dataset": "MATH"}
{"messages": [{"content": "Let RV = 1. Let N be the perpendicular bisector of segment RV. Point C lies on line N at distance 10 from point V. Point P is the reflection of point C across point R. Let M be the midpoint of segment RP. Point T is the midpoint of segment PV. Let I be the incenter of triangle MPV. Point K is the reflection of point T across segment RV. Point X is the midpoint of segment KR. What is 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": "7.54", "dataset": "MATH"}
{"messages": [{"content": "Let TR = 4. Let W be the rotation of point T by angle pi around point R. Let X be the reflection of point R across point T. Let U be the midpoint of segment RW. Let V be the vector from point U to point R. Point M is obtained by rotating point X by angle pi/12 about point U. Let E be point X translated by vector V. Let J be point E translated by vector V. Point A is obtained by rotating point J by angle pi/12 about point E. Circle N is the incircle of triangle AEM. Point D is the midpoint of segment EM. Let Z be the inversion of point D in circle N. Find the length of segment ZA. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.97", "dataset": "MATH"}
{"messages": [{"content": "CY = 6. Point S is obtained by rotating point C by angle 11pi/12 about point Y. Point H is obtained by rotating point C by angle 2pi/3 about point S. Point F is the orthocenter of triangle SCH. Let M be the line through point Y perpendicular to segment SC. Line Z passes through point S parallel to line M. Point W is the reflection of point F across line Z. Find 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": "11.9", "dataset": "MATH"}
{"messages": [{"content": "QS = 3. Let H be the rotation of point S by angle 3pi/4 around point Q. Let V be the reflection of point S across point H. Let A be the incenter of triangle HVQ. Let K be the midpoint of segment QH. Find 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": "1.47", "dataset": "MATH"}
{"messages": [{"content": "WD = 12. Let P be the rotation of point D by angle 2pi/3 around point W. Line H is the angle bisector of angle DPW. Point T is the reflection of point W across line H. Let Y be the midpoint of segment DW. What is the inradius of triangle PTY? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.11", "dataset": "MATH"}
{"messages": [{"content": "AX = 3. Point J is the reflection of point A across point X. Line N is the angle bisector of angle AJX. Let U be the rotation of point J by angle pi/4 around point X. Point H is the reflection of point U across line N. Point O is the midpoint of segment HU. Point I is the reflection of point J across point O. Point G is the midpoint of segment HJ. Find the measure of angle IUG, 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": "1.86", "dataset": "MATH"}
{"messages": [{"content": "Let IT = 11. Point C is the midpoint of segment TI. Let B be the rotation of point C by angle pi/12 around point I. Let H be the rotation of point T by angle 11pi/12 around point B. Let P be the rotation of point B by angle 11pi/12 around point C. Let W be the angle bisector of angle CPH. Point L is the reflection of point T across line W. Point F is the reflection of point H across point T. Line Z is the angle bisector of angle IFB. Point Q is the reflection of point H across line Z. Point R is the reflection of point Q across line W. What is the measure of angle LRC, 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": "ZF = 4. Let X be the reflection of point F across point Z. Point O is obtained by rotating point Z by angle pi/3 about point F. Point E is the centroid of triangle OXZ. Let C be the perpendicular bisector of segment EO. Point K is the reflection of point Z across line C. 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": "1.33", "dataset": "MATH"}
{"messages": [{"content": "Point R is the orthocenter of triangle J. Let Y be the circumcircle of triangle J. Circle T is the reflection of circle Y across point P. Let G be the inversion of point P in circle T. 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": "0.95", "dataset": "MATH"}
{"messages": [{"content": "Let TY = 1. Point U is the midpoint of segment YT. Let H be the rotation of point U by angle pi/12 around point Y. Point B is obtained by rotating point Y by angle pi/6 about point H. Let G be the rotation of point T by angle pi/2 around point U. Point L is the centroid of triangle TYB. Let J be the midpoint of segment BU. Let O be the orthocenter of triangle LUG. Find the circumradius of triangle JGO. Express your 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": "Let MV = 8. Let S be the rotation of point M by angle pi/6 around point V. Point J is obtained by rotating point V by angle pi about point M. Line N is the perpendicular bisector of segment VS. Point U is the reflection of point M across line N. Let P be the centroid of triangle MSU. What is 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": "10.05", "dataset": "MATH"}
{"messages": [{"content": "UR = 6. Point E is obtained by rotating point U by angle 5pi/12 about point R. Let G be the rotation of point U by angle pi/4 around point R. Point F is the reflection of point G across point E. Let W be the reflection of point U across point E. Let M be the orthocenter of triangle GFR. 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": "9.23", "dataset": "MATH"}
{"messages": [{"content": "Let DZ = 8. Let K be the rotation of point Z by angle 3pi/4 around point D. Let P be the reflection of point Z across point K. Let I be the reflection of point K across point D. Line A passes through point Z parallel to segment KI. Line B passes through point P parallel to segment IZ. Let O be the reflection of line B across line A. Line W is the reflection of line A across line O. Let N be the midpoint of segment IZ. Point T is the reflection of point N across line W. 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": "21.94", "dataset": "MATH"}
{"messages": [{"content": "BH = 2. Let F be the rotation of point B by angle 11pi/12 around point H. Let Y be the rotation of point H by angle 11pi/12 around point B. Point G is the reflection of point Y across point B. Point E is the midpoint of segment HF. Point U is the reflection of point G across segment HF. Point I is the orthocenter of triangle EUB. Let X be the centroid of triangle EUB. Find 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": "2.46", "dataset": "MATH"}
{"messages": [{"content": "MF = 6. Let K be the reflection of point F across point M. Line G passes through point K perpendicular to segment FM. Point Q lies on line G. S is constructed so that QS = 9. Point O lies on line G. Point E is the reflection of point O across segment FM. Let V be the midpoint of segment SE. Let T be the midpoint of segment EQ. What is 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": "4.5", "dataset": "MATH"}
{"messages": [{"content": "NX = 1. Let D be the rotation of point X by angle 3pi/4 around point N. Point P is obtained by rotating point X by angle pi/4 about point N. Let I be the midpoint of segment XD. Point M is the circumcenter of triangle NIP. Point B is the orthocenter of triangle NIP. 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": "1.51", "dataset": "MATH"}
{"messages": [{"content": "Let PM = 9. Point H is obtained by rotating point M by angle 2pi/3 about point P. Point W is the reflection of point H across point M. Point E is the incenter of triangle HMP. Point X is the reflection of point W across segment PH. What is 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": "26.1", "dataset": "MATH"}
{"messages": [{"content": "Let MA = 11. Let R be the vector from point A to point M. Point U is obtained by rotating point A by angle 7pi/12 about point M. Point D is the translation of point M by vector R. Let K be the line through point U perpendicular to segment UM. Let Z be the circumcenter of triangle AUM. Let Y be the line through point Z parallel to line K. Point S is the reflection of point D across line Y. What is the measure of angle MDS, 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": "XW = 3. Point J is the reflection of point X across point W. Let H be the midpoint of segment WX. Let S be the midpoint of segment HJ. Point F is obtained by rotating point X by angle pi/2 about point W. Let Q be the angle bisector of angle HFS. Let G be the reflection of point F across point X. Line N passes through point X parallel to line Q. Line N and segment HG intersect at point B. What is the distance between points B 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": "6.21", "dataset": "MATH"}
{"messages": [{"content": "Let VQ = 4. Let J be the perpendicular bisector of segment QV. Point W lies on line J. Let S be the midpoint of segment QW. Let X be the line through point Q parallel to line J. Point L is the reflection of point V across line X. Point D is the reflection of point V across point S. Point Z is the reflection of point W across segment LD. Find 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": "4.0", "dataset": "MATH"}
{"messages": [{"content": "Let HD = 7. Point V is obtained by rotating point D by angle pi/6 about point H. Point E is obtained by rotating point D by angle 5pi/12 about point V. Let I be the reflection of point H across point V. Let M be the circumcenter of triangle HDE. Point S is the midpoint of segment HD. What is the circumradius of triangle IMS? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26.43", "dataset": "MATH"}
{"messages": [{"content": "Let RW = 11. Point E is the midpoint of segment RW. Let Q be the rotation of point E by angle 2pi/3 around point R. Point L is the orthocenter of triangle WEQ. Let Y be the reflection of point R across point L. Let O be the midpoint of segment LW. Let P be the centroid of triangle WYL. 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": "12.33", "dataset": "MATH"}
{"messages": [{"content": "IG = 1. Let N be the reflection of point G across point I. Let R be the line through points G and I. Let Q be the rotation of point N by angle pi around point G. Let J be the reflection of point I across point G. Point C is obtained by rotating point Q by angle 5pi/12 about point J. Line H is the reflection of line R across point C. Let S be the reflection of line H through point N. Let L be the midpoint of segment IN. Point V is the reflection of point L across line H. Point D is the reflection of point V across line S. Point Z is the midpoint of segment CG. Find the measure of angle JDZ, 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": "2.04", "dataset": "MATH"}
{"messages": [{"content": "Let IL = 4. Point R is the reflection of point L across point I. Let G be the rotation of point R by angle pi/2 around point L. Point P is the centroid of triangle IGL. Point V is the incenter of triangle IGL. Point J is the reflection of point V across point I. What is the measure of angle JVP, 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": "2.52", "dataset": "MATH"}
{"messages": [{"content": "Let GB = 6. Point F is the midpoint of segment BG. Line Z is the perpendicular bisector of segment FB. Point U is the reflection of point F across point G. Point A is obtained by rotating point F by angle pi/12 about point U. Point H is the reflection of point U across line Z. Point Q is the reflection of point A across point H. Let S be the circumcenter of triangle GQA. 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": "5.43", "dataset": "MATH"}
{"messages": [{"content": "BT = 1. Let D be the reflection of point T across point B. Point K is the midpoint of segment BD. Point X is obtained by rotating point K by angle 2pi/3 about point B. Let Q be the reflection of point X across point T. Point O is the orthocenter of triangle QXD. Let W be the midpoint of segment DQ. Point P is the circumcenter of triangle QXD. Find the measure of angle OPW, 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": "0.09", "dataset": "MATH"}
{"messages": [{"content": "HD = 7. Let P be the line through points H and D. Let K be the rotation of point H by angle pi/4 around point D. Line Y passes through point K perpendicular to line P. Point F is obtained by rotating point H by angle 5pi/12 about point K. Point V is the reflection of point D across line Y. Point W is the midpoint of segment KV. Let U be the midpoint of segment WF. Point B is the reflection of point V across segment UF. What is the measure of angle BFK, 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": "0.69", "dataset": "MATH"}
{"messages": [{"content": "VN = 5. Point C is the midpoint of segment VN. Let R be the rotation of point C by angle pi/4 around point V. Point O is the reflection of point N across segment RC. Point U is the reflection of point R across point C. Find the inradius of triangle OUN. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.46", "dataset": "MATH"}
{"messages": [{"content": "Let IB = 5. Point U is obtained by rotating point I by angle 5pi/6 about point B. Let V be the centroid of triangle UIB. Point S is the midpoint of segment IB. Let M be the incircle of triangle UIB. Point E is the inversion of point S with respect to circle M. 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": "0.26", "dataset": "MATH"}
{"messages": [{"content": "Let OQ = 5. Point K is the midpoint of segment OQ. Let R be the rotation of point K by angle 5pi/6 around point O. Let W be the rotation of point O by angle 5pi/12 around point R. Let Y be the incircle of triangle WOR. Let E be the midpoint of segment OR. Point J is the reflection of point K across segment OR. Let I be the inversion of point J in circle Y. Let G be the circumcenter of triangle WOR. Find the measure of angle EIG, 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": "0.01", "dataset": "MATH"}
{"messages": [{"content": "Let QF = 5. Z is placed 4 units away from F. Point P is obtained by rotating point F by angle 11pi/12 about point Q. Let U be the midpoint of segment PF. Let O be the midpoint of segment QZ. Point Y is the midpoint of segment ZU. What is 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": "0.33", "dataset": "MATH"}
{"messages": [{"content": "BV = 1. Point E is the midpoint of segment BV. Let W be the rotation of point E by angle pi/2 around point B. Point K is obtained by rotating point V by angle 5pi/12 about point E. Circle G is the incircle of triangle KBW. Point F is the inversion of point B with respect to circle G. Let T be the incenter of triangle KBW. Let N be the circumcenter of triangle FKE. Find the length of segment TN. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.27", "dataset": "MATH"}
{"messages": [{"content": "JN = 12. Let Y be the rotation of point N by angle 7pi/12 around point J. Let Z be the reflection of point Y across point J. Let G be the centroid of triangle YJN. Let H be the midpoint of segment NG. Point B is the midpoint of segment JG. What is the inradius of triangle ZHB? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.4", "dataset": "MATH"}
{"messages": [{"content": "Let RH = 9. Point W is the midpoint of segment RH. Point B is obtained by rotating point W by angle pi/4 about point R. Line X is the angle bisector of angle WBR. Let G be the reflection of line X through point R. Point J is the reflection of point W across line G. Point U is the reflection of point R across line G. Find the circumradius of triangle UHJ. Express your 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": "MP = 12. Let H be the line through points P and M. Let T be the rotation of point M by angle pi/3 around point P. Point L is the reflection of point T across line H. Let V be the reflection of point M across point L. Let Z be the orthocenter of triangle MTV. 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": "34.64", "dataset": "MATH"}
{"messages": [{"content": "RK = 9. Let A be the rotation of point R by angle pi/4 around point K. Let W be the line through points A and R. Point M is obtained by rotating point K by angle 11pi/12 about point R. Point P is the reflection of point M across line W. Point X is the reflection of point A across segment RK. Let J be the orthocenter of triangle MPK. What is the inradius of triangle PXJ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.43", "dataset": "MATH"}
{"messages": [{"content": "Let WZ = 4. Let O be the reflection of point Z across point W. Let K be the rotation of point Z by angle 7pi/12 around point W. Let X be the perpendicular bisector of segment OW. Let D be the circumcenter of triangle KWO. Point A is the reflection of point D across line X. Let E be the midpoint of segment AO. What is the measure of angle ZEA, 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": "2.58", "dataset": "MATH"}
{"messages": [{"content": "Let EB = 12. Line U is the perpendicular bisector of segment EB. Point T is obtained by rotating point B by angle 11pi/12 about point E. Let A be the line through point B parallel to line U. Line P is the reflection of line A across line U. Point D is the reflection of point B across line P. Point Z is the circumcenter of triangle ETB. 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": "12.18", "dataset": "MATH"}
{"messages": [{"content": "Let EH = 7. Point M is obtained by rotating point E by angle 2pi/3 about point H. Point J is the reflection of point H across point E. Let O be the perpendicular bisector of segment HM. Point B is the reflection of point J across line O. Let W be the reflection of point J across point B. Find the measure of angle MWH, 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": "42.44", "dataset": "MATH"}
{"messages": [{"content": "IY = 7. Point H is obtained by rotating point I by angle 5pi/12 about point Y. Point C is the midpoint of segment IH. Let U be the midpoint of segment YI. Point L is the centroid of triangle HYI. Let N be the midpoint of segment LU. Point Z is the reflection of point Y across point N. 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": "2.32", "dataset": "MATH"}
{"messages": [{"content": "QL = 3. Let S be the rotation of point Q by angle pi/3 around point L. I is constructed so that LI = 12. Point K is the midpoint of segment IQ. Point U is the midpoint of segment KL. Let F be the circumcenter of triangle KUS. Find the length of segment FU. Express your 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 FX = 9. Let T be the rotation of point X by angle pi/2 around point F. Let C be the circumcenter of triangle XFT. Let Y be the midpoint of segment XC. Point J is the reflection of point T across point X. What is the distance between points Y 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": "15.95", "dataset": "MATH"}
{"messages": [{"content": "AT = 8. Let S be the perpendicular bisector of segment AT. Point K lies on line S at distance 12 from point A. Point Y is the incenter of triangle KAT. Let W be the circumcenter of triangle KAT. Point F is the midpoint of segment AY. 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": "3.25", "dataset": "MATH"}
{"messages": [{"content": "Let TE = 10. Let Y be the midpoint of segment TE. Point S is obtained by rotating point Y by angle 11pi/12 about point T. Let U be the midpoint of segment SE. Let N be the reflection of point T across point S. Point J is the reflection of point U across point Y. Point F is the reflection of point J across segment YS. What is 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": "17.35", "dataset": "MATH"}
{"messages": [{"content": "Let UC = 2. Point L is obtained by rotating point U by angle 2pi/3 about point C. Point P is the centroid of triangle LUC. Let N be the circumcircle of triangle LUC. Let I be the inversion of point P in circle N. 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": "2.65", "dataset": "MATH"}
{"messages": [{"content": "JL = 12. Point O is obtained by rotating point J by angle 2pi/3 about point L. Let D be the reflection of point O across point J. Let N be the circumcenter of triangle OJL. Find the inradius of triangle DLN. Express your 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 RU = 6. Let H be the midpoint of segment UR. Point C is obtained by rotating point R by angle pi/2 about point H. Point T is the reflection of point C across segment UR. Point K is the reflection of point R across point U. Let Y be the reflection of point T across point K. Let O be the midpoint of segment YR. What is the inradius of triangle YTO? Express your 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": "DY = 9. Point L is the midpoint of segment DY. Let G be the rotation of point L by angle 7pi/12 around point Y. Point C is the reflection of point G across point D. Point V is the reflection of point C across segment LD. Point M is the midpoint of segment CD. Find the inradius of triangle MLV. Express your 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 LU = 6. Let X be the rotation of point L by angle pi/12 around point U. Let Y be the rotation of point L by angle pi/6 around point U. Let K be the midpoint of segment YL. Point C is the midpoint of segment XY. Let R be the midpoint of segment LX. What is the measure of angle CRK, 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": "0.08", "dataset": "MATH"}
{"messages": [{"content": "XS = 10. Point E is the midpoint of segment XS. Let I be the rotation of point S by angle pi/12 around point E. Let Y be the perpendicular bisector of segment SX. Point B is the reflection of point I across line Y. Let Q be the circumcenter of triangle EXI. Let V be the orthocenter of triangle XBQ. 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": "1.07", "dataset": "MATH"}
{"messages": [{"content": "Let CO = 4. Let A be the rotation of point O by angle 7pi/12 around point C. Let Q be the incenter of triangle OAC. Let N be the reflection of point Q across point O. Point X is the orthocenter of triangle OAC. What is 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": "8.21", "dataset": "MATH"}
{"messages": [{"content": "Let HT = 8. Let M be the rotation of point H by angle 11pi/12 around point T. Point J is the midpoint of segment TM. Let I be the line through points H and J. Let B be the reflection of line I through point T. Line D passes through point J perpendicular to line I. Line P passes through point T parallel to line D. Point Z is the intersection of lines P and B. Find the measure of angle HMZ, 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": "13.79", "dataset": "MATH"}
{"messages": [{"content": "OB = 5. Line L is the perpendicular bisector of segment OB. Point N is obtained by rotating point B by angle 11pi/12 about point O. Let A be the rotation of point N by angle 7pi/12 around point B. Point T is obtained by rotating point A by angle pi/4 about point N. Point S is the reflection of point O across point N. Let Q be the rotation of point O by angle 11pi/12 around point N. Line J is the reflection of line L across point A. Point H is the reflection of point S across point T. Line J and segment OH intersect at point G. Find the length of segment GS. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "19.92", "dataset": "MATH"}
{"messages": [{"content": "DY = 2. Let Q be the rotation of point D by angle pi/6 around point Y. Point N is the midpoint of segment YQ. Point K is the midpoint of segment NY. Point A is the reflection of point Y across point D. Let W be the midpoint of segment KD. What is the measure of angle WDA, 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": "0.12", "dataset": "MATH"}
{"messages": [{"content": "XD = 2. Point O is obtained by rotating point X by angle 11pi/12 about point D. Let V be the midpoint of segment DO. Point T is the reflection of point V across segment XD. Let Q be the reflection of point T across point D. What is the inradius of triangle DQV? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.13", "dataset": "MATH"}
{"messages": [{"content": "JQ = 9. Let D be the rotation of point Q by angle pi/12 around point J. Let S be the angle bisector of angle JQD. Point K is the midpoint of segment DQ. Point G is obtained by rotating point J by angle pi/12 about point D. Let C be the orthocenter of triangle JQD. Point P is the reflection of point K across point Q. Point V is the reflection of point C across line S. Point M is the reflection of point J across point V. Find the inradius of triangle GMP. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.19", "dataset": "MATH"}
{"messages": [{"content": "Let ZH = 10. Let A be the rotation of point Z by angle 11pi/12 around point H. Point L is the orthocenter of triangle ZHA. Let W be the midpoint of segment ZH. Point Q is the centroid of triangle ZHA. Circle S is the incircle of triangle ZHA. Point D is the midpoint of segment WL. Let C be the inversion of point Q in circle S. What is the distance between points D 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": "39.37", "dataset": "MATH"}
{"messages": [{"content": "PZ = 10. Let U be the reflection of point Z across point P. Let O be the rotation of point U by angle pi/2 around point Z. Line A passes through points O and U. Let N be the line through point U perpendicular to line A. Let T be the line through point P perpendicular to line N. Let X be the reflection of line A across line T. Point F is the reflection of point P across line X. Let S be the midpoint of segment UP. What is the inradius of triangle OSF? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.16", "dataset": "MATH"}
{"messages": [{"content": "YD = 6. Let T be the line through points Y and D. Let R be the midpoint of segment DY. Point V is obtained by rotating point R by angle 5pi/6 about point Y. Point G is the midpoint of segment YV. Let K be the midpoint of segment YR. Line W is the perpendicular bisector of segment GD. Line A is the reflection of line W across point V. Let P be the reflection of line T across line A. Point Z is the reflection of point K across line P. Find the circumradius of triangle ZYG. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.25", "dataset": "MATH"}
{"messages": [{"content": "Line E passes through point F parallel to segment VI14. Point N is on line E with |NJ| = 1. Point C is obtained by rotating point J by angle 7pi/12 about point S. Point K is the reflection of point S across segment CV. Point H is the midpoint of segment SS. Let R be the midpoint of segment KH. 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": "1.16", "dataset": "MATH"}
{"messages": [{"content": "Let TY = 10. Point U is obtained by rotating point T by angle 3pi/4 about point Y. Point A is the midpoint of segment UY. Let K be the centroid of triangle YTU. Point E is the reflection of point A across point K. What is the length of segment UE? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "13.87", "dataset": "MATH"}
{"messages": [{"content": "Let RV = 2. Let J be the reflection of point R across point V. Let S be the rotation of point V by angle pi/2 around point J. Point U is the reflection of point V across point S. Point A is the circumcenter of triangle SUR. Point K is the reflection of point J across segment SU. What is the measure of angle RAK, 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": "1.33", "dataset": "MATH"}
{"messages": [{"content": "GR = 1. Let E be the rotation of point R by angle pi/12 around point G. Point N is the midpoint of segment ER. Let M be the line through point E perpendicular to segment GE. Line J is the perpendicular bisector of segment GN. Let L be the midpoint of segment GE. Let A be the line through point R parallel to line M. Let T be the line through point L perpendicular to line J. Let H be the reflection of line T through point G. Point W is the intersection of lines A and H. Point U is the midpoint of segment RL. What is the circumradius of triangle GWU? Express your 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": "IV = 6. Let X be the rotation of point V by angle pi/12 around point I. Let Q be the midpoint of segment VI. Let U be the reflection of point V across point X. Point B is the midpoint of segment QU. Circle D is the circumcircle of triangle BVU. Let T be the inversion of point I in circle D. Let L be the incenter of triangle BVU. Point Y is the reflection of point T across point V. 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": "3.36", "dataset": "MATH"}
{"messages": [{"content": "Let PL = 1. Let U be the vector from point L to point P. Point R is the translation of point P by vector U. Let I be the rotation of point P by angle pi/6 around point R. Point A is the orthocenter of triangle PLI. Circle O is the circumcircle of triangle PLI. Let N be the inversion of point R in circle O. What is the distance between points A 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": "0.74", "dataset": "MATH"}
{"messages": [{"content": "Let YB = 5. Point D is obtained by rotating point Y by angle 11pi/12 about point B. Let I be the angle bisector of angle YBD. Line M is the reflection of line I across point Y. Let T be the midpoint of segment BD. Point A is the reflection of point Y across line M. Let F be the circumcircle of triangle YBD. Let W be the inversion of point T in circle F. Find the distance between points A 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": "17.41", "dataset": "MATH"}
{"messages": [{"content": "Let KW = 8. Let R be the midpoint of segment KW. Point Q is obtained by rotating point W by angle 7pi/12 about point R. Point L is the midpoint of segment KQ. Let I be the incenter of triangle QWK. What is the inradius of triangle LIR? Express your 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": "PL = 5. Point F is the reflection of point P across point L. Let M be the rotation of point L by angle pi/3 around point F. Let Y be the reflection of point P across point F. Point V is the midpoint of segment LF. Let W be the circumcircle of triangle FML. Let N be the inversion of point Y in circle W. Point A is the reflection of point V across segment FM. Find the measure of angle ALN, 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": "0.85", "dataset": "MATH"}
{"messages": [{"content": "Let PZ = 3. Line Y is the perpendicular bisector of segment ZP. Point T lies on line Y at distance 2 from point Z. Point J is the midpoint of segment PZ. Point V is the reflection of point J across point T. Let I be the perpendicular bisector of segment ZV. Let W be the reflection of line I across line Y. Line W and segment ZV intersect at point Q. Find 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": "2.84", "dataset": "MATH"}
{"messages": [{"content": "Let NY = 9. Let U be the rotation of point Y by angle 5pi/12 around point N. Let K be the reflection of point N across point Y. Let C be the reflection of point K across point N. Let V be the midpoint of segment UY. Let F be the midpoint of segment CV. What is the circumradius of triangle FKV? Express your 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": "RF = 7. Let S be the rotation of point R by angle 5pi/6 around point F. Point I is the reflection of point S across segment FR. Let H be the angle bisector of angle ISF. Point M is the midpoint of segment FR. Point C is the reflection of point M across line H. Let B be the midpoint of segment CF. What is the measure of angle BIR, 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": "2.24", "dataset": "MATH"}
{"messages": [{"content": "Let LY = 11. Let D be the rotation of point Y by angle pi/12 around point L. Let F be the rotation of point L by angle pi/4 around point D. Line N is the angle bisector of angle YLF. Line W is the reflection of line N across point D. Point P is the centroid of triangle LYD. Point I is the reflection of point P across line W. What is 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": "25.69", "dataset": "MATH"}
{"messages": [{"content": "Let WV = 3. Point Q is the reflection of point W across point V. Let K be the midpoint of segment QV. Point F is obtained by rotating point K by angle pi/4 about point Q. Point U is the midpoint of segment FK. Let R be the line through point U perpendicular to segment FW. Point N is the reflection of point Q across line R. Point S is the reflection of point N across segment UF. Find the circumradius of triangle UNS. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.74", "dataset": "MATH"}
{"messages": [{"content": "Let IQ = 6. Point V is obtained by rotating point I by angle pi/3 about point Q. Circle W is the incircle of triangle VQI. Let O be the midpoint of segment QI. Point A is the inversion of point I with respect to circle W. Line U is the perpendicular bisector of segment IA. Point C is the reflection of point Q across line U. Find the measure of angle OAC, 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": "NR = 2. Let V be the reflection of point R across point N. Let S be the rotation of point R by angle 11pi/12 around point V. Let D be the line through points V and S. Line H is the angle bisector of angle VNS. Let K be the rotation of point R by angle pi/4 around point S. Point E is the reflection of point K across line D. Let Z be the midpoint of segment VE. Point B is the reflection of point Z across line H. Find the inradius of triangle KBN. Express your 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": "FS = 11. Point L is obtained by rotating point S by angle 7pi/12 about point F. Point B is the reflection of point S across point F. Let K be the reflection of point S across point L. Point T is the incenter of triangle FBL. What is 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": "21.81", "dataset": "MATH"}
{"messages": [{"content": "QP = 6. Let N be the rotation of point P by angle 3pi/4 around point Q. Let E be the reflection of point Q across point N. Let J be the midpoint of segment PQ. Line I is the perpendicular bisector of segment NE. Point M is the reflection of point J across line I. Find the inradius of triangle NME. Express your 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": "AR = 8. Point U is the reflection of point R across point A. Let C be the rotation of point R by angle pi/12 around point A. Circle F is the circumcircle of triangle ACR. Let G be the reflection of point R across point U. Point S is the midpoint of segment CR. Let Q be the inversion of point S in circle F. Find the distance between points Q 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": "32.16", "dataset": "MATH"}
{"messages": [{"content": "BP = 1. Point K is obtained by rotating point B by angle 7pi/12 about point P. Let H be the rotation of point P by angle pi/4 around point B. Let J be the circumcircle of triangle BHP. Point S is the inversion of point K with respect to circle J. Let L be the incenter of triangle BHP. What is the distance between points L 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": "0.22", "dataset": "MATH"}
{"messages": [{"content": "Let PG = 8. Point I is obtained by rotating point P by angle pi/4 about point G. Let N be the incenter of triangle IGP. Line S passes through point I perpendicular to segment IG. Point J is the reflection of point N across line S. Point K is the reflection of point P across point J. What is the measure of angle NKI, 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": "6.26", "dataset": "MATH"}
{"messages": [{"content": "SQ = 4. Point B is obtained by rotating point S by angle pi/2 about point Q. Let C be the vector from point B to point Q. Point J is the reflection of point S across point B. Let M be point S translated by vector C. Line H is the angle bisector of angle QMJ. Point E is the centroid of triangle BQS. Let Z be the midpoint of segment ES. Point X is the reflection of point J across line H. Let V be the reflection of point M across point X. What is the measure of angle VZJ, 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": "17.47", "dataset": "MATH"}
{"messages": [{"content": "Let TI = 1. Let V be the reflection of point I across point T. Let P be the rotation of point T by angle pi/4 around point I. Point K is the reflection of point P across point V. Let H be the circumcenter of triangle KIP. Let R be the reflection of point K across point H. Find the length of segment RH. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.08", "dataset": "MATH"}
{"messages": [{"content": "DG = 10. Let N be the rotation of point G by angle pi/2 around point D. Let I be the midpoint of segment DG. Let T be the rotation of point D by angle pi/4 around point I. Line Z is the perpendicular bisector of segment TN. Point J is the reflection of point G across line Z. Let S be the line through point T parallel to segment ID. Point V is the reflection of point J across line S. Point E is the midpoint of segment NV. Point B is the midpoint of segment ID. Find the measure of angle EBJ, 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": "17.54", "dataset": "MATH"}
{"messages": [{"content": "RU = 6. Let T be the rotation of point U by angle 7pi/12 around point R. Let L be the rotation of point T by angle pi around point R. Let O be the perpendicular bisector of segment UR. Point Y is the circumcenter of triangle UTL. Line F is the reflection of line O across point L. Point X is the reflection of point Y across line F. Find the distance between points Y 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": "3.79", "dataset": "MATH"}
{"messages": [{"content": "VY = 9. Point K is obtained by rotating point Y by angle pi/12 about point V. Let H be the midpoint of segment VY. Let C be the rotation of point K by angle pi/4 around point H. Point T is the midpoint of segment VC. Circle M is the incircle of triangle YVC. Let I be the circumcenter of triangle YVC. Point A is the inversion of point I with respect to circle M. What is the length of segment AT? Express your 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": "MU = 3. Let Z be the rotation of point M by angle pi/6 around point U. Point J is the reflection of point Z across point U. Point Y is the orthocenter of triangle UZM. Circle S is the circumcircle of triangle UZM. Line B passes through point Z perpendicular to segment UZ. Let A be the line through point J parallel to line B. Point P is the reflection of point U across line A. Let N be the line through point Y perpendicular to line A. Let C be the reflection of circle S across line N. Point Q is the inversion of point P with respect to circle C. What is 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": "2.05", "dataset": "MATH"}
{"messages": [{"content": "SB = 1. Line H passes through points B and S. Let C be the rotation of point B by angle 5pi/12 around point S. Point F is the midpoint of segment SB. Let K be the line through points C and F. Point W is the reflection of point B across line K. Point A is the reflection of point C across line H. Let P be the reflection of point A across point F. What is the measure of angle SPW, 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": "0.12", "dataset": "MATH"}
{"messages": [{"content": "DG = 5. Let N be the rotation of point G by angle pi/3 around point D. Let W be the midpoint of segment DG. Point M is the reflection of point W across point G. Circle S is the incircle of triangle GND. Let A be the inversion of point N in circle S. What is the length of segment AM? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.45", "dataset": "MATH"}
{"messages": [{"content": "Let TE = 7. Point A is obtained by rotating point T by angle 5pi/12 about point E. Point J is the reflection of point A across point T. Point B is obtained by rotating point J by angle pi/6 about point T. Point C is the midpoint of segment EJ. Point U is the orthocenter of triangle JTE. Point Z is the circumcenter of triangle JTE. Point V is the reflection of point C across point U. Point O is the midpoint of segment BV. What is the circumradius of triangle ZJO? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.39", "dataset": "MATH"}
{"messages": [{"content": "TB = 3. Point C is obtained by rotating point B by angle 2pi/3 about point T. Line J is the angle bisector of angle TBC. Let O be the reflection of point C across point T. Point H is on line J with |HB| = 4. Point I is the reflection of point T across line J. Let G be the line through point I perpendicular to segment TO. Point A is the reflection of point T across line G. What is the measure of angle IHA, 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": "0.6", "dataset": "MATH"}
{"messages": [{"content": "RJ = 3. Point D is obtained by rotating point R by angle pi/3 about point J. Point Z is the midpoint of segment RJ. Let N be the rotation of point D by angle 11pi/12 around point J. Line S passes through points D and N. Let T be the line through points Z and N. Let X be the reflection of line S across line T. Point H is the reflection of point J across line X. Point Y is the midpoint of segment DH. Point G is the reflection of point Y across line S. Point Q is the reflection of point G across line X. Point K is the midpoint of segment QZ. What is the measure of angle HKQ, 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": "0.41", "dataset": "MATH"}
{"messages": [{"content": "RY = 2. Line I passes through point R perpendicular to segment YR. Point O is on line I with |OY| = 9. Line L passes through point Y parallel to line I. Point W is the reflection of point R across line L. Let S be the reflection of point O across point Y. Point B is on line L. Find the measure of angle SBW, 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": "9.0", "dataset": "MATH"}
{"messages": [{"content": "KP = 11. Let Y be the midpoint of segment PK. Let Q be the rotation of point K by angle 7pi/12 around point Y. Let S be the midpoint of segment PQ. Point I is the reflection of point S across point P. Point L is the reflection of point S across segment QY. Find 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": "11.38", "dataset": "MATH"}
{"messages": [{"content": "Let ZX = 6. Point Q is the reflection of point Z across point X. Line J is the angle bisector of angle ZXQ. Point W is obtained by rotating point Q by angle 5pi/12 about point Z. Point E is the reflection of point Q across point W. Point U is obtained by rotating point Q by angle 7pi/12 about point X. Line M passes through point W parallel to line J. Let K be the reflection of line M through point U. Line K and segment QE intersect at point N. Point V is the reflection of point Z across point N. Find the measure of angle VUN, 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": "1.23", "dataset": "MATH"}
{"messages": [{"content": "VH = 10. Let Y be the midpoint of segment HV. Let G be the rotation of point V by angle pi/3 around point H. Let K be the rotation of point Y by angle 5pi/12 around point V. Let B be the incenter of triangle HKY. Let U be the angle bisector of angle KGV. Point X is the reflection of point K across line U. 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": "7.91", "dataset": "MATH"}
{"messages": [{"content": "PO = 3. Point E is obtained by rotating point P by angle pi/6 about point O. Let I be the midpoint of segment EP. Let C be the midpoint of segment IE. Point D is the reflection of point C across point E. What is the inradius of triangle OID? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.47", "dataset": "MATH"}
{"messages": [{"content": "Let CV = 12. Let K be the perpendicular bisector of segment CV. Point L is on line K with |LV| = 10. Point J is the reflection of point C across point L. Point F is the reflection of point L across segment VJ. Find the inradius of triangle LFV. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.4", "dataset": "MATH"}
{"messages": [{"content": "SI = 6. Let W be the rotation of point I by angle pi around point S. Let J be the vector from point W to point I. Let O be the rotation of point W by angle pi/3 around point I. Point B is the translation of point S by vector J. Let E be the perpendicular bisector of segment IO. Let A be the reflection of line E through point S. Point K is the reflection of point O across point B. Let R be the line through point S perpendicular to segment SI. Line P is the reflection of line R across line A. Point H is the reflection of point O across line P. Find the measure of angle HSK, 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": "187.06", "dataset": "MATH"}
{"messages": [{"content": "AF = 12. Let T be the reflection of point A across point F. Point Q is obtained by rotating point A by angle 11pi/12 about point F. Let B be the midpoint of segment QF. Point I is the reflection of point B across point T. Let L be the reflection of point I across point A. Let N be the midpoint of segment TL. Point R is the midpoint of segment NQ. Let P be the reflection of point I across point R. Let G be the reflection of point N across point P. Find the distance between points B 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": "35.18", "dataset": "MATH"}
{"messages": [{"content": "PM = 6. Point V is obtained by rotating point M by angle 2pi/3 about point P. Point W is the reflection of point M across point V. Line J is the perpendicular bisector of segment WV. Point S is the reflection of point P across line J. Point I is the orthocenter of triangle SPV. Find the distance between points I 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": "31.18", "dataset": "MATH"}
{"messages": [{"content": "NY = 10. Let B be the reflection of point Y across point N. Let E be the rotation of point N by angle pi/4 around point Y. Let J be the midpoint of segment EB. Let C be the midpoint of segment NE. Point R is the reflection of point C across segment YN. Let Z be the circumcenter of triangle NRJ. Find 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": "7.94", "dataset": "MATH"}
{"messages": [{"content": "QF = 11. Point Z is the reflection of point F across point Q. Let J be the rotation of point Z by angle 5pi/12 around point F. Point K is the reflection of point J across point Q. Let U be the midpoint of segment ZK. Point E is the reflection of point U across point Z. Find the circumradius of triangle EKF. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.03", "dataset": "MATH"}
{"messages": [{"content": "YP = 11. Let Q be the reflection of point Y across point P. Let N be the midpoint of segment YP. Let V be the rotation of point N by angle pi/6 around point P. Point J is the reflection of point N across point V. Line W is the angle bisector of angle YQV. Let L be the rotation of point Y by angle pi around point J. Point Z is on line W. Let S be the reflection of point L across point N. Let F be the midpoint of segment VS. What is the distance between points F 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": "20.77", "dataset": "MATH"}
{"messages": [{"content": "DG = 11. Let L be the line through points G and D. Point K is obtained by rotating point D by angle pi about point G. Let T be the rotation of point G by angle 2pi/3 around point K. Let V be the midpoint of segment GK. Point B is the reflection of point T across line L. Let E be the reflection of point B across point G. Point N is the midpoint of segment TE. Find the inradius of triangle VNB. Express your 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 VH = 9. Let G be the line through points V and H. Let M be the line through point V perpendicular to line G. Let W be the midpoint of segment VH. Point Z is on line M with |ZW| = 6. Point T is on line G. Line A is the angle bisector of angle TWZ. Line D is the reflection of line A across point Z. Point B is the reflection of point H across line D. Point R is the reflection of point Z across line D. What is the inradius of triangle BHR? Express your 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": "IG = 12. Point Y is obtained by rotating point I by angle 5pi/6 about point G. Point Q is the midpoint of segment YG. Point H is the orthocenter of triangle QIG. Point F is the reflection of point H across segment GQ. Find the measure of angle YIF, 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": "250.71", "dataset": "MATH"}
{"messages": [{"content": "Let NF = 9. Let L be the reflection of point N across point F. Let E be the reflection of point F across point L. Point U is obtained by rotating point N by angle 5pi/12 about point F. Let S be the rotation of point U by angle 5pi/12 around point L. Point O is the reflection of point S across segment EU. Point D is the midpoint of segment EL. Line A passes through point E perpendicular to segment EU. Line C is the reflection of line A across point D. Point Z is the reflection of point O across line C. Find the inradius of triangle UZF. Express your 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": "VC = 7. Point B is obtained by rotating point V by angle pi/4 about point C. Let Z be the rotation of point B by angle pi/3 around point C. Point J is the reflection of point V across point Z. Point F is the reflection of point Z across segment VB. Let T be the midpoint of segment VB. Let Y be the midpoint of segment JT. Let N be the reflection of point V across point Y. What is the measure of angle CTN, 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": "62.21", "dataset": "MATH"}
{"messages": [{"content": "Let TE = 3. Point Q is the midpoint of segment TE. Let G be the rotation of point E by angle pi/2 around point T. Let F be the perpendicular bisector of segment QE. Point P is the reflection of point G across line F. Let R be the orthocenter of triangle GPE. Point Z is the centroid of triangle GPE. Point X is the reflection of point T across point Z. Find 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": "3.2", "dataset": "MATH"}
{"messages": [{"content": "OC = 5. Let I be the reflection of point O across point C. Let P be the rotation of point O by angle pi/3 around point I. Point W is the reflection of point I across point O. Line E is the perpendicular bisector of segment CP. Let Y be the reflection of point C across point P. Point B is the reflection of point O across line E. Find the measure of angle BWY, 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": "21.65", "dataset": "MATH"}
{"messages": [{"content": "TP = 2. Line W is the perpendicular bisector of segment PT. Let G be the reflection of line W through point P. Point M is on line G with |MP| = 11. Let Z be the midpoint of segment MP. Line K is the perpendicular bisector of segment PZ. Point U is the midpoint of segment ZP. Line O passes through points T and M. Let D be the reflection of line K across line O. Point A is the reflection of point Z across line D. Point L is the midpoint of segment TZ. Find the circumradius of triangle AUL. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.08", "dataset": "MATH"}
{"messages": [{"content": "Line Q is the perpendicular bisector of segment OO. Point L is the reflection of point O across segment R46T. Point V is the reflection of point L across line Q. Let H be the circumcircle of triangle T. Point E is the inversion of point Z with respect to circle H. Point G is the reflection of point E across segment FR46. Let N be the midpoint of segment TV. 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": "1.32", "dataset": "MATH"}
{"messages": [{"content": "Let HF = 4. Let D be the rotation of point F by angle pi/2 around point H. Let Q be the centroid of triangle HDF. Let U be the incenter of triangle HDF. What is the inradius of triangle QUF? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.1", "dataset": "MATH"}
{"messages": [{"content": "QE = 12. Point P is obtained by rotating point E by angle 5pi/6 about point Q. Point S is the reflection of point Q across point E. Let V be the orthocenter of triangle PQE. Point W is the reflection of point E across point S. What is the length of segment WV? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "60.47", "dataset": "MATH"}
{"messages": [{"content": "Let XA = 2. Point J is the midpoint of segment AX. Let G be the rotation of point A by angle 5pi/12 around point J. Let M be the rotation of point G by angle pi/12 around point A. Let B be the centroid of triangle GJM. Let Q be the midpoint of segment MJ. Point R is the centroid of triangle AQB. Let Z be the incircle of triangle AQB. Point F is the inversion of point Q with respect to circle 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": "0.3", "dataset": "MATH"}
{"messages": [{"content": "Let VZ = 7. Let W be the rotation of point Z by angle 5pi/6 around point V. Point A is the midpoint of segment ZW. Line Q is the perpendicular bisector of segment VZ. Let U be the line through point W perpendicular to line Q. Point S is the reflection of point V across line U. Find the circumradius of triangle AWS. Express your 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 TL = 1. Point R is obtained by rotating point L by angle 5pi/12 about point T. Point J is the midpoint of segment LT. Let N be the reflection of point T across point R. Point W is the reflection of point N across point J. Find the distance between points W 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": "3.86", "dataset": "MATH"}
{"messages": [{"content": "Let AZ = 2. Let B be the reflection of point Z across point A. Let E be the rotation of point Z by angle 5pi/12 around point A. Point H is the midpoint of segment BE. Point F is the incenter of triangle ZBE. Point X is the reflection of point F across segment ZB. Point W is the reflection of point H across point B. What is the measure of angle XAW, 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": "1.49", "dataset": "MATH"}
{"messages": [{"content": "QC = 5. Point X is obtained by rotating point Q by angle pi/6 about point C. Point Z is the reflection of point C across point Q. Point O is the reflection of point Q across point Z. Let W be the circumcenter of triangle QXC. Let M be the midpoint of segment XC. Point T is the reflection of point O across point W. Find the measure of angle TQM, 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": "6.25", "dataset": "MATH"}
{"messages": [{"content": "Let EK = 11. Let D be the midpoint of segment EK. Let G be the rotation of point D by angle pi/12 around point E. Line W is the perpendicular bisector of segment GE. Line S is the perpendicular bisector of segment KE. Let P be the line through point E parallel to line S. Let C be the reflection of line W through point D. Line O passes through point K parallel to line P. Point V is the reflection of point G across segment KE. Point I is the intersection of lines C and O. What is the circumradius of triangle VID? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "9.46", "dataset": "MATH"}
{"messages": [{"content": "Let AV = 11. Line P is the perpendicular bisector of segment VA. Point O lies on line P at distance 1 from point A. Let X be the reflection of point A across point O. Let L be the vector from point X to point A. Point Y is the translation of point O by vector L. Line Z passes through points A and O. Let D be the midpoint of segment VY. Point W is the reflection of point V across line Z. Let Q be the incenter of triangle AWD. 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": "8.22", "dataset": "MATH"}
{"messages": [{"content": "Let QY = 3. Point Z is the midpoint of segment QY. Point B is obtained by rotating point Y by angle pi/4 about point Q. Let K be the centroid of triangle YBQ. Point I is the reflection of point Z across segment YB. Point D is the midpoint of segment KI. What is the measure of angle BDI, 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": "0.8", "dataset": "MATH"}
{"messages": [{"content": "HR = 1. Let W be the rotation of point R by angle 5pi/12 around point H. Point Q is obtained by rotating point W by angle pi/3 about point H. Point M is the reflection of point R across point H. Point S is the reflection of point Q across segment RH. Let P be the midpoint of segment WR. Find the inradius of triangle MSP. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.3", "dataset": "MATH"}
{"messages": [{"content": "MA = 3. Point P is the midpoint of segment AM. Let N be the rotation of point P by angle 11pi/12 around point M. Line R is the perpendicular bisector of segment NM. Point E is the incenter of triangle MPN. Let B be the line through point P parallel to line R. Point U is the reflection of point A across line R. Let S be the line through point M parallel to segment EN. Point C is the intersection of lines S and B. What is the inradius of triangle UEC? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.08", "dataset": "MATH"}
{"messages": [{"content": "Let VC = 4. Line G is the perpendicular bisector of segment VC. Line J passes through point V parallel to line G. Point Y is on line J with |YV| = 1. Line H is the reflection of line G across line J. Let L be the midpoint of segment YC. Let A be the midpoint of segment VC. Point I is the reflection of point L across line H. What is 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": "8.02", "dataset": "MATH"}
{"messages": [{"content": "Let XU = 9. Let V be the rotation of point X by angle 2pi/3 around point U. Point A is obtained by rotating point V by angle 11pi/12 about point X. Let Z be the reflection of point U across point V. Let O be the orthocenter of triangle AXZ. Point B is the circumcenter of triangle AXZ. Find the measure of angle XBO, 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": "752.12", "dataset": "MATH"}
{"messages": [{"content": "DE = 11. Point W is obtained by rotating point E by angle pi/12 about point D. Point M is the reflection of point E across point D. Let S be the rotation of point W by angle 3pi/4 around point M. Point P is the reflection of point S across point D. Point Z is the reflection of point P across segment ES. Find the inradius of triangle ZSM. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.55", "dataset": "MATH"}
{"messages": [{"content": "AT = 11. Point U is the reflection of point T across point A. Point L is obtained by rotating point T by angle 2pi/3 about point A. Point I is the midpoint of segment LT. Point X is the reflection of point I across point U. Point G is the midpoint of segment XT. Point O is the circumcenter of triangle TUX. Let V be the incircle of triangle GIL. Let S be the inversion of point U in circle V. Find the measure of angle OXS, 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": "38.51", "dataset": "MATH"}
{"messages": [{"content": "GJ = 11. Let H be the rotation of point G by angle 7pi/12 around point J. Point R is the midpoint of segment GH. Let M be the midpoint of segment HR. Point U is obtained by rotating point J by angle pi/12 about point G. Point T is obtained by rotating point U by angle pi/3 about point G. Line N is the perpendicular bisector of segment TR. Point Y is the midpoint of segment UM. Point A is the reflection of point T across segment UM. Line Z is the reflection of line N across point A. Point W is the reflection of point J across line Z. Let S be the reflection of point H across point W. What is the measure of angle YJS, 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": "32.96", "dataset": "MATH"}
{"messages": [{"content": "KG = 2. Let I be the midpoint of segment GK. Point O is obtained by rotating point G by angle 3pi/4 about point I. Y is placed 1 units away from G. Let U be the circumcenter of triangle GIO. Point J is the midpoint of segment IG. Let L be the midpoint of segment IO. Point A is the reflection of point L across segment UJ. Find the circumradius of triangle UKA. Express your 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": "UG = 6. Let H be the rotation of point G by angle 11pi/12 around point U. Point P is the reflection of point H across point G. Point S is the midpoint of segment GU. Let F be the midpoint of segment HG. Point Q is the centroid of triangle GSH. Point O is the reflection of point Q across point P. 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": "34.7", "dataset": "MATH"}
{"messages": [{"content": "XS = 2. Let V be the vector from point X to point S. Point Z is the reflection of point X across point S. Point A is the translation of point Z by vector V. Point Y is obtained by rotating point S by angle 3pi/4 about point A. Let B be the circumcenter of triangle ZXY. Line H is the perpendicular bisector of segment XY. Line K is the reflection of line H across point B. Point T is the midpoint of segment XY. Point L is the intersection of line K and segment YZ. 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": "1.28", "dataset": "MATH"}
{"messages": [{"content": "Let LB = 6. Let W be the rotation of point B by angle 11pi/12 around point L. Point P is the midpoint of segment LW. Point O is the reflection of point P across segment WB. Point T is the reflection of point O across point P. 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": "3.0", "dataset": "MATH"}
{"messages": [{"content": "Let YG = 7. Let B be the rotation of point Y by angle pi/3 around point G. Let K be the midpoint of segment YG. Point R is the reflection of point B across segment GY. Let X be the vector from point R to point K. Point E is the reflection of point G across point B. Point S is the translation of point G by vector X. Let J be the midpoint of segment BG. Line H passes through point J parallel to segment GY. Line U passes through points Y and R. Lines U and H intersect at point M. Find the circumradius of triangle ESM. Express your 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 RQ = 1. Point E is obtained by rotating point Q by angle 5pi/12 about point R. Let I be the midpoint of segment RQ. Point O is the centroid of triangle RQE. Circle V is the circumcircle of triangle RQE. Point B is the inversion of point I with respect to circle V. Find the distance between points O 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": "0.98", "dataset": "MATH"}
{"messages": [{"content": "Let ZQ = 4. Point E is the reflection of point Q across point Z. Let R be the rotation of point Z by angle 7pi/12 around point E. Let H be the rotation of point E by angle 5pi/12 around point Z. Point N is the reflection of point R across point H. Let K be the midpoint of segment QZ. Let L be the reflection of point K across point N. Let O be the midpoint of segment NH. What is the inradius of triangle KOL? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.83", "dataset": "MATH"}
{"messages": [{"content": "YB = 9. Let Q be the rotation of point B by angle 5pi/6 around point Y. Let H be the midpoint of segment YQ. Let E be the orthocenter of triangle YBQ. Point T is the midpoint of segment BQ. Point F is the reflection of point T across segment YB. What is the measure of angle HFE, 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": "79.28", "dataset": "MATH"}
{"messages": [{"content": "Let KJ = 4. Point W is obtained by rotating point J by angle 5pi/12 about point K. Line Y is the perpendicular bisector of segment WJ. Let I be the midpoint of segment WK. Point L is the midpoint of segment JW. Point M is the reflection of point I across line Y. 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": "2.0", "dataset": "MATH"}
{"messages": [{"content": "JF = 7. Line X is the perpendicular bisector of segment JF. Let O be the reflection of line X through point J. Let Y be the rotation of point J by angle 5pi/6 around point F. Let E be the orthocenter of triangle FJY. Point T is the midpoint of segment JF. Line R is the reflection of line O across point T. Segment YT and line R intersect at point Z. 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": "25.32", "dataset": "MATH"}
{"messages": [{"content": "HW = 3. Let E be the rotation of point H by angle 5pi/6 around point W. Let Y be the midpoint of segment WE. Line U is the perpendicular bisector of segment YE. Point P is the reflection of point H across line U. Point F is the circumcenter of triangle PHE. 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": "4.63", "dataset": "MATH"}
{"messages": [{"content": "Let SF = 3. Point K is obtained by rotating point F by angle 5pi/12 about point S. Circle Y is the circumcircle of triangle FSK. Circle H is the reflection of circle Y across point S. Point U is the inversion of point F with respect to circle H. Point J is the orthocenter of triangle FSK. Line A passes through point J parallel to segment KF. Point V is the intersection of segment SK and line A. Point R is the centroid of triangle FSK. Line A and segment UR intersect at point G. Point P is the incenter of triangle VUR. What is 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": "0.37", "dataset": "MATH"}
{"messages": [{"content": "PJ = 4. Let V be the rotation of point P by angle pi/3 around point J. Point X is the midpoint of segment JV. Circle H is the circumcircle of triangle VXP. Point U is the inversion of point J with respect to circle H. Point D is the centroid of triangle VXP. What is the circumradius of triangle UVD? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.76", "dataset": "MATH"}
{"messages": [{"content": "IM = 11. Let W be the rotation of point M by angle 11pi/12 around point I. Point D lies on segment IW. Line X is the perpendicular bisector of segment IW. Point V is the reflection of point M across line X. Find the measure of angle VMD, 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": "45.91", "dataset": "MATH"}
{"messages": [{"content": "OJ = 7. Let A be the reflection of point O across point J. Let Y be the perpendicular bisector of segment JA. Point E lies on line Y at distance 3 from point A. Point T is the reflection of point O across line Y. Point G is the circumcenter of triangle TEJ. Point S is the reflection of point G across line Y. Find 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": "12.87", "dataset": "MATH"}
{"messages": [{"content": "YC = 10. Point O is obtained by rotating point C by angle pi/12 about point Y. Let K be the reflection of point C across point Y. Let E be the midpoint of segment KY. Point M is the reflection of point K across segment OY. What is the measure of angle EMO, 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": "31.91", "dataset": "MATH"}
{"messages": [{"content": "FP = 2. Point R is obtained by rotating point P by angle 3pi/4 about point F. Let O be the centroid of triangle FPR. Point S is the midpoint of segment RO. Let C be the orthocenter of triangle FPR. Line J is the angle bisector of angle FOC. Point K is the reflection of point S across line J. What is the measure of angle CPK, 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": "1.94", "dataset": "MATH"}
{"messages": [{"content": "AO = 8. Point S is obtained by rotating point A by angle 5pi/12 about point O. Line G passes through points A and S. Point J is the reflection of point O across line G. Point Y is the midpoint of segment SO. Let N be the perpendicular bisector of segment SY. Point Q is the reflection of point A across line N. Point U is obtained by rotating point Q by angle pi about point O. Let F be the perpendicular bisector of segment AY. Point B is the reflection of point U across point J. Point I is the reflection of point O across line F. What is the inradius of triangle BJI? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.93", "dataset": "MATH"}
{"messages": [{"content": "YH = 2. Point W is the midpoint of segment HY. Let L be the reflection of point W across point H. Point E is obtained by rotating point H by angle 11pi/12 about point L. Let U be the rotation of point L by angle 2pi/3 around point W. Let K be the reflection of point W across point L. Point V is obtained by rotating point E by angle 7pi/12 about point H. Point D is the midpoint of segment YU. Point I is obtained by rotating point K by angle pi/6 about point V. Let Q be the reflection of point L across point Y. Let F be the midpoint of segment DI. What is the circumradius of triangle EFQ? Express your 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 BX = 11. Let T be the rotation of point B by angle 3pi/4 around point X. Let P be the rotation of point B by angle 5pi/12 around point X. Let Y be the midpoint of segment TX. Circle N is the incircle of triangle PBY. Let W be the orthocenter of triangle PBY. Let E be the inversion of point X in circle N. Find the circumradius of triangle ETW. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.25", "dataset": "MATH"}
{"messages": [{"content": "KD = 8. Let P be the rotation of point K by angle 3pi/4 around point D. Line V is the perpendicular bisector of segment KP. Point L is the midpoint of segment PD. Let U be the reflection of line V through point P. Point R is the midpoint of segment PL. Point M is the reflection of point R across line U. What is the measure of angle MLR, 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": "7.07", "dataset": "MATH"}
{"messages": [{"content": "QM = 1. Let C be the rotation of point M by angle pi/12 around point Q. Point A is the reflection of point C across point M. Let O be the angle bisector of angle AQC. Let B be the midpoint of segment CM. Point L is the reflection of point B across line O. Let U be the circumcenter of triangle CBQ. Find 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": "0.41", "dataset": "MATH"}
{"messages": [{"content": "UY = 11. Point Z is the reflection of point Y across point U. Let H be the vector from point Y to point Z. Point M is the midpoint of segment ZU. Let E be the rotation of point Z by angle pi/6 around point U. Let K be point Z translated by vector H. Let G be the rotation of point K by angle pi/12 around point M. Let J be the reflection of point G across point E. Point C is the reflection of point J across segment ME. Find the inradius of triangle KJC. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.88", "dataset": "MATH"}
{"messages": [{"content": "FB = 2. Let T be the midpoint of segment BF. Let Z be the reflection of point T across point F. Point O is obtained by rotating point T by angle pi/6 about point Z. Point H is the reflection of point T across segment FO. Line L is the angle bisector of angle TBH. Point C is the reflection of point O across line L. Find the measure of angle CFH, 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": "0.15", "dataset": "MATH"}
{"messages": [{"content": "VQ = 6. Let P be the midpoint of segment QV. Point L is the reflection of point V across point Q. Let C be the perpendicular bisector of segment QV. Line X passes through point V parallel to line C. Line O is the reflection of line X across line C. Point D is the reflection of point P across line X. Line K is the reflection of line O across point V. Let M be the midpoint of segment PV. Let E be the rotation of point V by angle 5pi/6 around point M. Let F be the line through points E and D. Point I is the reflection of point L across line F. Point Y is the reflection of point I across line K. What is the circumradius of triangle MIY? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "35.49", "dataset": "MATH"}
{"messages": [{"content": "Let AO = 10. Let S be the rotation of point O by angle 3pi/4 around point A. Point M is the midpoint of segment SA. Let V be the circumcenter of triangle SOA. Point P is the reflection of point M across segment OA. What is the measure of angle SVP, 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": "34.34", "dataset": "MATH"}
{"messages": [{"content": "Let LX = 12. Let E be the rotation of point L by angle 11pi/12 around point X. Let A be the rotation of point L by angle 3pi/4 around point E. Point V is the midpoint of segment XL. Point S is the circumcenter of triangle EAX. What is 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": "20.35", "dataset": "MATH"}
{"messages": [{"content": "NY = 12. Let P be the rotation of point N by angle 11pi/12 around point Y. Point H is the orthocenter of triangle NYP. Let W be the incircle of triangle NYP. Let U be the circumcenter of triangle NYP. Point I is the midpoint of segment PN. Let D be the reflection of circle W through point I. Point J is the midpoint of segment YP. Point Q is the inversion of point J with respect to circle D. Let C be the midpoint of segment UH. 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": "46.2", "dataset": "MATH"}
{"messages": [{"content": "Let FM = 10. Let R be the reflection of point F across point M. Point C is obtained by rotating point R by angle 3pi/4 about point F. Point P is obtained by rotating point M by angle 11pi/12 about point R. Point O is the reflection of point F across point C. Line V is the perpendicular bisector of segment MC. Let G be the reflection of line V through point C. Point Z is the reflection of point O across line G. What is the circumradius of triangle PZF? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "38.69", "dataset": "MATH"}
{"messages": [{"content": "Let XQ = 2. Let J be the rotation of point Q by angle 3pi/4 around point X. Point F is the incenter of triangle QJX. Point Y is the reflection of point F across point Q. Point K is the centroid of triangle QJX. Let W be the midpoint of segment QJ. What is the circumradius of triangle KYW? Express your 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 VE = 11. Point G is obtained by rotating point V by angle pi/12 about point E. Let U be the circumcenter of triangle EGV. Point N is the reflection of point U across segment EG. What is the inradius of triangle GNU? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.74", "dataset": "MATH"}
{"messages": [{"content": "NR = 11. Line Z is the perpendicular bisector of segment NR. Point M is the reflection of point N across point R. Point B is on line Z with |BR| = 5. Point W is the reflection of point M across point N. Line E is the angle bisector of angle MNB. Line Y passes through point M perpendicular to line E. Line I passes through point B perpendicular to line Y. Point T is the reflection of point M across line I. What is the circumradius of triangle WBT? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "27.49", "dataset": "MATH"}
{"messages": [{"content": "Let IL = 3. Point N is obtained by rotating point I by angle 7pi/12 about point L. Point A is the midpoint of segment NI. Let Z be the midpoint of segment AL. Let F be the midpoint of segment NA. Find the length of segment ZF. Express your 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": "NR = 8. Let W be the reflection of point N across point R. Let C be the rotation of point R by angle 7pi/12 around point W. Let F be the vector from point N to point R. Let P be the midpoint of segment RN. Point S is the translation of point P by vector F. Let V be the angle bisector of angle NCP. Point D is the midpoint of segment SC. Point K is the reflection of point D across line V. Let O be the midpoint of segment PK. Let H be the midpoint of segment WC. What is the inradius of triangle CHO? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.44", "dataset": "MATH"}
{"messages": [{"content": "ZQ = 9. Point V is the midpoint of segment QZ. Line S passes through points Q and V. Line W passes through point V perpendicular to line S. Point A lies on line W at distance 7 from point Q. Let D be the midpoint of segment VA. Point P is the reflection of point Q across point A. Point I is the reflection of point D across line S. Let M be the orthocenter of triangle PAZ. 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": "12.29", "dataset": "MATH"}
{"messages": [{"content": "OK = 7. Let Q be the rotation of point K by angle 7pi/12 around point O. Let T be the reflection of point Q across point O. Point U is the reflection of point K across point T. Point H is obtained by rotating point K by angle pi/3 about point O. Let W be the incircle of triangle KTH. Circle M is the reflection of circle W across point K. Point X is the inversion of point K with respect to circle M. 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": "17.45", "dataset": "MATH"}
{"messages": [{"content": "ED = 8. Point X is obtained by rotating point D by angle pi/6 about point E. Line G passes through points X and D. Point S is the reflection of point E across line G. Let L be the orthocenter of triangle EXS. Let I be the midpoint of segment SL. 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": "14.93", "dataset": "MATH"}
{"messages": [{"content": "ZE = 5. Point Y is obtained by rotating point Z by angle 2pi/3 about point E. Point P is obtained by rotating point E by angle pi/12 about point Z. Let H be the rotation of point Y by angle pi/3 around point Z. Let F be the incenter of triangle HZP. Let A be the incircle of triangle HZP. Point W is the inversion of point Z with respect to circle A. What is the measure of angle EFW, 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": "0.08", "dataset": "MATH"}
{"messages": [{"content": "Let PJ = 7. Let L be the rotation of point J by angle pi/6 around point P. Point A is the midpoint of segment JP. Point V is the incenter of triangle JPL. Point D is the orthocenter of triangle JPL. Line T is the perpendicular bisector of segment VD. Point N is the reflection of point A across line T. Find 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": "4.89", "dataset": "MATH"}
{"messages": [{"content": "TL = 6. Let D be the reflection of point T across point L. Let H be the rotation of point T by angle 2pi/3 around point D. Let S be the rotation of point T by angle pi/12 around point L. Point B is the reflection of point S across point H. Let X be the midpoint of segment BL. Point I is the reflection of point S across point B. Let V be the reflection of point L across point S. What is the circumradius of triangle XIV? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "166.9", "dataset": "MATH"}
{"messages": [{"content": "KE = 7. Let T be the rotation of point E by angle pi/4 around point K. Let D be the midpoint of segment TK. Point P is the midpoint of segment ET. Let A be the perpendicular bisector of segment ED. Let S be the line through point E perpendicular to line A. Point X is the reflection of point P across segment TK. Point M is the reflection of point X across line S. 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": "6.14", "dataset": "MATH"}
{"messages": [{"content": "Let RC = 7. Let S be the line through points R and C. Let U be the midpoint of segment CR. Point X is obtained by rotating point C by angle 5pi/6 about point R. Let A be the midpoint of segment XR. Point Z is the reflection of point X across line S. Let L be the midpoint of segment ZU. Let P be the line through point A parallel to line S. Let M be the midpoint of segment ZC. Point J is on line P. What is the measure of angle LMJ, 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": "3.06", "dataset": "MATH"}
{"messages": [{"content": "Let ZX = 12. Line C is the perpendicular bisector of segment XZ. Let T be the reflection of point X across point Z. Line E passes through point X parallel to line C. Let B be the rotation of point T by angle pi/3 around point Z. Let J be the midpoint of segment TZ. Let Y be the reflection of line E through point J. Point Q is the reflection of point X across line Y. Find the circumradius of triangle TBQ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31.75", "dataset": "MATH"}
{"messages": [{"content": "LK = 1. Point A is obtained by rotating point L by angle 7pi/12 about point K. Let T be the midpoint of segment AL. Let S be the centroid of triangle LAK. Let W be the incircle of triangle LAK. Point D is the inversion of point S with respect to circle W. 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": "0.82", "dataset": "MATH"}
{"messages": [{"content": "KA = 10. Point E is obtained by rotating point A by angle 2pi/3 about point K. Let O be the midpoint of segment EA. Point L is obtained by rotating point A by angle pi/6 about point O. Point D is the reflection of point E across segment AK. Point H is the midpoint of segment OD. Point V is the reflection of point L across segment DE. Point U is the midpoint of segment VH. Point Z is the midpoint of segment AH. Find 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": "15.73", "dataset": "MATH"}
{"messages": [{"content": "XE = 6. Point Q is obtained by rotating point X by angle 5pi/6 about point E. Point I is the incenter of triangle EQX. Point F is the midpoint of segment EQ. Let J be the perpendicular bisector of segment XE. Let M be the midpoint of segment QX. Let C be the midpoint of segment IM. Point A is the reflection of point F across line J. Find the length of segment AC. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.3", "dataset": "MATH"}
{"messages": [{"content": "Let AM = 4. Point H is the midpoint of segment AM. Let K be the reflection of point A across point M. Let B be the perpendicular bisector of segment KA. Point C lies on line B at distance 1 from point A. Let R be the reflection of line B through point A. Point X lies on line R. Let Q be the circumcenter of triangle CXA. What is the measure of angle HQX, 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": "1.0", "dataset": "MATH"}
{"messages": [{"content": "Let PN = 1. Let O be the reflection of point P across point N. Point X is obtained by rotating point N by angle 7pi/12 about point P. Point I is the midpoint of segment XP. Let Z be the circumcenter of triangle OIX. Point C is the midpoint of segment ZI. What is the length of segment CZ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.33", "dataset": "MATH"}
{"messages": [{"content": "GP = 3. Let C be the midpoint of segment PG. Let W be the line through point P perpendicular to segment GC. Point E is obtained by rotating point P by angle pi/12 about point G. Point V is the reflection of point E across line W. Let R be the line through points E and C. Let J be the rotation of point E by angle pi around point P. Line A passes through point P perpendicular to line R. Point M is the reflection of point E across line A. Point D is the midpoint of segment EV. Find the measure of angle JDM, 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": "0.48", "dataset": "MATH"}
{"messages": [{"content": "VE = 4. Let N be the rotation of point V by angle pi/12 around point E. Point W is the midpoint of segment EN. Let J be the midpoint of segment WE. Let Y be the angle bisector of angle JVN. Let L be the line through point W perpendicular to line Y. Let X be the line through point W perpendicular to line L. Point Z is the reflection of point V across line X. Find the circumradius of triangle JEZ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.66", "dataset": "MATH"}
{"messages": [{"content": "ZH = 3. Let U be the rotation of point Z by angle 7pi/12 around point H. Let R be the incenter of triangle UHZ. Let Q be the midpoint of segment HR. Point A is the midpoint of segment ZQ. Let Y be the line through point R parallel to segment HZ. Point N is the intersection of segment UH and line Y. Find the distance between points A 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": "1.97", "dataset": "MATH"}
{"messages": [{"content": "AP = 2. Let Q be the rotation of point P by angle 5pi/6 around point A. Let D be the rotation of point Q by angle 11pi/12 around point A. Point R is the reflection of point Q across point D. Let W be the incenter of triangle QAP. Let S be the orthocenter of triangle QAP. Line K passes through points W and S. Point J is the reflection of point A across segment PQ. Point T is the reflection of point R across line K. Find 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": "6.46", "dataset": "MATH"}
{"messages": [{"content": "Let ZV = 8. Line D passes through points V and Z. Point L is the midpoint of segment ZV. Point A lies on line D at distance 12 from point V. Point J is obtained by rotating point Z by angle 2pi/3 about point V. Point X is the midpoint of segment AJ. Let P be the midpoint of segment JX. Point I is the midpoint of segment ZX. Let O be the midpoint of segment JP. Let B be the midpoint of segment JZ. Point T is the reflection of point L across point I. Find the measure of angle BOT, 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": "7.79", "dataset": "MATH"}
{"messages": [{"content": "Let MQ = 6. Let K be the rotation of point M by angle pi/4 around point Q. Let W be the vector from point K to point M. Let E be the reflection of point Q across point K. Point L is the orthocenter of triangle QKM. Point I is the circumcenter of triangle QKM. Let C be the midpoint of segment QK. Let U be the line through point E perpendicular to segment MQ. Point S is the translation of point M by vector W. Let P be the line through point M parallel to line U. Point J is the reflection of point L across line P. Line N is the perpendicular bisector of segment EJ. Let T be the reflection of line N across line P. Point Z is the reflection of point C across line T. What is the measure of angle IZS, 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": "8.24", "dataset": "MATH"}
{"messages": [{"content": "Let NY = 7. Let E be the midpoint of segment NY. Let L be the midpoint of segment EY. Line S is the perpendicular bisector of segment NE. Point G is on line S with |GE| = 7. Let W be the reflection of point L across point N. Point O is the midpoint of segment EG. Let C be the midpoint of segment GN. Let Q be the midpoint of segment WO. What is the circumradius of triangle QCL? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.48", "dataset": "MATH"}
{"messages": [{"content": "JP = 11. Let L be the line through points J and P. Point W is obtained by rotating point J by angle 5pi/12 about point P. Point E is obtained by rotating point J by angle pi/12 about point W. Point R is the reflection of point W across line L. Point Y is the reflection of point P across segment EW. Find the inradius of triangle RYW. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.34", "dataset": "MATH"}
{"messages": [{"content": "Let WT = 5. Point A is obtained by rotating point W by angle 7pi/12 about point T. Point I is the reflection of point A across segment TW. Line V passes through points I and T. Point N is the intersection of segment WA and line V. 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": "5.23", "dataset": "MATH"}
{"messages": [{"content": "VU = 7. Let J be the vector from point V to point U. Let S be point U translated by vector J. Let W be the rotation of point U by angle 7pi/12 around point S. Point T is the reflection of point W across point V. Let O be the midpoint of segment UV. Circle Y is the circumcircle of triangle OTW. Let X be the orthocenter of triangle OTW. Point B is the inversion of point X with respect to circle Y. Find the measure of angle UWB, 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": "388.31", "dataset": "MATH"}
{"messages": [{"content": "LV = 9. Point T is obtained by rotating point L by angle 3pi/4 about point V. Line W is the perpendicular bisector of segment TL. Point F is the midpoint of segment VL. Point S is the reflection of point F across line W. Point X is the reflection of point F across point T. Let I be the reflection of point S across point F. Find the measure of angle XSI, 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": "28.64", "dataset": "MATH"}
{"messages": [{"content": "Let AO = 6. Let W be the perpendicular bisector of segment OA. Let L be the rotation of point A by angle pi around point O. Let I be the midpoint of segment AO. Let U be the perpendicular bisector of segment LI. Point B is obtained by rotating point I by angle pi/4 about point A. Line Q is the reflection of line U across line W. Point P is the reflection of point L across line Q. Point M is obtained by rotating point B by angle 11pi/12 about point I. Let V be the midpoint of segment OB. Point F is the incenter of triangle MBP. Let R be the orthocenter of triangle MBP. Find the measure of angle VFR, 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": "6.95", "dataset": "MATH"}
{"messages": [{"content": "NQ = 1. Let T be the rotation of point N by angle pi/6 around point Q. Line S is the perpendicular bisector of segment QT. Line Z is the reflection of line S across point Q. Point Y is the reflection of point T across line Z. What is the inradius of triangle NYT? Express your 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": "BL = 4. Let Y be the rotation of point L by angle pi/12 around point B. Let I be the orthocenter of triangle BYL. Let X be the midpoint of segment LB. Let K be the reflection of point Y across point B. What is the measure of angle IKX, 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": "0.53", "dataset": "MATH"}
{"messages": [{"content": "Let BF = 7. Let V be the reflection of point F across point B. Point D is obtained by rotating point V by angle 7pi/12 about point F. Point L is the reflection of point B across point D. Point I is the midpoint of segment BV. Let G be the reflection of point L across point I. What is the circumradius of triangle GDL? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "374.18", "dataset": "MATH"}
{"messages": [{"content": "Let RU = 7. Let Y be the rotation of point U by angle pi/3 around point R. Point M is obtained by rotating point Y by angle 5pi/6 about point R. Point K is the reflection of point R across point U. Let H be the circumcircle of triangle MRY. Let N be the inversion of point K in circle H. What is the circumradius of triangle NKY? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "23.42", "dataset": "MATH"}
{"messages": [{"content": "JT = 9. Point R is the reflection of point T across point J. Line Y is the angle bisector of angle RTJ. Let E be the rotation of point J by angle 3pi/4 around point R. Point X is the midpoint of segment ER. Line C is the perpendicular bisector of segment XR. Point F lies on line Y. Point D is the reflection of point J across line C. Point A is the midpoint of segment ED. Find the distance between points A 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": "3.32", "dataset": "MATH"}
{"messages": [{"content": "Let QO = 3. Point Y is obtained by rotating point O by angle pi/12 about point Q. Let V be the midpoint of segment QO. Let H be the midpoint of segment YO. Let G be the rotation of point Y by angle pi/4 around point V. Line W is the angle bisector of angle GQY. Point J is the reflection of point H across point Q. Let M be the line through points O and G. Point K is the reflection of point Y across line M. Line P passes through points K and Y. Line B is the reflection of line W across line P. Point Z is the intersection of lines B and P. Point A is the midpoint of segment GJ. What is the measure of angle GZA, 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": "0.07", "dataset": "MATH"}
{"messages": [{"content": "Let YT = 4. Let D be the rotation of point Y by angle 7pi/12 around point T. Point X is obtained by rotating point T by angle pi/12 about point D. Let F be the incenter of triangle DXY. Line B is the perpendicular bisector of segment XY. Line V passes through point F parallel to line B. Point L is the centroid of triangle DXY. Point Z is the reflection of point T across line V. Find the measure of angle YZL, 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": "1.51", "dataset": "MATH"}
{"messages": [{"content": "Let YM = 5. Point Z is obtained by rotating point Y by angle 11pi/12 about point M. Point G is obtained by rotating point M by angle pi/12 about point Y. Let Q be the reflection of point Y across point Z. Let R be the reflection of point Z across point Q. Let K be the midpoint of segment ZG. Find the circumradius of triangle ZRK. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "33.1", "dataset": "MATH"}
{"messages": [{"content": "Let MN = 12. Let Z be the rotation of point M by angle 5pi/6 around point N. Let O be the midpoint of segment ZM. Let L be the circumcenter of triangle ONZ. Let V be the incenter of triangle ONZ. What is the distance between points L 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": "2.53", "dataset": "MATH"}
{"messages": [{"content": "Let ON = 7. Point W is the reflection of point N across point O. Let K be the rotation of point O by angle pi/12 around point N. Line U is the perpendicular bisector of segment NW. Point M is the reflection of point N across segment WK. Point A is the reflection of point M across line U. Find the length of segment AM. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.7", "dataset": "MATH"}
{"messages": [{"content": "QU = 12. Let E be the reflection of point Q across point U. Point J is obtained by rotating point E by angle 3pi/4 about point U. Let Z be the reflection of point Q across point J. Point N is the midpoint of segment ZJ. Point X is the reflection of point U across segment EZ. Find the inradius of triangle XNE. Express your 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": "WF = 2. Let D be the reflection of point W across point F. Let O be the rotation of point F by angle pi/2 around point D. Point H is obtained by rotating point D by angle pi/4 about point F. Let A be the centroid of triangle OWF. Point U is the reflection of point O across segment WF. Let G be the incenter of triangle OWF. Point J is the reflection of point H across segment UA. Find the inradius of triangle WGJ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.47", "dataset": "MATH"}
{"messages": [{"content": "LP = 9. Point R is obtained by rotating point L by angle pi/4 about point P. Line X is the perpendicular bisector of segment RP. Let W be the midpoint of segment PL. Line N is the reflection of line X across point L. Let F be the midpoint of segment WR. Let O be the angle bisector of angle FLW. Let T be the reflection of line N across line O. Line H is the reflection of line T across line N. Point Q is the reflection of point L across line H. What is the circumradius of triangle RQW? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.82", "dataset": "MATH"}
{"messages": [{"content": "XE = 9. Point S is obtained by rotating point E by angle 5pi/6 about point X. Let C be the reflection of point X across point S. Let N be the vector from point S to point X. Let H be point E translated by vector N. Line W is the angle bisector of angle HSC. Point I is the midpoint of segment HX. Point M is the midpoint of segment IX. Point U is the reflection of point M across line W. Let Q be the midpoint of segment UM. What is the measure of angle HIQ, 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": "10.05", "dataset": "MATH"}
{"messages": [{"content": "Let SE = 5. Let W be the rotation of point E by angle pi/12 around point S. Let V be the rotation of point W by angle pi/12 around point S. Point T is obtained by rotating point W by angle pi/3 about point S. Let P be the reflection of point T across point E. Let J be the midpoint of segment VW. Find the inradius of triangle PTJ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.41", "dataset": "MATH"}
{"messages": [{"content": "OY = 9. Point W is obtained by rotating point O by angle pi/2 about point Y. Let P be the rotation of point Y by angle 7pi/12 around point O. Point V is the reflection of point O across point Y. Let U be the angle bisector of angle OPY. Point D is the reflection of point W across line U. What is the inradius of triangle WVD? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.08", "dataset": "MATH"}
{"messages": [{"content": "Let IF = 8. Point B is obtained by rotating point I by angle 5pi/12 about point F. Point D is the incenter of triangle IBF. Point Z is the midpoint of segment DB. Let R be the circumcenter of triangle IBF. Find 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": "2.6", "dataset": "MATH"}
{"messages": [{"content": "Let YA = 7. Point K is obtained by rotating point Y by angle pi/6 about point A. Point T is the reflection of point K across segment AY. Let V be the centroid of triangle KTA. What is the distance between points Y 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": "2.96", "dataset": "MATH"}
{"messages": [{"content": "WP = 2. Line T is the perpendicular bisector of segment WP. Point A lies on line T at distance 10 from point W. Point U is the midpoint of segment PW. Let M be the midpoint of segment UA. Point L is the orthocenter of triangle WAP. What is 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": "4.9", "dataset": "MATH"}
{"messages": [{"content": "VY = 12. Point C is the midpoint of segment YV. Point E is obtained by rotating point V by angle pi/2 about point C. Point Q is obtained by rotating point Y by angle pi/12 about point C. Let D be the incircle of triangle QVE. Let G be the reflection of circle D through point Y. Point B is the inversion of point Y with respect to circle G. Find the distance between points B 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": "11.34", "dataset": "MATH"}
{"messages": [{"content": "CP = 6. Point A is obtained by rotating point C by angle 11pi/12 about point P. Let F be the perpendicular bisector of segment AC. Let L be the reflection of point A across point C. Point M is the reflection of point L across line F. What is 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": "35.69", "dataset": "MATH"}
{"messages": [{"content": "KU = 5. Point A is obtained by rotating point U by angle 5pi/12 about point K. Point S is the reflection of point K across point U. Let T be the perpendicular bisector of segment AS. Let L be the circumcenter of triangle KAU. Let H be the line through point L parallel to line T. Point E is the reflection of point U across line H. Find 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": "3.32", "dataset": "MATH"}
{"messages": [{"content": "Let CJ = 2. Point H is obtained by rotating point C by angle pi/3 about point J. Point D is the reflection of point C across point J. Point W is the reflection of point H across segment CJ. Let S be the circumcenter of triangle HCJ. What is the measure of angle DWS, 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": "QA = 10. Let V be the reflection of point Q across point A. Point P is obtained by rotating point V by angle pi/2 about point Q. Let F be the midpoint of segment QP. Point E is the reflection of point A across point Q. Line T passes through point E parallel to segment QP. Point C is the reflection of point F across line T. Find the inradius of triangle EPC. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.1", "dataset": "MATH"}
{"messages": [{"content": "Let PZ = 1. Point G is the midpoint of segment ZP. Point Y is obtained by rotating point Z by angle 3pi/4 about point G. Let S be the midpoint of segment YZ. Let Q be the midpoint of segment GZ. Let H be the angle bisector of angle SGY. Let V be the midpoint of segment YG. Point A is the reflection of point Q across line H. Let L be the midpoint of segment AV. Find the distance between points V 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": "0.14", "dataset": "MATH"}
{"messages": [{"content": "Let GA = 11. Let S be the vector from point G to point A. Let P be the rotation of point A by angle 11pi/12 around point G. Let Q be the midpoint of segment GP. Point T is the translation of point P by vector S. Circle B is the circumcircle of triangle TQA. Point H is the incenter of triangle TQA. Let K be the reflection of circle B through point A. Point U is the centroid of triangle TQA. Point W is the reflection of point H across segment TQ. Point C is the inversion of point G with respect to circle K. What is the measure of angle WUC, 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": "16.59", "dataset": "MATH"}
{"messages": [{"content": "Let MU = 2. Let K be the rotation of point M by angle pi/3 around point U. Let P be the rotation of point K by angle 5pi/6 around point M. Let C be the perpendicular bisector of segment PK. Point F is the reflection of point U across line C. Let S be the midpoint of segment UM. Let H be the reflection of point S across point K. Find the inradius of triangle FHK. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.14", "dataset": "MATH"}
{"messages": [{"content": "PU = 7. Point H is obtained by rotating point P by angle 2pi/3 about point U. Let I be the midpoint of segment PU. Point Q is the midpoint of segment HP. Let W be the midpoint of segment HU. What is the circumradius of triangle IWQ? Express your 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": "Let JI = 2. Point T is obtained by rotating point J by angle 2pi/3 about point I. Let L be the midpoint of segment TJ. Circle R is the circumcircle of triangle JIT. Point K is the inversion of point L with respect to circle R. Point U is the incenter of triangle JIT. Let Q be the midpoint of segment KJ. Find the measure of angle QLU, 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 HZ = 5. Line Y is the perpendicular bisector of segment HZ. Point C is the reflection of point H across point Z. Point K is on line Y with |KZ| = 5. Let Q be the reflection of point H across point C. Point J is the reflection of point Q across segment KH. Point N is the reflection of point Q across line Y. Find the measure of angle NJK, 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": "132.5", "dataset": "MATH"}
{"messages": [{"content": "Let KU = 10. Let J be the vector from point K to point U. Point M is obtained by rotating point K by angle pi/3 about point U. Point P is the translation of point U by vector J. Let I be the midpoint of segment UP. Point D is the reflection of point I across point P. Let V be the incenter of triangle MKP. Let G be the line through point V parallel to segment KP. Point B is the midpoint of segment PM. Let T be the reflection of point B across point P. Point F is the reflection of point T across line G. What is the measure of angle DVF, 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": "113.28", "dataset": "MATH"}
{"messages": [{"content": "EG = 3. Let L be the perpendicular bisector of segment GE. Point D is on line L with |DG| = 1. Line A is the angle bisector of angle EGD. Line U is the reflection of line L across line A. Line Z is the reflection of line A across point D. Point M is the reflection of point G across line Z. Segment EG and line U intersect at point X. Let I be the incenter of triangle XMD. 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": "0.96", "dataset": "MATH"}
{"messages": [{"content": "Let VT = 7. Point J is obtained by rotating point V by angle 2pi/3 about point T. Let C be the incenter of triangle TVJ. Let I be the reflection of point T across point V. Let Z be the midpoint of segment IC. Find 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": "0.94", "dataset": "MATH"}
{"messages": [{"content": "PS = 4. Let J be the rotation of point S by angle pi/12 around point P. Let V be the midpoint of segment PJ. Let O be the midpoint of segment VJ. Line T is the perpendicular bisector of segment OS. Point A is the reflection of point V across line T. Point E is the midpoint of segment JA. Point Q is the reflection of point A across segment JS. Find the circumradius of triangle PQE. Express your 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": "NI = 3. Let V be the rotation of point N by angle 7pi/12 around point I. Point W is obtained by rotating point I by angle 2pi/3 about point N. Point F is the midpoint of segment WV. Point R is the reflection of point V across segment NI. Let Z be the line through points F and R. Line Z and segment VN intersect at point T. 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": "4.61", "dataset": "MATH"}
{"messages": [{"content": "DN = 8. Let I be the midpoint of segment DN. Let P be the rotation of point D by angle pi/2 around point N. Point E is the reflection of point P across point N. Point M is obtained by rotating point P by angle 11pi/12 about point N. Point L is the midpoint of segment IM. Point X is the midpoint of segment ML. Let G be the midpoint of segment LD. What is the measure of angle EXG, 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": "6.62", "dataset": "MATH"}
{"messages": [{"content": "Let LB = 8. Point J is obtained by rotating point L by angle 3pi/4 about point B. Let C be the circumcenter of triangle LJB. Point A is the reflection of point J across point L. Let M be the midpoint of segment BL. Find the measure of angle MCA, 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": "7.54", "dataset": "MATH"}
{"messages": [{"content": "AY = 5. Let R be the rotation of point A by angle 11pi/12 around point Y. Point L is the reflection of point R across point A. Let M be the reflection of point Y across point L. Point V is the midpoint of segment AM. Let J be the centroid of triangle RAV. Let X be the incenter of triangle RAV. What is the measure of angle JMX, 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": "0.21", "dataset": "MATH"}
{"messages": [{"content": "Let TH = 8. Let Z be the rotation of point H by angle pi/12 around point T. Point F is the reflection of point T across point H. Circle R is the incircle of triangle FZT. Point I is the reflection of point H across segment ZT. Point A is the circumcenter of triangle FZT. Let N be the reflection of circle R through point A. Point U is the inversion of point I with respect to circle N. What is the length of segment UA? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.11", "dataset": "MATH"}
{"messages": [{"content": "Let UG = 12. Point Z is the reflection of point U across point G. UR = 12. Point B is the reflection of point R across point Z. Let W be the circumcenter of triangle UZB. Point S is the reflection of point W across segment UZ. What is 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": "4.0", "dataset": "MATH"}
{"messages": [{"content": "US = 4. Point D is obtained by rotating point U by angle pi/3 about point S. Let H be the orthocenter of triangle DUS. Point I is the reflection of point D across segment US. Point L is the reflection of point I across segment DU. Find the length of segment LH. Express your 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": "HS = 2. Point M is obtained by rotating point S by angle 5pi/12 about point H. Let G be the line through point M parallel to segment SH. Point O is the reflection of point H across line G. Point D is the reflection of point M across segment SH. What is the circumradius of triangle DOS? Express your 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 BN = 3. Point H is the reflection of point B across point N. Let T be the reflection of point H across point B. Line J is the angle bisector of angle HBT. Point M lies on line J at distance 5 from point N. Point Y is the reflection of point B across point H. Point P is obtained by rotating point Y by angle pi about point M. Line K passes through point N perpendicular to segment NT. Point R is the reflection of point P across line K. Let X be the incenter of triangle NPR. Find the length of segment XH. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.23", "dataset": "MATH"}
{"messages": [{"content": "Let XY = 2. Line Q is the perpendicular bisector of segment XY. Point F is on line Q with |FX| = 11. Let G be the reflection of point F across point X. Let D be the line through point Y perpendicular to segment XY. Let H be the line through point G perpendicular to segment XY. Point W is the reflection of point F across line D. Point O is the reflection of point X across line H. Find the circumradius of triangle WFO. Express your 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 SR = 7. Point K is obtained by rotating point R by angle 5pi/6 about point S. Let L be the midpoint of segment KR. Point M is the midpoint of segment SK. Point U is the reflection of point L across point M. Find the circumradius of triangle UKR. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.82", "dataset": "MATH"}
{"messages": [{"content": "Let MC = 12. Point T is the reflection of point C across point M. Let I be the rotation of point M by angle 3pi/4 around point C. Let F be the perpendicular bisector of segment MI. Let A be the centroid of triangle TMI. Point P is the midpoint of segment AC. Point J is the reflection of point P across line F. 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": "26.58", "dataset": "MATH"}
{"messages": [{"content": "BR = 6. Let X be the rotation of point R by angle pi around point B. Point S is the reflection of point R across point X. Point J is the reflection of point X across point R. Point V is obtained by rotating point B by angle 3pi/4 about point J. Let D be the line through points B and V. Point G is the reflection of point S across line D. Let P be the midpoint of segment BR. Find the inradius of triangle PVG. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.51", "dataset": "MATH"}
{"messages": [{"content": "ZJ = 5. Point G is the reflection of point Z across point J. Line Y is the perpendicular bisector of segment ZG. Point T is the midpoint of segment GJ. Point X is on line Y with |XT| = 5. Let Q be the line through point Z parallel to line Y. Point O is the reflection of point T across line Q. Let V be the reflection of point O across point X. Point P is the midpoint of segment OG. What is the measure of angle VPT, 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": "31.25", "dataset": "MATH"}
{"messages": [{"content": "RJ = 1. Point K is obtained by rotating point J by angle pi/4 about point R. Line C is the perpendicular bisector of segment JR. Let M be the rotation of point J by angle 3pi/4 around point R. Let T be the midpoint of segment KM. Point O is the reflection of point T across line C. Point A is the midpoint of segment OJ. Line V passes through point M parallel to segment KR. Let E be the reflection of line V through point R. Point L is the reflection of point T across line E. 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": "2.09", "dataset": "MATH"}
{"messages": [{"content": "Let TE = 10. Point P is the midpoint of segment TE. Let I be the rotation of point E by angle 7pi/12 around point T. Let L be the midpoint of segment IP. Let N be the midpoint of segment EI. Point B is the reflection of point N across segment PE. Let R be the midpoint of segment BI. What is the distance between points R 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": "2.5", "dataset": "MATH"}
{"messages": [{"content": "IL = 5. Let A be the midpoint of segment IL. Line M is the perpendicular bisector of segment IL. Let F be the line through point L parallel to line M. Point S is the midpoint of segment IA. Let K be the rotation of point I by angle 5pi/12 around point S. Let W be the reflection of point L across point K. Point D is the reflection of point K across line F. Line X passes through points I and W. Point U is the reflection of point A across line X. Let R be the midpoint of segment UL. Find the circumradius of triangle IDR. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.56", "dataset": "MATH"}
{"messages": [{"content": "CJ = 11. Point I is obtained by rotating point C by angle 3pi/4 about point J. Let V be the rotation of point C by angle 11pi/12 around point J. Point L is the reflection of point C across point V. Let K be the circumcenter of triangle IVL. Point H is the orthocenter of triangle IVL. What is the measure of angle HKL, 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": "96.89", "dataset": "MATH"}
{"messages": [{"content": "CG = 7. Point X is obtained by rotating point C by angle pi/6 about point G. Point E is the reflection of point G across point C. Let R be the rotation of point G by angle pi/2 around point X. Let U be the angle bisector of angle GXR. Point B is the reflection of point C across line U. Point T is the midpoint of segment GX. What is the circumradius of triangle TBE? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.79", "dataset": "MATH"}
{"messages": [{"content": "EN = 11. Point F is the reflection of point N across point E. Let T be the reflection of point F across point N. Line L is the perpendicular bisector of segment ET. Point O lies on line L at distance 4 from point N. Point G lies on line L at distance 6 from point F. Point P is the reflection of point G across point N. Point H is the reflection of point P across line L. Point R is the reflection of point F across segment OH. 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": "38.96", "dataset": "MATH"}
{"messages": [{"content": "EM = 2. Point D is obtained by rotating point M by angle pi/6 about point E. Let A be the midpoint of segment ME. Point Z is the centroid of triangle DAE. Point U is the incenter of triangle DAE. What is the length of segment ZU? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.1", "dataset": "MATH"}
{"messages": [{"content": "AO = 4. Let E be the rotation of point O by angle pi/3 around point A. Point X is obtained by rotating point E by angle pi/4 about point O. Let Z be the midpoint of segment OX. Point Y is the reflection of point X across segment AZ. Point I is the incenter of triangle EAZ. What is the circumradius of triangle OIY? Express your 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 WG = 3. Let Q be the vector from point G to point W. Point Y is the translation of point W by vector Q. Line M is the perpendicular bisector of segment WG. Point O is the midpoint of segment YW. Let J be the rotation of point G by angle 2pi/3 around point O. Let L be the line through point O parallel to line M. Point T is obtained by rotating point J by angle 3pi/4 about point Y. Line I is the angle bisector of angle GTW. Line V is the reflection of line I across point Y. Line U is the reflection of line M across line V. Lines L and U intersect at point X. Find the inradius of triangle YXT. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.46", "dataset": "MATH"}
{"messages": [{"content": "Let SN = 2. Let P be the perpendicular bisector of segment SN. Point Y is obtained by rotating point S by angle pi/2 about point N. Point K is obtained by rotating point N by angle 5pi/6 about point Y. Point G is the reflection of point Y across line P. Point Q is the reflection of point K across segment SG. What is the circumradius of triangle QGY? Express your 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 HS = 2. Point T is obtained by rotating point H by angle pi/6 about point S. Line W passes through points S and T. Point C is the reflection of point H across line W. Point E is the reflection of point C across point T. Point R is the midpoint of segment HE. What is the circumradius of triangle SRH? Express your 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 FI = 12. Let R be the rotation of point I by angle 2pi/3 around point F. Let P be the circumcenter of triangle RIF. Let H be the incenter of triangle RIF. Point G is the midpoint of segment IH. Let T be the midpoint of segment PF. Point Q is the midpoint of segment TP. 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": "5.44", "dataset": "MATH"}
{"messages": [{"content": "Let NG = 2. Let V be the rotation of point G by angle 5pi/6 around point N. Point Z is the reflection of point G across point N. Let H be the line through point N perpendicular to segment GN. Point T is the midpoint of segment VZ. Point Q is the reflection of point T across line H. Point E is the reflection of point Q across segment GN. Find the inradius of triangle EQG. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.07", "dataset": "MATH"}
{"messages": [{"content": "WY = 9. Point A is the midpoint of segment YW. Point T is obtained by rotating point W by angle pi/12 about point A. Point N is the reflection of point Y across segment TA. Let I be the midpoint of segment NT. 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": "4.46", "dataset": "MATH"}
{"messages": [{"content": "Let CS = 8. Let I be the rotation of point S by angle pi/6 around point C. Point K is the reflection of point I across point C. Let D be the angle bisector of angle CSK. Point G is the reflection of point K across line D. Let X be the reflection of line D through point K. Point V is the midpoint of segment GC. Let P be the midpoint of segment VS. Point Q is the reflection of point K across line X. What is the circumradius of triangle QPG? Express your 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": "RM = 2. Point C is obtained by rotating point R by angle pi/6 about point M. Line H is the perpendicular bisector of segment MR. Let T be the midpoint of segment RC. Let I be the line through point T parallel to line H. Point K is the reflection of point C across segment MR. Point L is the reflection of point K across line I. Point U is the reflection of point L across line H. Point W is the midpoint of segment KC. Find the measure of angle TWU, 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": "0.37", "dataset": "MATH"}
{"messages": [{"content": "Let DQ = 6. Line A is the perpendicular bisector of segment DQ. Let O be the rotation of point Q by angle 7pi/12 around point D. Let F be the reflection of point Q across point O. Let I be the reflection of point D across point Q. Point G is the reflection of point I across line A. What is the inradius of triangle FGQ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.23", "dataset": "MATH"}
{"messages": [{"content": "Let BR = 4. Line M is the perpendicular bisector of segment BR. Point O is on line M with |OB| = 5. Line N passes through point R parallel to line M. Let S be the perpendicular bisector of segment OB. Let E be the reflection of line S across line N. Let A be the reflection of point O across point B. Point X is the orthocenter of triangle ORB. Point U is the reflection of point X across line E. Let P be the midpoint of segment OR. What is the measure of angle AUP, 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": "2.57", "dataset": "MATH"}
{"messages": [{"content": "FJ = 7. Let I be the rotation of point F by angle 5pi/12 around point J. Line U passes through points I and F. Point W is obtained by rotating point I by angle pi/4 about point F. Let X be the line through point W perpendicular to line U. Point Y is the midpoint of segment FW. Point Z is the reflection of point I across line X. Find the circumradius of triangle ZYW. Express your 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": "CA = 8. Let G be the rotation of point A by angle pi/3 around point C. Point E is the midpoint of segment CA. Point H is the reflection of point E across point C. Let M be the reflection of point C across point H. Let I be the midpoint of segment GE. Find the measure of angle MHI, 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": "6.93", "dataset": "MATH"}
{"messages": [{"content": "SN = 2. Let L be the rotation of point S by angle 2pi/3 around point N. Let D be the reflection of point N across point L. Let X be the rotation of point N by angle 5pi/12 around point L. Let A be the midpoint of segment LX. Line T is the angle bisector of angle XSN. Point E is the reflection of point D across line T. Point Y is the orthocenter of triangle AEX. What is the distance between points Y 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 ZD = 3. Let U be the midpoint of segment ZD. Let W be the rotation of point U by angle 11pi/12 around point D. Let I be the rotation of point Z by angle 5pi/12 around point W. Point F is the reflection of point Z across point D. Point Y is the midpoint of segment DF. Point K is the midpoint of segment DW. Point E is the reflection of point Y across point F. Point R is the midpoint of segment IK. 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": "3.59", "dataset": "MATH"}
{"messages": [{"content": "SP = 2. Let A be the rotation of point P by angle 3pi/4 around point S. Let E be the midpoint of segment PS. Let B be the incircle of triangle PAS. Let V be the inversion of point E in circle B. Let F be the midpoint of segment AE. What is the measure of angle FVP, 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": "AB = 4. Point F is the midpoint of segment BA. Point L is obtained by rotating point A by angle 3pi/4 about point F. Point V is obtained by rotating point L by angle pi/6 about point A. Point J is obtained by rotating point B by angle 11pi/12 about point A. Let Y be the angle bisector of angle BVJ. Let R be the line through point A parallel to line Y. Point T is the reflection of point B across line R. What is the inradius of triangle JAT? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.01", "dataset": "MATH"}
{"messages": [{"content": "Let BE = 11. Point Y is the midpoint of segment EB. Let G be the rotation of point E by angle 11pi/12 around point B. Point W is the reflection of point B across point G. Let I be the angle bisector of angle BWE. Point D is the reflection of point E across line I. Point A is the reflection of point G across segment DE. What is the measure of angle AYD, 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": "20.56", "dataset": "MATH"}
{"messages": [{"content": "Let JF = 3. Let V be the rotation of point J by angle pi/3 around point F. Let P be the midpoint of segment JF. Point Q is the reflection of point V across point J. Point Y is the incenter of triangle VPJ. What is the circumradius of triangle YQF? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.82", "dataset": "MATH"}
{"messages": [{"content": "Let FJ = 8. Let N be the rotation of point J by angle 5pi/6 around point F. Let S be the centroid of triangle NFJ. Let R be the rotation of point N by angle pi/12 around point F. Let W be the orthocenter of triangle NFJ. Let A be the midpoint of segment RF. Point O is obtained by rotating point W by angle pi/6 about point N. Let T be the rotation of point O by angle pi/6 around point J. Point Z is the reflection of point S across point O. Line X passes through point Z parallel to segment JN. Line P passes through point T perpendicular to line X. Line B is the perpendicular bisector of segment SA. Line I is the reflection of line P across line B. Lines I and X intersect at point Y. Let K be the midpoint of segment JN. Find the measure of angle YKA, 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": "80.45", "dataset": "MATH"}
{"messages": [{"content": "YQ = 6. Let X be the reflection of point Q across point Y. Let H be the rotation of point X by angle 11pi/12 around point Q. Point Z is obtained by rotating point H by angle 5pi/6 about point X. Let F be the midpoint of segment XH. Point N is the circumcenter of triangle HZF. Point R is the reflection of point X across segment ZF. Find the measure of angle RNQ, 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": "10.1", "dataset": "MATH"}
{"messages": [{"content": "Let MN = 10. Let G be the rotation of point N by angle 5pi/12 around point M. Point C is the circumcenter of triangle NMG. Let J be the circumcircle of triangle NMG. Let V be the reflection of point C across point G. Let U be the inversion of point V in circle J. Find the length of segment UC. Express your 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 QV = 4. Let C be the rotation of point V by angle 11pi/12 around point Q. Point B is obtained by rotating point C by angle pi/6 about point Q. Let Z be the reflection of point Q across point V. Let U be the midpoint of segment BV. Point M is the midpoint of segment BC. Point W is the reflection of point U across segment ZQ. What is the measure of angle UMW, 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 KJ = 5. Let P be the midpoint of segment JK. Let W be the rotation of point K by angle 11pi/12 around point P. Let L be the reflection of point P across point K. Let N be the reflection of point W across point K. Point U is the midpoint of segment PW. Let Z be the reflection of point U across point N. Find the length of segment LZ. Express your 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": "GQ = 3. Point F is obtained by rotating point G by angle 3pi/4 about point Q. Let P be the reflection of point Q across point G. Point H is the reflection of point G across point Q. Line Y is the perpendicular bisector of segment QF. Point M is obtained by rotating point F by angle 2pi/3 about point G. Let T be the reflection of point M across point G. Let Z be the incircle of triangle FPQ. Line I passes through point T perpendicular to line Y. Point R is the reflection of point F across segment PQ. Let L be the line through point G parallel to line I. Point J is the reflection of point R across line L. Let W be the inversion of point H in circle Z. Find 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": "6.66", "dataset": "MATH"}
{"messages": [{"content": "MX = 3. Point H is the midpoint of segment XM. Let W be the rotation of point M by angle 5pi/12 around point H. Let A be the rotation of point X by angle pi/3 around point M. Point I is the reflection of point H across segment WA. Point O is the midpoint of segment MW. Let V be the incenter of triangle XWA. Point K is the centroid of triangle VOI. What is the distance between points H 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": "0.34", "dataset": "MATH"}
{"messages": [{"content": "Let OZ = 11. Let D be the rotation of point Z by angle 2pi/3 around point O. Let U be the midpoint of segment ZO. Point V is the reflection of point O across segment UD. Point W is the reflection of point Z across segment UD. Find the circumradius of triangle VWO. Express your 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": "AK = 2. Point N is obtained by rotating point A by angle 7pi/12 about point K. Point E is the reflection of point A across point N. Point Q is the midpoint of segment EN. Let O be the midpoint of segment QN. Point W is the incenter of triangle ENK. Find 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": "0.77", "dataset": "MATH"}
{"messages": [{"content": "OV = 4. Let B be the line through points O and V. Line R passes through point V perpendicular to line B. Point X is the reflection of point O across line R. Let C be the reflection of point O across point X. Point L lies on line R at distance 8 from point V. Point Q is the reflection of point L across point X. Line D is the reflection of line B across point L. Point U is the reflection of point C across line D. Let I be the reflection of point V across point Q. What is the circumradius of triangle XIU? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "31.03", "dataset": "MATH"}
{"messages": [{"content": "FW = 12. Let P be the rotation of point F by angle pi/6 around point W. Let I be the angle bisector of angle PFW. Point J is the reflection of point W across line I. Let Q be the midpoint of segment WJ. 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": "7.31", "dataset": "MATH"}
{"messages": [{"content": "Let JW = 9. Point I is obtained by rotating point J by angle pi/3 about point W. Point Y is the reflection of point I across point J. Let T be the incenter of triangle WIY. Point H is the centroid of triangle WIY. What is the length of segment HT? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.92", "dataset": "MATH"}
{"messages": [{"content": "PJ = 8. V is constructed so that JV = 8. Point Q is the midpoint of segment PJ. Let B be the midpoint of segment VP. Point Y is the centroid of triangle VPQ. What is the length of segment YB? Express your 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 GJ = 1. Point F is obtained by rotating point J by angle pi/6 about point G. Point M is the circumcenter of triangle GJF. Let P be the line through point M parallel to segment JF. Point R is the reflection of point J across line P. 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": "0.41", "dataset": "MATH"}
{"messages": [{"content": "Let GW = 4. Let S be the rotation of point W by angle pi/2 around point G. Let R be the perpendicular bisector of segment WG. Circle C is the incircle of triangle WGS. Let N be the reflection of circle C across line R. Point J is the midpoint of segment SW. Let O be the inversion of point W in circle N. Find the distance between points J 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": "2.0", "dataset": "MATH"}
{"messages": [{"content": "WN = 8. Point C is obtained by rotating point N by angle pi/3 about point W. Point B is the reflection of point C across point N. Point X is obtained by rotating point N by angle 5pi/6 about point C. Point H is the reflection of point W across point N. Let Q be the reflection of point H across point C. Point L is the reflection of point W across segment NX. Let S be the reflection of point C across point B. What is the measure of angle LQS, 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": "259.14", "dataset": "MATH"}
{"messages": [{"content": "Let IF = 9. Let T be the midpoint of segment IF. Point X is obtained by rotating point I by angle pi/6 about point F. Point H is the reflection of point F across point X. Line L is the angle bisector of angle FHT. Point O is the midpoint of segment TF. Point R is the reflection of point O across line L. What is the measure of angle HTR, 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": "8.03", "dataset": "MATH"}
{"messages": [{"content": "Let RM = 12. Let N be the rotation of point R by angle 7pi/12 around point M. Point J is the midpoint of segment RM. Let K be the circumcenter of triangle MNJ. Let S be the reflection of point R across point K. Point F is the reflection of point N across point S. 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": "29.29", "dataset": "MATH"}
{"messages": [{"content": "Let EA = 11. Let O be the rotation of point E by angle 7pi/12 around point A. Let R be the rotation of point E by angle 5pi/6 around point A. Point W is the incenter of triangle ARO. Point N is the reflection of point W across segment AR. Point C is the reflection of point W across point N. Let D be the midpoint of segment AR. What is the length of segment DC? Express your 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 OH = 3. Let R be the rotation of point O by angle pi/2 around point H. Let W be the reflection of point R across point O. Point C is the reflection of point R across point W. Point T is the reflection of point R across segment HO. Point D is the reflection of point W across segment HO. Find the measure of angle TDC, 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": "54.0", "dataset": "MATH"}
{"messages": [{"content": "Let UK = 6. Let D be the midpoint of segment UK. Let V be the rotation of point K by angle pi/6 around point D. Point X is the midpoint of segment VU. Let G be the line through points U and K. Point R is the reflection of point X across line G. Find the inradius of triangle DXR. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.1", "dataset": "MATH"}
{"messages": [{"content": "JB = 1. Let L be the rotation of point J by angle 3pi/4 around point B. Point T is the midpoint of segment LB. Let C be the circumcircle of triangle BJL. Let F be the circumcenter of triangle BJL. Let H be the inversion of point F in circle C. What is the length of segment HT? Express your 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": "WY = 1. Let V be the reflection of point W across point Y. Let X be the angle bisector of angle VYW. Line S is the perpendicular bisector of segment YW. Point L is obtained by rotating point Y by angle pi/4 about point W. Point F is the reflection of point L across point W. Point Q is obtained by rotating point L by angle pi/2 about point F. Let I be the midpoint of segment VL. Let M be the reflection of line X through point Q. Point A is on line M. Let P be the midpoint of segment WQ. Point B is the reflection of point P across line S. Find the measure of angle FBI, 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": "0.38", "dataset": "MATH"}
{"messages": [{"content": "Let BN = 1. Let L be the rotation of point B by angle 11pi/12 around point N. Point I is obtained by rotating point N by angle pi/2 about point B. Let J be the reflection of point B across point I. Let Q be the midpoint of segment LN. Point E is the reflection of point Q across segment JB. Let U be the incenter of triangle BLJ. What is the measure of angle JUE, 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": "1.72", "dataset": "MATH"}
{"messages": [{"content": "Let LN = 6. Point D is obtained by rotating point L by angle pi/2 about point N. Let A be the midpoint of segment LD. Line Q passes through point L perpendicular to segment ND. Circle S is the circumcircle of triangle DAN. Point M is the reflection of point D across line Q. Point W is the circumcenter of triangle DAN. Point T is the reflection of point M across segment DA. Point B is the inversion of point L with respect to circle S. What is the measure of angle WTB, 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": "3.6", "dataset": "MATH"}
{"messages": [{"content": "Let LC = 2. Let Q be the line through points L and C. Point A is the reflection of point L across point C. Line D is the perpendicular bisector of segment AC. Point R is on line D with |RC| = 9. Point T is the reflection of point R across line Q. Point F is the reflection of point A across point T. Point K is the reflection of point C across point F. What is the inradius of triangle KRL? Express your 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": "Let VR = 8. Point Q is obtained by rotating point V by angle 3pi/4 about point R. Let K be the midpoint of segment RQ. Let W be the orthocenter of triangle KVQ. Let P be the reflection of point R across point V. 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": "35.09", "dataset": "MATH"}
{"messages": [{"content": "HZ = 2. Point I is obtained by rotating point H by angle pi/3 about point Z. Point P is the reflection of point I across point H. Line E is the perpendicular bisector of segment IP. Let J be the midpoint of segment PZ. Point W is the reflection of point Z across line E. Find the circumradius of triangle WZJ. Express your 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 ZY = 6. Point A is obtained by rotating point Y by angle 3pi/4 about point Z. Line E is the perpendicular bisector of segment AZ. Point S is the reflection of point Z across point A. Point W is the reflection of point S across line E. Point N is the midpoint of segment WY. Let L be the reflection of point Y across point A. Find the measure of angle ZNL, 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": "6.36", "dataset": "MATH"}
{"messages": [{"content": "Let BN = 3. Point J is the reflection of point B across point N. Point A is obtained by rotating point B by angle pi/6 about point N. Let M be the reflection of point N across point B. Let U be the orthocenter of triangle AMJ. Point P is the midpoint of segment AM. Find the measure of angle NUP, 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": "26.32", "dataset": "MATH"}
{"messages": [{"content": "PG = 4. Let K be the rotation of point G by angle pi/4 around point P. Point F is obtained by rotating point P by angle pi/12 about point K. Let C be the orthocenter of triangle GFP. Point M is the incenter of triangle GFP. Point Q is the reflection of point M across segment FK. What is the length of segment QC? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.74", "dataset": "MATH"}
{"messages": [{"content": "Let HL = 4. Point R is obtained by rotating point H by angle 2pi/3 about point L. Point D is the reflection of point R across point L. Point K is the circumcenter of triangle LHR. Let S be the midpoint of segment KD. What is the distance between points S 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": "2.4", "dataset": "MATH"}
{"messages": [{"content": "Let UK = 2. Let H be the rotation of point U by angle 5pi/6 around point K. Let P be the midpoint of segment KH. Let C be the circumcircle of triangle UHK. Let N be the reflection of circle C through point P. Let Z be the inversion of point P in circle N. Let R be the incenter of triangle UHK. 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": "0.93", "dataset": "MATH"}
{"messages": [{"content": "Let CA = 8. Line M is the perpendicular bisector of segment CA. Line W is the reflection of line M across point A. Point U lies on line M at distance 5 from point A. Point G is the reflection of point A across line W. Point T is the midpoint of segment AU. Let V be the orthocenter of triangle UAG. Find the circumradius of triangle VGT. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.71", "dataset": "MATH"}
{"messages": [{"content": "Let KD = 4. Let C be the rotation of point K by angle pi around point D. Line G is the angle bisector of angle CKD. Let A be the rotation of point C by angle pi/3 around point K. Let M be the rotation of point K by angle pi/2 around point C. Let F be the midpoint of segment MK. Point R is the midpoint of segment AC. Point Y is the reflection of point M across line G. What is the circumradius of triangle YRF? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "41.37", "dataset": "MATH"}
{"messages": [{"content": "UT = 7. Point E is the midpoint of segment TU. Let J be the rotation of point T by angle pi/4 around point E. Point R is obtained by rotating point J by angle pi about point U. Let V be the rotation of point J by angle pi/2 around point U. Point P is obtained by rotating point R by angle 3pi/4 about point T. Let K be the midpoint of segment VE. Point I is the incenter of triangle VJK. Let H be the reflection of point P across point I. Find 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": "20.39", "dataset": "MATH"}
{"messages": [{"content": "Let NY = 10. Line T is the perpendicular bisector of segment NY. Point U lies on line T at distance 1 from point Y. Let L be the midpoint of segment NU. Line D passes through point N parallel to line T. Let E be the incenter of triangle YNU. Point R is the circumcenter of triangle YNU. Point X is the reflection of point E across line D. Find the measure of angle RXL, 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 WC = 3. Let E be the midpoint of segment WC. Let D be the rotation of point W by angle pi/6 around point C. Point H is the reflection of point D across point W. Point A is the midpoint of segment DE. Let T be the incenter of triangle CHA. Find 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": "2.18", "dataset": "MATH"}
{"messages": [{"content": "EI = 10. Point F is obtained by rotating point I by angle pi/2 about point E. Point T is the midpoint of segment FE. Point R is the incenter of triangle TFI. Let K be the orthocenter of triangle TFI. 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": "8.97", "dataset": "MATH"}
{"messages": [{"content": "Let BH = 4. Line O is the perpendicular bisector of segment HB. Point C is on line O with |CB| = 8. Point F is the reflection of point H across point B. Point Y is the reflection of point F across line O. Point D is the centroid of triangle HCF. 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": "7.8", "dataset": "MATH"}
{"messages": [{"content": "CB = 11. Point H is the midpoint of segment CB. Point Y is the reflection of point B across point C. Let S be the rotation of point B by angle 3pi/4 around point Y. Line Q passes through point B perpendicular to segment SH. Line U passes through point C parallel to line Q. Point G is the intersection of segment SH and line U. Find the distance between points C 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": "2.4", "dataset": "MATH"}
{"messages": [{"content": "Let MH = 8. Let B be the rotation of point H by angle pi/4 around point M. Line V passes through points B and H. Point G is obtained by rotating point B by angle 5pi/6 about point H. Point I is the reflection of point M across line V. Line K passes through point M parallel to line V. Point P is the midpoint of segment MB. Point X is the reflection of point G across line K. Find the measure of angle XIP, 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": "60.99", "dataset": "MATH"}
{"messages": [{"content": "RI = 7. Point V is obtained by rotating point I by angle pi/4 about point R. Point W is the reflection of point V across point R. Let A be the perpendicular bisector of segment RV. Point O is the reflection of point W across line A. Let J be the circumcenter of triangle WVI. Let X be the angle bisector of angle JIW. Point Y is the reflection of point W across line X. 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": "18.67", "dataset": "MATH"}
{"messages": [{"content": "NI = 9. Let P be the reflection of point I across point N. Let O be the reflection of point P across point I. Let M be the midpoint of segment NO. Let B be the rotation of point M by angle 11pi/12 around point O. Line C passes through points B and O. Point H is the reflection of point N across line C. Point E is the reflection of point N across point I. What is the circumradius of triangle MEH? Express your 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 RL = 7. Let U be the rotation of point R by angle pi/6 around point L. Let W be the rotation of point R by angle pi/2 around point U. Point P is the incenter of triangle WLU. Point H is the midpoint of segment PW. What is the circumradius of triangle HRW? Express your 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": "XN = 7. Let K be the rotation of point X by angle pi/3 around point N. Let J be the rotation of point X by angle pi/2 around point K. Point S is the reflection of point K across point J. Let Q be the reflection of point N across point J. Point M is the reflection of point N across point S. Find the circumradius of triangle MQX. Express your 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": "Let SX = 10. Point D is the midpoint of segment SX. Let W be the perpendicular bisector of segment XD. Let I be the line through point X parallel to line W. Let T be the midpoint of segment DS. Point Y is the reflection of point S across line I. Point B is obtained by rotating point X by angle 5pi/12 about point T. Point V is the midpoint of segment SB. Point J is the reflection of point T across segment VY. Find the inradius of triangle TJB. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.21", "dataset": "MATH"}
{"messages": [{"content": "ID = 1. Let S be the rotation of point I by angle 5pi/12 around point D. Let B be the circumcenter of triangle SDI. Point O is the reflection of point B across segment SD. Point R is the reflection of point B across segment IS. Find the measure of angle ROI, 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": "0.35", "dataset": "MATH"}
{"messages": [{"content": "RH = 3. Point K is obtained by rotating point R by angle 5pi/6 about point H. Let Y be the reflection of point K across point R. Let Q be the midpoint of segment RK. Let S be the midpoint of segment QY. Find 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": "4.35", "dataset": "MATH"}
{"messages": [{"content": "EC = 11. Let N be the perpendicular bisector of segment EC. Let D be the reflection of point C across point E. Point B is on line N with |BC| = 12. Point I is the reflection of point D across line N. Point S is the reflection of point B across segment DC. Point U is the reflection of point I across segment CB. Find the distance between points U 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": "4.03", "dataset": "MATH"}
{"messages": [{"content": "Let QF = 5. Point U is obtained by rotating point F by angle 7pi/12 about point Q. Let Y be the centroid of triangle UFQ. Line X passes through point F parallel to segment QU. Let A be the midpoint of segment QU. Line I passes through point Y parallel to line X. Point S is the intersection of line I and segment FQ. Point L is the midpoint of segment US. What is the measure of angle LAQ, 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": "QH = 2. Point S is the reflection of point H across point Q. Let R be the rotation of point H by angle 5pi/12 around point S. Let W be the rotation of point S by angle 2pi/3 around point H. Let G be the midpoint of segment SR. Let M be the angle bisector of angle GWQ. Point K is the reflection of point Q across line M. Find the circumradius of triangle KHR. Express your 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": "LX = 3. Line N passes through points L and X. Point U is obtained by rotating point L by angle 3pi/4 about point X. Let G be the midpoint of segment XU. Point J is the reflection of point G across line N. Let R be the midpoint of segment JL. Point D is the orthocenter of triangle XJR. Point Q is the centroid of triangle XJR. What is the measure of angle QDG, 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": "0.48", "dataset": "MATH"}
{"messages": [{"content": "Let MA = 12. Point Z is the midpoint of segment MA. Let K be the rotation of point A by angle 11pi/12 around point Z. Point S is the incenter of triangle MKZ. Point X is the midpoint of segment ZS. Point Y is the centroid of triangle MKZ. What is the distance between points Y 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": "1.33", "dataset": "MATH"}
{"messages": [{"content": "Let HD = 11. Point F is obtained by rotating point H by angle pi/2 about point D. Line Z is the perpendicular bisector of segment FH. Point T is the circumcenter of triangle DFH. Let N be the reflection of point F across point T. Point K is the reflection of point N across line Z. Let U be the incircle of triangle DFH. Point X is the inversion of point H with respect to circle U. 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": "9.24", "dataset": "MATH"}
{"messages": [{"content": "FT = 4. Point E is the reflection of point F across point T. Point G is obtained by rotating point E by angle pi/2 about point F. Let C be the rotation of point G by angle 3pi/4 around point F. Point I is the reflection of point C across point F. Point N is the reflection of point I across point E. Point Q is the reflection of point I across segment GN. Point Y is the centroid of triangle CGN. What is the measure of angle QYF, 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": "1.9", "dataset": "MATH"}
{"messages": [{"content": "RJ = 3. Point M is obtained by rotating point J by angle pi/12 about point R. Let N be the rotation of point M by angle 2pi/3 around point R. Let T be the midpoint of segment JR. Point Q is the reflection of point N across point T. Find the inradius of triangle JNQ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.38", "dataset": "MATH"}
{"messages": [{"content": "RW = 10. Point M is obtained by rotating point W by angle pi/4 about point R. Point G is the midpoint of segment WM. Let K be the reflection of point G across point R. Let B be the incircle of triangle WKM. Point U is the inversion of point M with respect to circle B. Let L be the line through point M parallel to segment WK. Let H be the centroid of triangle WKM. Point N is the reflection of point U across line L. Point J is the orthocenter of triangle NKH. 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": "14.18", "dataset": "MATH"}
{"messages": [{"content": "TH = 10. Point B is obtained by rotating point T by angle 3pi/4 about point H. Let Z be the perpendicular bisector of segment TB. Point O is obtained by rotating point T by angle pi about point B. Circle V is the circumcircle of triangle OHB. Let K be the reflection of circle V across line Z. Point L is the inversion of point B with respect to circle K. Find 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": "15.03", "dataset": "MATH"}
{"messages": [{"content": "Let R be the midpoint of segment Z92L. Point B is the orthocenter of triangle L. Let F be the midpoint of segment UR. Let Z be the reflection of point L across point F. Point G is the midpoint of segment BF. What is 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": "0.89", "dataset": "MATH"}
{"messages": [{"content": "YH = 11. Point D is the reflection of point Y across point H. Point K is obtained by rotating point H by angle pi/2 about point D. Line V passes through points D and H. Let N be the line through points K and H. Line B is the reflection of line N across line V. Point X is the midpoint of segment DH. Point C is the reflection of point Y across line B. What is the measure of angle KCX, 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": "60.5", "dataset": "MATH"}
{"messages": [{"content": "Let VK = 8. Let F be the rotation of point K by angle 5pi/6 around point V. Let Q be the midpoint of segment FK. Point N is the midpoint of segment QK. Let J be the midpoint of segment NK. Point O is the centroid of triangle QVN. Find the distance between points J 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": "4.56", "dataset": "MATH"}
{"messages": [{"content": "HC = 12. Point N is obtained by rotating point C by angle 3pi/4 about point H. Point M is the midpoint of segment CN. Point B is the incenter of triangle NHC. Let T be the midpoint of segment BM. Point P lies on segment NH. What is the measure of angle PHT, 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 UK = 3. Let V be the rotation of point K by angle pi/3 around point U. Point G is the midpoint of segment UK. Point B is the reflection of point G across segment VK. Find the inradius of triangle KGB. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.35", "dataset": "MATH"}
{"messages": [{"content": "GP = 6. Point D is obtained by rotating point G by angle 11pi/12 about point P. Let A be the midpoint of segment DP. Let N be the reflection of point G across point P. Let T be the midpoint of segment NA. Point V is the midpoint of segment PA. What is the measure of angle VGT, 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": "0.58", "dataset": "MATH"}
{"messages": [{"content": "Let UL = 10. Let B be the reflection of point U across point L. Let N be the rotation of point B by angle 5pi/6 around point U. Line O passes through points U and N. Let UT = 4. Point S is the reflection of point B across line O. Let E be the perpendicular bisector of segment BL. Point F is the reflection of point S across line E. 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": "17.32", "dataset": "MATH"}
{"messages": [{"content": "TK = 2. Point N is the midpoint of segment KT. Let P be the rotation of point K by angle 2pi/3 around point N. Point Z is the midpoint of segment PN. Point O is the reflection of point N across segment PT. Point D is the circumcenter of triangle ZPT. Find the measure of angle DPO, 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": "0.22", "dataset": "MATH"}
{"messages": [{"content": "JP = 5. Point N is the reflection of point P across point J. Let U be the rotation of point P by angle pi/6 around point J. Line D is the angle bisector of angle UPJ. Line O is the angle bisector of angle UJP. Let A be the midpoint of segment JN. Let Y be the midpoint of segment JU. Line W passes through point A perpendicular to line O. Let I be the line through point N parallel to line W. Line L passes through point J parallel to line D. Point E is the intersection of lines L and I. Find the measure of angle EUY, 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": "9.16", "dataset": "MATH"}
{"messages": [{"content": "Let UL = 9. Point M is the midpoint of segment UL. Let E be the angle bisector of angle UML. Point H is obtained by rotating point L by angle pi/12 about point U. Point P is obtained by rotating point M by angle 3pi/4 about point H. Line I passes through point L parallel to line E. Point G is the reflection of point P across line I. Let V be the midpoint of segment LG. Find the measure of angle MVU, 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 DL = 3. Let K be the rotation of point L by angle pi/4 around point D. Point P is the midpoint of segment LK. Let B be the reflection of point P across point K. Let A be the circumcenter of triangle DKL. Let I be the midpoint of segment BA. Find the distance between points D 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": "2.58", "dataset": "MATH"}
{"messages": [{"content": "TX = 3. Line N passes through points X and T. Point C is obtained by rotating point T by angle 5pi/12 about point X. Let G be the reflection of point X across point C. Let Y be the line through point C perpendicular to line N. Point O is on line Y. Let V be the midpoint of segment TG. Point W is the reflection of point C across point V. Point A is the reflection of point C across segment TG. What is the length of segment WA? Express your 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": "MI = 2. Let J be the rotation of point I by angle pi/6 around point M. Point X is obtained by rotating point I by angle pi/12 about point M. Point B is the midpoint of segment MJ. Point E is the reflection of point X across segment IM. Point A is the reflection of point B across segment IM. Point L is the midpoint of segment MX. What is the inradius of triangle ELA? Express your 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 JP = 6. Let D be the rotation of point P by angle pi/4 around point J. M is placed 1 units away from D. Let C be the rotation of point M by angle pi/6 around point P. Point S is the reflection of point C across segment PM. Point U is the reflection of point P across point S. Point R is the midpoint of segment CS. What is the measure of angle RSU, 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 UZ = 1. Let E be the reflection of point Z across point U. Let K be the rotation of point Z by angle pi/2 around point E. Point R is the circumcenter of triangle EKU. Point L is the orthocenter of triangle KRZ. Point X is the centroid of triangle KRZ. What is the measure of angle KXL, 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": "1.7", "dataset": "MATH"}
{"messages": [{"content": "RO = 11. Let C be the line through points R and O. Point Y is obtained by rotating point O by angle 7pi/12 about point R. Let U be the perpendicular bisector of segment YR. Point L is the reflection of point Y across line C. Let I be the rotation of point Y by angle pi around point R. Let G be the rotation of point I by angle 2pi/3 around point R. Let Q be the reflection of line U through point O. Let A be the circumcircle of triangle YGL. Let V be the reflection of circle A across line Q. Point K is the inversion of point Y with respect to circle V. What is the measure of angle KYL, 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": "LK = 8. Point V is the reflection of point L across point K. Point D is obtained by rotating point K by angle pi/12 about point V. Point O is the midpoint of segment KD. Let M be the line through point O parallel to segment LK. Point Z is the reflection of point L across line M. Find the circumradius of triangle OVZ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1859.51", "dataset": "MATH"}
{"messages": [{"content": "SB = 9. Let W be the rotation of point S by angle 7pi/12 around point B. Let C be the rotation of point W by angle pi/6 around point B. Point K is the midpoint of segment WS. Line N is the perpendicular bisector of segment KC. Point D is the reflection of point W across line N. Line I is the angle bisector of angle CDS. Point T is obtained by rotating point D by angle pi about point W. Line G is the reflection of line I across point C. Line Z passes through point T perpendicular to line I. Point E is the reflection of point C across line Z. Point P is the reflection of point E across line G. What is the measure of angle PEK, 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": "35.67", "dataset": "MATH"}
{"messages": [{"content": "Let LT = 1. Let R be the vector from point L to point T. Point E is obtained by rotating point T by angle 5pi/12 about point L. Point K is the translation of point T by vector R. Line F is the perpendicular bisector of segment LK. Let C be the midpoint of segment LE. Let O be the midpoint of segment LC. Point I is the reflection of point O across line F. What is the inradius of triangle KTI? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.11", "dataset": "MATH"}
{"messages": [{"content": "Let LG = 3. Point P is the reflection of point G across point L. Point J is obtained by rotating point P by angle 5pi/12 about point G. Point V is obtained by rotating point J by angle pi/6 about point P. Let Y be the perpendicular bisector of segment PV. Point O is the orthocenter of triangle PVG. Point Q is the reflection of point G across line Y. 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": "4.26", "dataset": "MATH"}
{"messages": [{"content": "ZU = 9. Let H be the reflection of point Z across point U. Point M is obtained by rotating point H by angle pi/4 about point Z. Point V is the centroid of triangle HMZ. Point Q is the midpoint of segment ZM. Point S is the midpoint of segment UV. Point Y is the reflection of point S across segment MZ. 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": "5.35", "dataset": "MATH"}
{"messages": [{"content": "ZG = 11. Point Q is the reflection of point Z across point G. Point J is obtained by rotating point G by angle 7pi/12 about point Z. Let F be the rotation of point Q by angle 11pi/12 around point G. Point X is the orthocenter of triangle FJG. Point W is the incenter of triangle FJG. Point D is the centroid of triangle FJG. Find the inradius of triangle WXD. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.38", "dataset": "MATH"}
{"messages": [{"content": "Let HP = 12. Let I be the rotation of point H by angle pi/12 around point P. Point C is the midpoint of segment IP. Let Z be the midpoint of segment IC. Point T is the circumcenter of triangle ICH. What is the measure of angle TCZ, 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 HS = 9. Line Z passes through points H and S. Point J is obtained by rotating point S by angle 11pi/12 about point H. Point G is the reflection of point J across point S. Point Y is the incenter of triangle GHJ. Let I be the reflection of line Z through point Y. Point C is the reflection of point J across line I. What is the distance between points G 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": "35.46", "dataset": "MATH"}
{"messages": [{"content": "Let WO = 3. Point L is obtained by rotating point W by angle 11pi/12 about point O. Point S is the midpoint of segment OW. Let B be the incenter of triangle LOW. Point X is the reflection of point B across segment WL. Point U is the midpoint of segment SX. 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": "0.77", "dataset": "MATH"}
{"messages": [{"content": "Let BC = 9. Let L be the rotation of point C by angle 5pi/6 around point B. Let Q be the midpoint of segment CL. Let S be the midpoint of segment QL. Point Z is the reflection of point Q across segment BL. Find the measure of angle SBZ, 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": "5.74", "dataset": "MATH"}
{"messages": [{"content": "UZ = 2. Let I be the reflection of point Z across point U. Let J be the vector from point Z to point I. Let K be point I translated by vector J. Point D is obtained by rotating point U by angle 5pi/12 about point Z. Let F be the vector from point U to point I. Point R is the translation of point K by vector J. Let E be point D translated by vector F. Let G be the rotation of point R by angle pi around point E. Point M is obtained by rotating point G by angle pi/3 about point R. Point X is the midpoint of segment DZ. Point C is the midpoint of segment MX. Point B is the midpoint of segment KD. Find the inradius of triangle CBU. Express your 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": "Let CH = 12. Let W be the rotation of point C by angle 5pi/12 around point H. Line N passes through point H parallel to segment CW. Line U is the reflection of line N across point W. Point E is the midpoint of segment HC. Point T is the reflection of point W across line U. Point L is the reflection of point E across line N. What is the measure of angle TEL, 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": "52.16", "dataset": "MATH"}
{"messages": [{"content": "WE = 2. Let X be the rotation of point E by angle 3pi/4 around point W. Point S is obtained by rotating point X by angle 3pi/4 about point E. Let J be the reflection of point W across point E. Let Y be the line through points X and E. Let U be the line through point X perpendicular to line Y. Let V be the midpoint of segment SE. Point M is the midpoint of segment XV. Let F be the midpoint of segment ME. Point I is the reflection of point W across line U. Find the measure of angle IFJ, 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": "2.16", "dataset": "MATH"}
{"messages": [{"content": "AW = 7. Point M is obtained by rotating point A by angle 2pi/3 about point W. Point V is obtained by rotating point W by angle pi/12 about point M. Point E is the reflection of point W across point V. Let F be the circumcenter of triangle EMV. Let T be the incenter of triangle EMV. What is the measure of angle TEF, 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": "0.73", "dataset": "MATH"}
{"messages": [{"content": "Let XD = 2. Point Z is the reflection of point X across point D. Point U is obtained by rotating point Z by angle 3pi/4 about point D. Let Y be the perpendicular bisector of segment DZ. Point H is the centroid of triangle XUZ. Point I is the reflection of point H across line Y. Let F be the incenter of triangle XUZ. What is the inradius of triangle IFH? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.06", "dataset": "MATH"}
{"messages": [{"content": "OU = 3. Let G be the rotation of point U by angle pi/4 around point O. Let Y be the incircle of triangle UGO. Point K is the incenter of triangle UGO. Let P be the inversion of point U in circle 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": "0.43", "dataset": "MATH"}
{"messages": [{"content": "Let YJ = 4. Let N be the reflection of point Y across point J. Line L is the perpendicular bisector of segment JY. Point O is obtained by rotating point N by angle pi/4 about point Y. Let D be the angle bisector of angle OJN. Let R be the line through point J parallel to line L. Point W is the reflection of point O across line R. Point K is the reflection of point Y across line D. Find the circumradius of triangle KJW. Express your 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": "QZ = 12. Let U be the vector from point Z to point Q. Let P be point Q translated by vector U. Let J be the rotation of point Z by angle 11pi/12 around point Q. Let R be the reflection of point Z across point J. Point N is the reflection of point Q across segment RZ. Point I is the midpoint of segment QJ. Find the distance between points P 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": "6.4", "dataset": "MATH"}
{"messages": [{"content": "SX = 1. Point J is obtained by rotating point X by angle pi/12 about point S. Let H be the midpoint of segment SX. Point B is the reflection of point H across point J. Let F be the line through point B perpendicular to segment XJ. Point I is the reflection of point S across line F. 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": "1.12", "dataset": "MATH"}
{"messages": [{"content": "Let FK = 12. Let E be the rotation of point K by angle pi/3 around point F. Let V be the midpoint of segment FE. Point J is the centroid of triangle FEK. Point N is the midpoint of segment JV. Point Y is the reflection of point K across point N. Point P is the reflection of point Y across segment KF. What is 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": "17.32", "dataset": "MATH"}
{"messages": [{"content": "NX = 8. Point E is obtained by rotating point N by angle 7pi/12 about point X. Let I be the angle bisector of angle NXE. Line F is the reflection of line I across point N. Point H is the reflection of point N across line F. Point J is the midpoint of segment HN. What is the inradius of triangle JHX? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.78", "dataset": "MATH"}
{"messages": [{"content": "Let XQ = 8. Let G be the rotation of point Q by angle 5pi/6 around point X. Point D is the reflection of point G across point Q. Let J be the circumcenter of triangle DQX. Point N is the incenter of triangle DQX. Point T is the midpoint of segment DQ. What is the inradius of triangle TNJ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.11", "dataset": "MATH"}
{"messages": [{"content": "GV = 2. Point A is obtained by rotating point V by angle pi/3 about point G. Let C be the reflection of point A across point G. Let D be the reflection of point G across point A. Point O is the reflection of point A across point V. What is the inradius of triangle COD? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.36", "dataset": "MATH"}
{"messages": [{"content": "Let FC = 10. Let L be the rotation of point F by angle pi/3 around point C. Point T is the midpoint of segment LC. Point A is the circumcenter of triangle CFT. Let E be the midpoint of segment TA. Point R is the midpoint of segment TC. Find the measure of angle ERF, 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": "5.41", "dataset": "MATH"}
{"messages": [{"content": "Let SC = 3. Point T is the midpoint of segment CS. Line E is the angle bisector of angle CTS. Line Y is the perpendicular bisector of segment CT. Point H lies on line Y at distance 10 from point S. Let F be the reflection of point T across point H. Let A be the reflection of line Y across line E. Point L is the reflection of point F across line A. Point N is the reflection of point F across segment ST. Point X is the reflection of point N across line E. What is the circumradius of triangle LXN? Express your 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 XK = 2. Point Z is obtained by rotating point K by angle pi/12 about point X. Let S be the perpendicular bisector of segment ZK. Point Y is the reflection of point K across point X. Let N be the line through point K parallel to line S. Let Q be the midpoint of segment YX. Point L is the reflection of point Q across line N. Find the inradius of triangle KLY. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.37", "dataset": "MATH"}
{"messages": [{"content": "RZ = 5. Let N be the midpoint of segment ZR. Let K be the midpoint of segment RN. Point Q is the midpoint of segment ZK. Point I is obtained by rotating point N by angle pi/6 about point Z. Point Y is obtained by rotating point Z by angle pi/4 about point R. Let J be the midpoint of segment QI. Line X is the angle bisector of angle JRQ. Let H be the reflection of line X through point N. Point M is the midpoint of segment KJ. Point L is the reflection of point M across line H. Find the circumradius of triangle YNL. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.74", "dataset": "MATH"}
{"messages": [{"content": "Let YC = 5. Point J is the reflection of point C across point Y. Let F be the rotation of point Y by angle 7pi/12 around point C. Line G is the perpendicular bisector of segment JY. Let E be the centroid of triangle YCF. Point U is the orthocenter of triangle YCF. Point A is the reflection of point E across line G. Point B is the reflection of point A across segment CF. What is the measure of angle BUF, 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": "37.24", "dataset": "MATH"}
{"messages": [{"content": "Let CZ = 1. Let D be the rotation of point Z by angle pi/12 around point C. Let W be the angle bisector of angle DZC. Point P is the reflection of point C across line W. Point Y is the reflection of point D across line W. Point J is on line W. Point G is the reflection of point J across segment ZD. Point S is the midpoint of segment DY. Point K is the reflection of point S across segment CZ. Find 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": "1.12", "dataset": "MATH"}
{"messages": [{"content": "ST = 1. Line D passes through points T and S. Point H is obtained by rotating point T by angle 5pi/12 about point S. Let A be the midpoint of segment SH. Point I is the reflection of point H across line D. Point O is the midpoint of segment TA. Let K be the reflection of line D through point I. Point F is the reflection of point T across line K. Let Z be the midpoint of segment SO. What is the measure of angle FTZ, 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": "1.39", "dataset": "MATH"}
{"messages": [{"content": "Let EV = 9. Point Y is obtained by rotating point E by angle 7pi/12 about point V. Circle F is the incircle of triangle EYV. Point R is the centroid of triangle EYV. Let S be the inversion of point Y in circle F. What is the distance between points S 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": "0.82", "dataset": "MATH"}
{"messages": [{"content": "Let FO = 11. Point P is obtained by rotating point O by angle 11pi/12 about point F. Let B be the rotation of point F by angle pi/3 around point O. Line E is the perpendicular bisector of segment PF. Point T is the reflection of point B across line E. Point W is the midpoint of segment PO. Find the inradius of triangle FTW. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.66", "dataset": "MATH"}
{"messages": [{"content": "CO = 3. Point B is obtained by rotating point O by angle pi/2 about point C. Let L be the incenter of triangle COB. Point M is the reflection of point C across segment OB. Point I is the reflection of point L across segment CO. 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": "4.42", "dataset": "MATH"}
{"messages": [{"content": "ZN = 8. Point R is obtained by rotating point N by angle pi/3 about point Z. Point K is the orthocenter of triangle RNZ. Let B be the midpoint of segment ZR. Let X be the midpoint of segment RB. What is the measure of angle XKN, 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": "4.62", "dataset": "MATH"}
{"messages": [{"content": "CD = 11. Point S is obtained by rotating point C by angle 3pi/4 about point D. Point R is the incenter of triangle DSC. Let Z be the midpoint of segment DS. Point L is the midpoint of segment ZR. Let K be the midpoint of segment ZS. 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": "5.18", "dataset": "MATH"}
{"messages": [{"content": "Let BM = 10. Let P be the rotation of point B by angle 11pi/12 around point M. Line Q passes through point P perpendicular to segment BM. Point U is obtained by rotating point P by angle 3pi/4 about point B. Point E is the reflection of point U across segment BM. Point G is the reflection of point M across line Q. Point X is the reflection of point E across line Q. Point Z is the reflection of point M across point G. Let R be the reflection of point X across point Z. Point C is the reflection of point X across segment BM. What is the measure of angle CRG, 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": "43.3", "dataset": "MATH"}
{"messages": [{"content": "GJ = 10. Let N be the rotation of point J by angle 3pi/4 around point G. Let E be the reflection of point G across point N. Point U is the orthocenter of triangle JEN. Let R be the circumcenter of triangle JEN. Let B be the line through point R parallel to segment EN. Point Z is the intersection of segment JE and line B. What is the measure of angle ZNU, 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": "11.79", "dataset": "MATH"}
{"messages": [{"content": "ER = 5. Point P is obtained by rotating point E by angle 7pi/12 about point R. Point C is the reflection of point P across point E. Point V is the incenter of triangle ECR. Point O is the midpoint of segment EC. Find 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": "3.77", "dataset": "MATH"}
{"messages": [{"content": "RD = 8. Point M is obtained by rotating point D by angle pi/4 about point R. Let H be the rotation of point R by angle 3pi/4 around point D. Let O be the midpoint of segment RH. Point C is the circumcenter of triangle MRO. Point Z is the incenter of triangle MRO. Point W is the incenter of triangle CDZ. Find the length of segment WM. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.02", "dataset": "MATH"}
{"messages": [{"content": "Let OK = 5. Point S is obtained by rotating point K by angle 3pi/4 about point O. Line M passes through point O perpendicular to segment SK. Let E be the reflection of point K across point O. Let Z be the circumcircle of triangle KOS. Let U be the rotation of point O by angle pi/3 around point E. Let X be the inversion of point U in circle Z. Point J is the reflection of point X across line M. Point P is the orthocenter of triangle KJX. What is the measure of angle XSP, 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": "11.79", "dataset": "MATH"}
{"messages": [{"content": "EB = 4. Point J is the reflection of point B across point E. Let Y be the rotation of point J by angle 7pi/12 around point B. Point P is the midpoint of segment EY. Point Z is the circumcenter of triangle JPE. Let Q be the angle bisector of angle JZB. Point M is the midpoint of segment PE. Point D is the reflection of point P across line Q. Find 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": "5.6", "dataset": "MATH"}
{"messages": [{"content": "Let DW = 3. Point R is obtained by rotating point W by angle pi/2 about point D. Let C be the perpendicular bisector of segment RD. Let A be the line through point W perpendicular to line C. Point U is obtained by rotating point D by angle pi/6 about point W. Let G be the circumcenter of triangle RWD. Let H be the midpoint of segment RU. Point E is the reflection of point U across line A. Point S is the reflection of point G across point E. What is the distance between points S 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": "11.07", "dataset": "MATH"}
{"messages": [{"content": "QJ = 2. Point P is obtained by rotating point Q by angle 3pi/4 about point J. Let C be the circumcenter of triangle QJP. Point I is the midpoint of segment PQ. Let W be the midpoint of segment CP. Find 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": "0.93", "dataset": "MATH"}
{"messages": [{"content": "SD = 10. Point P is obtained by rotating point S by angle 11pi/12 about point D. Let X be the midpoint of segment PS. Point O is the midpoint of segment PD. Let J be the reflection of point X across point O. Let L be the midpoint of segment PO. Find the measure of angle LJX, 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": "3.24", "dataset": "MATH"}
{"messages": [{"content": "Let RB = 2. Let K be the rotation of point B by angle pi/4 around point R. Let E be the reflection of point R across point B. Point W is the orthocenter of triangle KRE. Let L be the centroid of triangle KRE. What is the distance between points W 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": "2.15", "dataset": "MATH"}
{"messages": [{"content": "Let OC = 4. Point Q is obtained by rotating point O by angle 11pi/12 about point C. Point E is the midpoint of segment QO. Let K be the incircle of triangle CEO. Point N is the inversion of point E with respect to circle K. Find 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": "4.09", "dataset": "MATH"}
{"messages": [{"content": "WY = 11. Line D is the perpendicular bisector of segment WY. Line H passes through point W perpendicular to line D. Point K is obtained by rotating point W by angle pi/12 about point Y. Point I is obtained by rotating point W by angle pi/12 about point K. Let B be the midpoint of segment IW. Let N be the line through point B parallel to line H. Point T is the reflection of point W across line N. Find the circumradius of triangle TBW. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.19", "dataset": "MATH"}
{"messages": [{"content": "EP = 11. Point T is obtained by rotating point E by angle 11pi/12 about point P. Point J is the centroid of triangle PTE. Line O passes through point E parallel to segment PT. Point A is the reflection of point J across line O. Find the distance between points E 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": "10.92", "dataset": "MATH"}
{"messages": [{"content": "Let UW = 11. Let H be the rotation of point W by angle 2pi/3 around point U. Point M is the incenter of triangle WUH. Let T be the orthocenter of triangle WUH. Point B is the reflection of point M across segment UH. Point O is the reflection of point T across point B. Find 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": "9.53", "dataset": "MATH"}
{"messages": [{"content": "AK = 7. Point I is obtained by rotating point A by angle pi/2 about point K. Circle Y is the circumcircle of triangle AIK. Let R be the circumcenter of triangle AIK. Circle N is the reflection of circle Y across point R. Let S be the reflection of point K across point I. Let F be the inversion of point A in circle N. Find 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": "14.6", "dataset": "MATH"}
{"messages": [{"content": "YS = 4. Point N is the midpoint of segment YS. Let T be the line through point S perpendicular to segment SY. Point V is obtained by rotating point N by angle 5pi/6 about point S. Line A passes through point N parallel to line T. Point O is the reflection of point V across line A. Point I is the midpoint of segment ON. What is the circumradius of triangle NIV? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.62", "dataset": "MATH"}
{"messages": [{"content": "Let XJ = 9. Point B is obtained by rotating point J by angle 7pi/12 about point X. Let U be the reflection of point B across point J. Point T is the orthocenter of triangle BUX. Point O is the circumcenter of triangle BUX. Point A is the reflection of point U across segment TO. Let D be the midpoint of segment AX. Find the inradius of triangle ABD. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.76", "dataset": "MATH"}
{"messages": [{"content": "Let DH = 9. Point Y is obtained by rotating point H by angle pi/6 about point D. Point V is obtained by rotating point H by angle pi/2 about point Y. Let R be the rotation of point Y by angle 5pi/6 around point H. Let T be the line through points H and Y. Line J is the perpendicular bisector of segment DV. Let Q be the midpoint of segment RY. Let O be the reflection of line T through point R. Line U is the reflection of line J across line O. Let I be the midpoint of segment DV. Point M is the reflection of point Q across line U. What is the measure of angle IYM, 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": "6.14", "dataset": "MATH"}
{"messages": [{"content": "PQ = 9. Point K is the midpoint of segment PQ. Let A be the perpendicular bisector of segment KP. Point J lies on line A at distance 8 from point Q. Point R is the midpoint of segment QJ. Point E is the reflection of point K across point J. Let S be the reflection of point R across point P. What is the inradius of triangle ESR? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.76", "dataset": "MATH"}
{"messages": [{"content": "Let OQ = 4. Let H be the vector from point O to point Q. Let M be the perpendicular bisector of segment OQ. Let J be the line through point O perpendicular to line M. Let U be the rotation of point O by angle pi/6 around point Q. Let L be the reflection of point U across point O. Point X is obtained by rotating point Q by angle 7pi/12 about point O. Point W is the midpoint of segment LQ. Let A be the perpendicular bisector of segment UX. Point F is the translation of point W by vector H. Line V is the reflection of line A across line J. Point T is the reflection of point L across point F. Let R be the midpoint of segment QU. Point N is the reflection of point L across line V. What is the inradius of triangle TNR? Express your 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": "TA = 5. Point Y is the midpoint of segment TA. Point M is the midpoint of segment AY. Let B be the rotation of point Y by angle pi/2 around point T. Point W is the reflection of point T across point M. Point H is the midpoint of segment WB. Point D is the reflection of point A across point H. Find the circumradius of triangle BWD. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.84", "dataset": "MATH"}
{"messages": [{"content": "Let DL = 1. Let H be the rotation of point D by angle pi/2 around point L. Let K be the midpoint of segment DH. Let S be the reflection of point L across point H. Line F passes through points D and K. Let X be the reflection of line F through point L. Point Y is the reflection of point S across point L. Let V be the reflection of line X across line F. Line Z is the perpendicular bisector of segment HD. Point G is the reflection of point L across line V. Point Q is the reflection of point K across point D. Line B is the angle bisector of angle SDY. Point N is the reflection of point K across line B. Line I passes through point D parallel to line Z. Point O lies on segment QN. Point C is the reflection of point G across line I. Find the measure of angle CSO, 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": "5.0", "dataset": "MATH"}
{"messages": [{"content": "VG = 5. Let X be the rotation of point G by angle 11pi/12 around point V. Let Y be the line through point G perpendicular to segment VX. Point N is the reflection of point V across line Y. Point F is the orthocenter of triangle VXG. 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": "37.0", "dataset": "MATH"}
{"messages": [{"content": "Let TK = 10. Point E is the reflection of point K across point T. Let C be the rotation of point T by angle 2pi/3 around point E. Point F is the orthocenter of triangle KEC. Let Q be the line through point E parallel to segment CK. Let N be the reflection of point K across point E. Point Z is the reflection of point N across line Q. What is the circumradius of triangle EFZ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.28", "dataset": "MATH"}
{"messages": [{"content": "Let QA = 5. Point K is obtained by rotating point Q by angle 5pi/12 about point A. Let Y be the reflection of point A across point Q. Let B be the line through point Q perpendicular to segment QK. Point T is the reflection of point A across line B. Point W is the midpoint of segment QY. Point C is the reflection of point W across segment QK. 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": "2.5", "dataset": "MATH"}
{"messages": [{"content": "OF = 10. Let L be the rotation of point O by angle pi/3 around point F. Let V be the circumcircle of triangle OLF. Let Y be the centroid of triangle OLF. Let E be the angle bisector of angle YFO. Let X be the reflection of line E through point L. Circle G is the reflection of circle V across line X. Let M be the inversion of point O in circle G. Find the inradius of triangle MFY. Express your 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": "BE = 8. Point Z is obtained by rotating point E by angle 3pi/4 about point B. Let Q be the incircle of triangle ZBE. Point R is the orthocenter of triangle ZBE. Point J is the inversion of point R with respect to circle Q. 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": "7.56", "dataset": "MATH"}
{"messages": [{"content": "Let IW = 8. Let L be the rotation of point I by angle 3pi/4 around point W. Let F be the rotation of point L by angle pi/2 around point W. Line R passes through points W and F. Let C be the line through point L parallel to line R. Let O be the reflection of point L across point F. Point M is the reflection of point W across line C. Find the measure of angle MIO, 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": "241.14", "dataset": "MATH"}
{"messages": [{"content": "Let PU = 1. Let Z be the rotation of point P by angle 3pi/4 around point U. Let G be the midpoint of segment PU. Let S be the reflection of point Z across point U. Point O is the midpoint of segment PG. Find the circumradius of triangle SZO. Express your 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 JZ = 7. Let G be the reflection of point Z across point J. Let U be the line through point G perpendicular to segment JZ. Line I is the perpendicular bisector of segment GZ. Let X be the midpoint of segment JZ. Let W be the reflection of line I across line U. Let M be the reflection of line W across line I. Line K is the reflection of line M across line W. Point R is on line U with |RJ| = 8. Let H be the line through points X and R. Point E is the intersection of lines K and H. Point S is the reflection of point G across line H. Point T is the reflection of point E across line W. Find the measure of angle STX, 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": "101.45", "dataset": "MATH"}
{"messages": [{"content": "UR = 4. Let X be the rotation of point R by angle pi/4 around point U. Let F be the perpendicular bisector of segment UX. Line O passes through point R parallel to line F. Let I be the centroid of triangle RUX. Point K is the reflection of point X across line O. What is the distance between points I 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": "1.13", "dataset": "MATH"}
{"messages": [{"content": "GF = 10. Let S be the rotation of point F by angle pi/6 around point G. Let D be the orthocenter of triangle GSF. Point N is the reflection of point D across segment FG. Let U be the midpoint of segment NF. 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": "3.54", "dataset": "MATH"}
{"messages": [{"content": "IB = 4. Let E be the rotation of point I by angle pi around point B. Point P is obtained by rotating point I by angle 5pi/6 about point E. Let L be the midpoint of segment PE. Let V be the midpoint of segment LI. Point Y is the reflection of point L across point B. Let S be the line through points Y and V. Point R is the reflection of point E across line S. Find the inradius of triangle RVL. Express your 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": "HA = 10. Point K is obtained by rotating point H by angle 11pi/12 about point A. Let P be the perpendicular bisector of segment AK. Let L be the angle bisector of angle AKH. Point T is the reflection of point H across line P. Point M is the reflection of point A across line L. Let B be the incircle of triangle HMT. Point F is the inversion of point M with respect to circle B. Let Y be the midpoint of segment HM. 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": "4.91", "dataset": "MATH"}
{"messages": [{"content": "XB = 11. Point Y is the reflection of point B across point X. Let I be the midpoint of segment XY. Let F be the midpoint of segment IY. Point Z is obtained by rotating point F by angle 5pi/6 about point X. Point K is the reflection of point Z across point X. Point G is the reflection of point X across point K. Point R is the reflection of point G across segment ZY. Find the measure of angle GYR, 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": "36.84", "dataset": "MATH"}
{"messages": [{"content": "YK = 9. Let L be the rotation of point K by angle pi/6 around point Y. Point F is the incenter of triangle LKY. Line H passes through point F parallel to segment YL. Point E is the intersection of segment KY and line H. What is the measure of angle FEK, 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 KI = 1. Let E be the rotation of point K by angle 5pi/6 around point I. Let Z be the midpoint of segment EI. Point Q is the reflection of point E across point K. Point B is the centroid of triangle KEI. What is the measure of angle ZBQ, 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": "0.04", "dataset": "MATH"}
{"messages": [{"content": "ON = 5. Point L is obtained by rotating point O by angle pi/12 about point N. Point Y is obtained by rotating point L by angle 5pi/6 about point N. Point Q is the reflection of point L across point N. Point C is the centroid of triangle LNY. Point X is the reflection of point Q across segment YL. Find the measure of angle COX, 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": "5.47", "dataset": "MATH"}
{"messages": [{"content": "LM = 1. Let P be the rotation of point M by angle 11pi/12 around point L. Circle W is the incircle of triangle LPM. Point B is the inversion of point L with respect to circle W. Let E be the midpoint of segment BP. Let Q be the centroid of triangle LPM. 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": "0.5", "dataset": "MATH"}
{"messages": [{"content": "VS = 2. Let I be the rotation of point S by angle 5pi/6 around point V. Point G is the reflection of point I across point V. Point Z is the reflection of point I across segment SG. Point P is the incenter of triangle GIS. What is the measure of angle PZG, 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": "1.36", "dataset": "MATH"}
{"messages": [{"content": "Let UW = 11. Let A be the rotation of point W by angle pi/12 around point U. Let X be the midpoint of segment WU. Point L is obtained by rotating point U by angle 2pi/3 about point A. Point S is obtained by rotating point W by angle pi about point L. Point E is the reflection of point S across point A. Let C be the midpoint of segment XL. What is the inradius of triangle EAC? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.28", "dataset": "MATH"}
{"messages": [{"content": "Let Y be the perpendicular bisector of segment BB. Let U be the reflection of point B across point B. Let I be the reflection of point B across point U. Point O is the reflection of point I across line Y. What is the distance between points B 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": "2.0", "dataset": "MATH"}
{"messages": [{"content": "DB = 11. Let C be the rotation of point B by angle 5pi/6 around point D. Point S is the centroid of triangle BDC. Let Y be the midpoint of segment BD. Point V is the midpoint of segment BY. Point M is the reflection of point S across segment CB. Segments CB and SM intersect at point Q. What is the length of segment QV? Express your 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": "JD = 2. Let P be the rotation of point D by angle pi/2 around point J. Let A be the rotation of point P by angle pi/4 around point D. Point V is the midpoint of segment JP. Let X be the reflection of point D across point P. Let O be the line through point P parallel to segment AX. Point Z is the midpoint of segment DV. Point S is the reflection of point J across line O. What is the inradius of triangle SJZ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.09", "dataset": "MATH"}
{"messages": [{"content": "Let AF = 8. Point K is obtained by rotating point A by angle pi/3 about point F. Let X be the perpendicular bisector of segment AF. Line S is the reflection of line X across point F. Point E is the reflection of point K across line S. Point Y is the circumcenter of triangle EFK. Find 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": "7.06", "dataset": "MATH"}
{"messages": [{"content": "HA = 2. Point X is obtained by rotating point A by angle pi/4 about point H. Let J be the reflection of point A across point X. Point P is obtained by rotating point X by angle 3pi/4 about point A. Let U be the orthocenter of triangle JXH. Let D be the line through point J parallel to segment XH. Let N be the line through point P parallel to line D. Let Y be the line through points A and U. Lines N and Y intersect at point Z. Point L is the centroid of triangle JXH. Find 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": "3.25", "dataset": "MATH"}
{"messages": [{"content": "Let FP = 4. Let R be the rotation of point F by angle 5pi/12 around point P. Point U is the midpoint of segment PR. Point S is the orthocenter of triangle PRF. Let G be the centroid of triangle PRF. Point M is the reflection of point U across point G. Find the length of segment SM. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.81", "dataset": "MATH"}
{"messages": [{"content": "Let OY = 6. Line D is the perpendicular bisector of segment YO. Point A is the midpoint of segment YO. Point P lies on line D at distance 4 from point O. Let R be the perpendicular bisector of segment AP. Point M is the reflection of point O across line R. Point F is the reflection of point M across point O. What is the inradius of triangle FMY? Express your 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": "Let XK = 10. Let Q be the midpoint of segment KX. Let R be the midpoint of segment QX. Let T be the rotation of point R by angle 3pi/4 around point Q. Let P be the rotation of point K by angle 3pi/4 around point R. Point O is the midpoint of segment TX. Point U is the reflection of point Q across segment KO. Find the inradius of triangle UPQ. Express your 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": "XW = 7. Point U is obtained by rotating point W by angle pi about point X. Let S be the rotation of point W by angle 5pi/6 around point U. Let Q be the reflection of point X across point S. Point Y is the reflection of point Q across point W. Point T is the midpoint of segment WS. Let H be the midpoint of segment YX. What is the circumradius of triangle TWH? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "190.28", "dataset": "MATH"}
{"messages": [{"content": "Let ZS = 6. Let X be the rotation of point Z by angle pi/3 around point S. Let D be the circumcircle of triangle ZXS. Point H is the midpoint of segment ZX. Point A is the inversion of point H with respect to circle D. Point C is the midpoint of segment SZ. What is the measure of angle XAC, 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": "15.59", "dataset": "MATH"}
{"messages": [{"content": "Let OS = 1. Let W be the rotation of point S by angle 2pi/3 around point O. Point U is obtained by rotating point S by angle pi/4 about point O. Let N be the midpoint of segment WS. Let E be the rotation of point S by angle 7pi/12 around point U. Point K is the reflection of point U across point E. Line Y is the perpendicular bisector of segment EK. Point G is the reflection of point U across line Y. Find the inradius of triangle ENG. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.02", "dataset": "MATH"}
{"messages": [{"content": "Let BE = 9. Point S is obtained by rotating point B by angle pi/12 about point E. Point F is obtained by rotating point E by angle 7pi/12 about point B. Let X be the centroid of triangle BES. Circle Z is the circumcircle of triangle BES. Let L be the inversion of point F in circle Z. Point O is the reflection of point L across segment FX. What is the length of segment SO? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.41", "dataset": "MATH"}
{"messages": [{"content": "Let XB = 2. Let S be the midpoint of segment XB. Let E be the rotation of point B by angle 5pi/6 around point X. Let U be the line through points B and E. Point H is the reflection of point S across line U. Let C be the angle bisector of angle EHS. Point I is the reflection of point X across line C. Point T is the incenter of triangle HIX. 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": "0.25", "dataset": "MATH"}
{"messages": [{"content": "Let QL = 8. Point I is the midpoint of segment LQ. Point S is the midpoint of segment LI. Line D is the perpendicular bisector of segment LS. Point W is the reflection of point I across line D. Point V is obtained by rotating point L by angle pi/3 about point S. Let B be the reflection of point V across point Q. Let R be the line through point S parallel to segment IB. Point G is the reflection of point V across line R. Point H is the circumcenter of triangle GVS. What is the measure of angle GHW, 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": "2.29", "dataset": "MATH"}
{"messages": [{"content": "JT = 9. Point M is obtained by rotating point T by angle 5pi/6 about point J. Let R be the reflection of point T across point M. Let V be the rotation of point M by angle pi/3 around point R. Let E be the line through points J and R. Line P passes through point V parallel to segment MJ. Point U is the intersection of lines P and E. 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": "80.42", "dataset": "MATH"}
{"messages": [{"content": "HX = 10. Let P be the rotation of point X by angle 2pi/3 around point H. Point R is the midpoint of segment PX. Let Q be the midpoint of segment XH. Point E is the orthocenter of triangle RHQ. Find the distance between points P 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": "10.41", "dataset": "MATH"}
{"messages": [{"content": "Let DG = 10. Point W is obtained by rotating point G by angle pi/2 about point D. Point U is the reflection of point D across segment WG. Point S is obtained by rotating point G by angle pi/4 about point U. Point R is the reflection of point S across segment WG. Let N be the reflection of point G across point R. Let B be the midpoint of segment WG. Point F is the reflection of point N across segment WG. What is the measure of angle BNF, 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": "182.84", "dataset": "MATH"}
{"messages": [{"content": "Let VL = 1. Point C is obtained by rotating point L by angle 11pi/12 about point V. Point R lies on segment VL. Let A be the reflection of point R across point L. Let F be the midpoint of segment VL. Point P is the midpoint of segment CF. Line U passes through point P parallel to segment VL. Point D is the reflection of point L across line U. What is the measure of angle ACD, 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": "0.25", "dataset": "MATH"}
{"messages": [{"content": "Point S is the midpoint of segment D52H. Let V be the circumcircle of triangle H. Circle O is the reflection of circle V across point H. Let G be the reflection of circle O through point R. Let U be the inversion of point H in circle G. Point F is the reflection of point H across segment HN. Point Q is the reflection of point S across point F. What is the measure of angle UQF, 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": "0.67", "dataset": "MATH"}
{"messages": [{"content": "Let AX = 6. Let T be the rotation of point A by angle 5pi/12 around point X. Let U be the reflection of point A across point T. Let B be the orthocenter of triangle TUX. Point M is the midpoint of segment BT. Let Q be the centroid of triangle TUX. Find the measure of angle MXQ, 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": "18.93", "dataset": "MATH"}
{"messages": [{"content": "KZ = 2. Point O is obtained by rotating point K by angle pi/4 about point Z. Let X be the rotation of point Z by angle 5pi/6 around point K. Let A be the midpoint of segment XO. Let U be the perpendicular bisector of segment ZO. Point Q is on line U. Point B lies on line U at distance 4 from point Q. Point T is the midpoint of segment AK. Point L is the midpoint of segment OK. Point N is the midpoint of segment TA. What is 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": "0.91", "dataset": "MATH"}
{"messages": [{"content": "Let XP = 5. Let W be the rotation of point P by angle pi/12 around point X. Let C be the midpoint of segment PW. Let Y be the rotation of point P by angle 2pi/3 around point C. Point Z is the reflection of point Y across segment WP. Let U be the incircle of triangle WPX. Let K be the inversion of point Z in circle U. Point J is the reflection of point W across segment PX. What is the measure of angle WKJ, 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": "0.77", "dataset": "MATH"}
{"messages": [{"content": "Let GW = 2. Let J be the rotation of point G by angle pi around point W. Point F is obtained by rotating point J by angle pi/4 about point W. Let M be the rotation of point J by angle 2pi/3 around point W. Point V is the reflection of point J across point M. Point Q is the midpoint of segment GW. Point B is the orthocenter of triangle GFV. 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": "1.46", "dataset": "MATH"}
{"messages": [{"content": "MQ = 1. Point L is the midpoint of segment MQ. Point P is obtained by rotating point L by angle pi/2 about point M. Point A is obtained by rotating point M by angle 5pi/12 about point P. Point U is the reflection of point M across point P. Line D is the perpendicular bisector of segment UA. Point G is the reflection of point L across line D. Point H is the incenter of triangle UGQ. Find the distance between points U 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": "0.07", "dataset": "MATH"}
{"messages": [{"content": "Let JS = 3. Let L be the rotation of point S by angle 7pi/12 around point J. Let U be the rotation of point J by angle pi/4 around point S. Point X is the reflection of point U across point J. Point M is the incenter of triangle LSU. Line P is the perpendicular bisector of segment LS. Point R is the reflection of point X across line P. What is the measure of angle URM, 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": "0.97", "dataset": "MATH"}
{"messages": [{"content": "VE = 3. Point T is the reflection of point E across point V. Let N be the rotation of point E by angle pi/12 around point V. Point W is obtained by rotating point N by angle pi/6 about point T. Point U is the centroid of triangle VWT. Point J is the midpoint of segment TU. Let B be the orthocenter of triangle VWT. Line I passes through points J and E. Point Q is the reflection of point B across line I. Point Y is the incenter of triangle VWT. 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": "3.3", "dataset": "MATH"}
{"messages": [{"content": "BH = 4. Line L passes through points B and H. Let F be the rotation of point H by angle 5pi/12 around point B. Point P is the reflection of point F across line L. Let E be the angle bisector of angle FPH. Line U is the reflection of line E across line L. Point R is the reflection of point H across line U. Let C be the centroid of triangle RHP. What is the measure of angle CBR, 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": "0.18", "dataset": "MATH"}
{"messages": [{"content": "Let OH = 10. Line L passes through points O and H. Point G is obtained by rotating point O by angle pi/12 about point H. Let C be the line through points O and G. Let K be the reflection of point H across point O. Let R be the midpoint of segment GH. Let A be the line through point G perpendicular to line C. Let B be the reflection of point O across point K. Point N is the reflection of point R across line L. Point E is the reflection of point O across line A. Find the measure of angle EBN, 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": "78.53", "dataset": "MATH"}
{"messages": [{"content": "ZF = 8. Let N be the midpoint of segment FZ. Point K is obtained by rotating point F by angle 7pi/12 about point N. Point U is obtained by rotating point F by angle 2pi/3 about point K. Point C is the reflection of point U across point N. Point Q is the reflection of point Z across segment UC. Point V is the orthocenter of triangle ZQC. Let Y be the reflection of point U across point V. Find the measure of angle YKZ, 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": "27.69", "dataset": "MATH"}
{"messages": [{"content": "Let WG = 7. Let D be the rotation of point G by angle 3pi/4 around point W. Let X be the midpoint of segment WD. Point M is the reflection of point W across segment DG. Let O be the circumcenter of triangle DGX. Find the length of segment OM. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.31", "dataset": "MATH"}
{"messages": [{"content": "XJ = 1. Let Z be the rotation of point J by angle pi/4 around point X. Point R is the centroid of triangle JXZ. Line G passes through point J parallel to segment XZ. Let V be the midpoint of segment XZ. Point S is the reflection of point V across line G. Find the measure of angle RVS, 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": "0.05", "dataset": "MATH"}
{"messages": [{"content": "Let VJ = 11. Point X is obtained by rotating point V by angle 3pi/4 about point J. Let S be the reflection of point J across point X. Let W be the incircle of triangle JVS. Point Y is the inversion of point X with respect to circle W. Find the length of segment YV. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.93", "dataset": "MATH"}
{"messages": [{"content": "Let ZN = 5. Line B passes through points Z and N. Point J is obtained by rotating point Z by angle 5pi/12 about point N. Let H be the midpoint of segment NZ. Line K passes through point J parallel to line B. Point R is the reflection of point H across line K. Let W be the midpoint of segment ZR. Let F be the circumcenter of triangle HWR. Find the inradius of triangle FNR. Express your 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": "VD = 11. Let R be the reflection of point V across point D. Let X be the perpendicular bisector of segment RV. Point W lies on line X at distance 9 from point V. Line K passes through points W and R. Line G is the reflection of line K across point D. Let A be the reflection of line K across line G. Point J is the reflection of point D across line A. What is the measure of angle JVR, 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": "355.81", "dataset": "MATH"}
{"messages": [{"content": "MJ = 5. Point U is obtained by rotating point M by angle pi/2 about point J. Let Y be the line through point M perpendicular to segment MJ. Let C be the rotation of point J by angle pi/4 around point M. Let Q be the angle bisector of angle JCM. Line T is the reflection of line Y across line Q. Point A is the centroid of triangle JUM. Point R is the reflection of point M across line T. Let X be the midpoint of segment AU. 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": "5.96", "dataset": "MATH"}
{"messages": [{"content": "Let BR = 6. Point Q is the midpoint of segment BR. Point V is obtained by rotating point Q by angle 2pi/3 about point R. Let C be the rotation of point R by angle pi/6 around point V. Point D is the midpoint of segment QR. Let T be the rotation of point B by angle 5pi/12 around point D. Line I is the perpendicular bisector of segment BD. Let O be the line through point T perpendicular to segment QR. Let K be the reflection of line O across line I. Point Y is the reflection of point C across line K. What is the distance between points Y 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": "7.16", "dataset": "MATH"}
{"messages": [{"content": "XO = 9. Point M is the midpoint of segment OX. Line B is the perpendicular bisector of segment MO. Point R is the reflection of point X across line B. Point Z is on line B with |ZX| = 5. Point L is the reflection of point X across point Z. Let P be the midpoint of segment RL. Let Y be the reflection of point M across point R. What is the circumradius of triangle YLP? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.7", "dataset": "MATH"}
{"messages": [{"content": "Let NB = 5. Let V be the rotation of point N by angle pi/4 around point B. Point A is obtained by rotating point N by angle 5pi/6 about point V. Let K be the line through point N parallel to segment AB. Point X is the reflection of point B across line K. Point O is the orthocenter of triangle VAB. Find the length of segment XO. Express your 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 DL = 5. Let K be the reflection of point L across point D. Let Y be the reflection of point D across point L. Let Q be the rotation of point D by angle pi/6 around point Y. Line S is the perpendicular bisector of segment KQ. Point A is the reflection of point L across line S. Let W be the circumcenter of triangle YAQ. What is the distance between points W 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": "6.55", "dataset": "MATH"}
{"messages": [{"content": "NA = 11. Point G is obtained by rotating point N by angle pi/3 about point A. Point T is obtained by rotating point G by angle 11pi/12 about point N. Let I be the reflection of point G across point N. Point B is the reflection of point A across segment TI. What is the circumradius of triangle IBA? Express your 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 AJ = 1. Point Z is the midpoint of segment JA. Point Q is obtained by rotating point J by angle 3pi/4 about point Z. Point F is the midpoint of segment ZQ. Let W be the midpoint of segment QA. Let U be the angle bisector of angle JWQ. Point I is the reflection of point Q across line U. Find the measure of angle FAI, 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": "0.0", "dataset": "MATH"}
{"messages": [{"content": "GN = 8. Let P be the midpoint of segment NG. Let Z be the rotation of point P by angle pi/2 around point G. Let W be the reflection of point P across point Z. Point D is the reflection of point P across point N. Point I is the reflection of point D across segment PW. Find the distance between points I 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": "17.89", "dataset": "MATH"}
{"messages": [{"content": "Let BY = 8. Point P is obtained by rotating point Y by angle pi/6 about point B. Point E is obtained by rotating point Y by angle 3pi/4 about point P. Line W is the angle bisector of angle BPY. Let R be the midpoint of segment BY. Point A is the midpoint of segment RY. Point D is the reflection of point E across line W. What is the measure of angle ADP, 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": "0.29", "dataset": "MATH"}
{"messages": [{"content": "TF = 11. Point W is obtained by rotating point F by angle 5pi/6 about point T. Point B is the midpoint of segment FW. Point G is the reflection of point F across point T. Let M be the centroid of triangle FBT. Find the distance between points M 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": "14.94", "dataset": "MATH"}
{"messages": [{"content": "PY = 6. Let D be the rotation of point Y by angle 5pi/6 around point P. Point G is the reflection of point D across point P. Point E is the reflection of point Y across point P. Let U be the midpoint of segment YG. Let W be the reflection of point G across point D. Find the circumradius of triangle EUW. Express your 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": "Let AV = 12. Let B be the midpoint of segment VA. Let Y be the rotation of point B by angle 5pi/6 around point A. Circle X is the incircle of triangle BYV. Let O be the midpoint of segment YV. Point G is the inversion of point O with respect to circle X. What is 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": "2.75", "dataset": "MATH"}
{"messages": [{"content": "Let RJ = 4. Point E is obtained by rotating point R by angle 7pi/12 about point J. Let Q be the incenter of triangle EJR. Let S be the midpoint of segment RQ. Let B be the midpoint of segment JR. 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": "0.68", "dataset": "MATH"}
{"messages": [{"content": "ME = 12. Point Q is obtained by rotating point E by angle 5pi/12 about point M. Point X is the midpoint of segment QM. Let T be the reflection of point E across point X. Let I be the midpoint of segment QT. Point P is the midpoint of segment QI. What is the circumradius of triangle PMI? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.97", "dataset": "MATH"}
{"messages": [{"content": "WR = 4. Point P is the midpoint of segment RW. Let L be the line through points W and P. Point X is obtained by rotating point P by angle 3pi/4 about point R. Line Y is the perpendicular bisector of segment WX. Line E passes through point R parallel to line Y. Let Z be the reflection of line L through point X. Point U is the reflection of point X across line Z. Let C be the rotation of point X by angle pi/2 around point P. Point S is the reflection of point C across line E. Let J be the midpoint of segment UC. Let N be the midpoint of segment PJ. Find the length of segment WN. Express your 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": "SI = 2. Point R is obtained by rotating point I by angle pi/3 about point S. Let D be the orthocenter of triangle SRI. Line F is the angle bisector of angle DIS. Point Q is the reflection of point R across line F. Find 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": "1.04", "dataset": "MATH"}
{"messages": [{"content": "BM = 2. Let F be the rotation of point B by angle pi/6 around point M. Point T is the midpoint of segment MB. Line R is the perpendicular bisector of segment BT. Point O is the reflection of point F across line R. Let I be the midpoint of segment MF. What is the measure of angle TIO, 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": "0.13", "dataset": "MATH"}
{"messages": [{"content": "Let SV = 7. Point F is obtained by rotating point S by angle 3pi/4 about point V. Point B is the reflection of point V across segment FS. Point Y is obtained by rotating point F by angle pi/2 about point B. Line K passes through point Y parallel to segment FS. Point C is the circumcenter of triangle VBY. Point A is the reflection of point C across line K. What is the length of segment AF? Express your 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": "UY = 4. Let O be the rotation of point Y by angle 11pi/12 around point U. Let X be the reflection of point U across point Y. Point S is the reflection of point U across segment OX. Let H be the midpoint of segment XY. 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": "6.04", "dataset": "MATH"}
{"messages": [{"content": "Let UZ = 4. Let W be the rotation of point U by angle 7pi/12 around point Z. Point R is obtained by rotating point W by angle 3pi/4 about point U. Let F be the midpoint of segment UZ. Line D is the angle bisector of angle URF. Let M be the line through point W parallel to line D. Point K is the reflection of point R across line M. Point Q is the midpoint of segment WK. Let P be the midpoint of segment FQ. What is the distance between points P 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": "6.97", "dataset": "MATH"}
{"messages": [{"content": "XF = 5. Point U is obtained by rotating point X by angle pi/12 about point F. Point L is the midpoint of segment XF. Point P is the reflection of point L across segment XU. Point S is the midpoint of segment PU. Point K is the reflection of point P across segment UF. What is the measure of angle XSK, 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": "0.92", "dataset": "MATH"}
{"messages": [{"content": "Let OA = 6. Let D be the rotation of point A by angle pi/3 around point O. Point X is obtained by rotating point O by angle pi/3 about point A. Point H is the reflection of point D across point O. Let V be the reflection of point O across point H. Let R be the reflection of point H across point A. Find the inradius of triangle VRX. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.59", "dataset": "MATH"}
{"messages": [{"content": "Let FG = 12. Point A is obtained by rotating point G by angle 3pi/4 about point F. Let Q be the orthocenter of triangle AGF. Point P is the circumcenter of triangle AGF. Line X passes through point P parallel to segment GF. Point L is the reflection of point A across line X. Point D is the reflection of point L across segment FA. What is the measure of angle DQG, 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": "180.85", "dataset": "MATH"}
{"messages": [{"content": "DK = 7. Point X is the midpoint of segment KD. Line P is the perpendicular bisector of segment DK. Point U is obtained by rotating point X by angle 5pi/6 about point D. Let M be the reflection of line P through point D. Let F be the reflection of line P through point U. Point J is obtained by rotating point X by angle 11pi/12 about point K. Point R is the reflection of point D across point K. Point N is the reflection of point R across line M. Point G is the reflection of point U across point N. Point C is the reflection of point G across line F. Find the measure of angle KCJ, 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 PZ = 12. Point T is obtained by rotating point P by angle 2pi/3 about point Z. Point U is the midpoint of segment ZP. Let W be the midpoint of segment PT. Line N is the perpendicular bisector of segment WU. Point R is the reflection of point P across line N. Find 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": "10.39", "dataset": "MATH"}
{"messages": [{"content": "Let GP = 8. Let I be the midpoint of segment GP. Let D be the rotation of point P by angle 2pi/3 around point I. Line B is the perpendicular bisector of segment PD. Point F is the reflection of point G across line B. Point C is the reflection of point D across segment IP. What is the distance between points F 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": "4.0", "dataset": "MATH"}
{"messages": [{"content": "Let KY = 11. Point Z is obtained by rotating point Y by angle 7pi/12 about point K. Point F is the reflection of point Z across segment YK. Let G be the incenter of triangle YKZ. Let J be the reflection of point G across point F. Find the measure of angle JZF, 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": "54.4", "dataset": "MATH"}
{"messages": [{"content": "Let RZ = 11. Let S be the rotation of point Z by angle 5pi/12 around point R. Circle V is the circumcircle of triangle ZRS. Point P is the midpoint of segment RS. Point B is the inversion of point P with respect to circle V. Find 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": "9.03", "dataset": "MATH"}
{"messages": [{"content": "Let PN = 7. Let E be the rotation of point N by angle pi/4 around point P. Point G is the incenter of triangle PEN. Point M is the reflection of point G across segment NP. Let Z be the orthocenter of triangle PEN. Find 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": "3.89", "dataset": "MATH"}
{"messages": [{"content": "Let OJ = 9. Let Q be the rotation of point O by angle pi/3 around point J. Point F is the midpoint of segment JO. Let N be the rotation of point J by angle pi/3 around point F. Let H be the rotation of point F by angle pi/2 around point J. Let E be the angle bisector of angle QJH. Let L be the centroid of triangle OHN. Point R is the reflection of point Q across line E. Find the measure of angle RQL, 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": "17.54", "dataset": "MATH"}
{"messages": [{"content": "LB = 4. Let S be the rotation of point L by angle 3pi/4 around point B. Let E be the centroid of triangle SLB. Point X is the incenter of triangle SLB. Let K be the midpoint of segment EX. What is 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": "3.75", "dataset": "MATH"}
{"messages": [{"content": "PW = 3. Let Z be the midpoint of segment WP. Let A be the rotation of point Z by angle pi/12 around point W. Let I be the midpoint of segment AP. Point V is the centroid of triangle PWA. Point B is the reflection of point V across segment PW. Find the length of segment BI. Express your 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 DN = 4. Let V be the rotation of point N by angle pi/12 around point D. Let M be the rotation of point D by angle 5pi/12 around point V. Point K is the orthocenter of triangle NVM. Let C be the reflection of point N across point K. Point Y is the midpoint of segment VM. Let I be the circumcenter of triangle NVM. What is the measure of angle ICY, 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": "7.2", "dataset": "MATH"}
{"messages": [{"content": "BU = 12. Point F is obtained by rotating point B by angle 5pi/6 about point U. Point J is the circumcenter of triangle FUB. Let Z be the incenter of triangle FUB. What is the circumradius of triangle JZB? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.87", "dataset": "MATH"}
{"messages": [{"content": "TH = 4. Let J be the reflection of point H across point T. Let Q be the rotation of point H by angle 7pi/12 around point J. Let A be the midpoint of segment TQ. Line P is the angle bisector of angle AHT. Point C is the midpoint of segment QH. Point W is the reflection of point J across line P. Point S is the midpoint of segment AW. Find the length of segment CS. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.99", "dataset": "MATH"}
{"messages": [{"content": "Let RJ = 3. Let M be the perpendicular bisector of segment JR. Let C be the rotation of point J by angle 2pi/3 around point R. Let U be the reflection of line M through point C. Let H be the centroid of triangle JCR. Point A is the reflection of point R across line U. 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": "9.54", "dataset": "MATH"}
{"messages": [{"content": "RO = 5. Let W be the rotation of point R by angle 3pi/4 around point O. Line U is the perpendicular bisector of segment RW. Line Y is the perpendicular bisector of segment WO. Let A be the line through point O perpendicular to line U. Line X is the reflection of line Y across line A. Line X and segment WO intersect at point L. Point V is the midpoint of segment OL. Find 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": "3.23", "dataset": "MATH"}
{"messages": [{"content": "Let TQ = 8. Let G be the midpoint of segment QT. Point C is obtained by rotating point G by angle 11pi/12 about point T. Line Z is the angle bisector of angle GCQ. Point Y is the reflection of point T across line Z. Line B is the angle bisector of angle CGY. Point F is the reflection of point C across line B. Point S is the circumcenter of triangle YFC. 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": "1.41", "dataset": "MATH"}
{"messages": [{"content": "TG = 1. Point J is obtained by rotating point G by angle pi/12 about point T. Let B be the midpoint of segment GT. Line I is the angle bisector of angle JGT. Point U is the reflection of point B across line I. What is the inradius of triangle UGB? Express your 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": "Let CA = 9. Point K is obtained by rotating point C by angle pi/6 about point A. Point F is obtained by rotating point A by angle pi/2 about point K. Point W is the circumcenter of triangle KAF. Point L is the midpoint of segment FC. Point P is the midpoint of segment LF. 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": "5.57", "dataset": "MATH"}
{"messages": [{"content": "Let LM = 2. Point Z is the midpoint of segment ML. Let R be the rotation of point L by angle pi/3 around point Z. Line N is the perpendicular bisector of segment ZM. Line X passes through point Z parallel to line N. Point C is the reflection of point R across line X. What is the inradius of triangle ZRC? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.29", "dataset": "MATH"}
{"messages": [{"content": "Let KM = 5. Point O is obtained by rotating point M by angle 3pi/4 about point K. Circle T is the incircle of triangle MKO. Point P is the inversion of point K with respect to circle T. Point W is the circumcenter of triangle MKO. What is 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": "1.13", "dataset": "MATH"}
{"messages": [{"content": "CN = 4. Let L be the midpoint of segment CN. Point P is obtained by rotating point L by angle 11pi/12 about point N. Let R be the rotation of point P by angle pi/3 around point L. Let U be the midpoint of segment RL. Point O is the midpoint of segment NU. Point V is the midpoint of segment ON. Point I is the midpoint of segment CP. Find the measure of angle IOL, 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": "0.59", "dataset": "MATH"}
{"messages": [{"content": "Let WT = 8. Point V is the reflection of point T across point W. Point S is the reflection of point V across point T. Point Y is obtained by rotating point T by angle pi/4 about point W. Line R passes through points S and T. Line P is the reflection of line R across point Y. Line L is the reflection of line P across line R. Point B is the reflection of point S across line L. Find the measure of angle YVB, 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": "245.02", "dataset": "MATH"}
{"messages": [{"content": "Let H be the reflection of point J across point F. Point G is the centroid of triangle J. Let D be the reflection of point H across point G. Let I be the midpoint of segment JS. Find the distance between points I 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": "1.76", "dataset": "MATH"}
{"messages": [{"content": "Let GV = 1. Point M is obtained by rotating point V by angle pi/2 about point G. Let K be the reflection of point V across point M. Let F be the reflection of point M across point V. Let R be the midpoint of segment VG. Line J is the perpendicular bisector of segment VG. Point L is the reflection of point K across line J. Find the measure of angle RFL, 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": "2.25", "dataset": "MATH"}
{"messages": [{"content": "NW = 6. Point S is the midpoint of segment WN. Point T is obtained by rotating point W by angle pi/12 about point S. Line G is the perpendicular bisector of segment SN. Line A is the reflection of line G across point S. Point L is the reflection of point T across line G. Point Q is the reflection of point L across line A. What is the measure of angle NLQ, 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": "5.74", "dataset": "MATH"}
{"messages": [{"content": "LU = 1. Point R is obtained by rotating point L by angle pi/12 about point U. Point Q is the midpoint of segment LU. Point F is the midpoint of segment UQ. Let I be the midpoint of segment RF. Point M is the reflection of point L across segment RF. What is the measure of angle UMI, 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": "0.09", "dataset": "MATH"}
{"messages": [{"content": "WE = 3. Point P is obtained by rotating point E by angle 5pi/12 about point W. Point Z is the reflection of point P across point W. Let V be the rotation of point P by angle pi/4 around point Z. Point F is the midpoint of segment EP. Line D passes through points V and F. Let R be the centroid of triangle FWE. Point X is the reflection of point R across line D. What is the measure of angle EXZ, 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": "6.18", "dataset": "MATH"}
{"messages": [{"content": "JW = 4. Point K is obtained by rotating point J by angle 7pi/12 about point W. Let D be the rotation of point K by angle pi around point W. Point I is the reflection of point J across point K. Let N be the line through points I and J. Point G is the reflection of point W across line N. Let E be the reflection of point J across point D. Find the measure of angle EGK, 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": "23.18", "dataset": "MATH"}
{"messages": [{"content": "Let XO = 2. Point G is the reflection of point O across point X. Point H is obtained by rotating point G by angle pi/3 about point O. Let E be the midpoint of segment HG. Point S is the midpoint of segment XH. Point L is the reflection of point S across segment HO. What is the distance between points E 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": "2.65", "dataset": "MATH"}
{"messages": [{"content": "BV = 3. Point Y is the reflection of point V across point B. Point P is obtained by rotating point V by angle 2pi/3 about point Y. Point N is the midpoint of segment VB. Let M be the reflection of point P across point B. Point Q is obtained by rotating point B by angle 7pi/12 about point M. Let K be the reflection of point Y across point P. Let G be the reflection of point N across point K. Let D be the circumcircle of triangle VPQ. Point R is the inversion of point G with respect to circle D. 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": "7.66", "dataset": "MATH"}
{"messages": [{"content": "XY = 6. Point R is obtained by rotating point X by angle 5pi/12 about point Y. Let Z be the rotation of point R by angle pi/3 around point Y. Point B is the reflection of point R across segment YX. Point S is the reflection of point X across point R. Let L be the line through points S and Y. Point T is the reflection of point B across line L. Find the inradius of triangle TZB. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.1", "dataset": "MATH"}
{"messages": [{"content": "TI = 12. Point S is obtained by rotating point T by angle pi/12 about point I. Let K be the angle bisector of angle ITS. Point Z is the reflection of point S across line K. Point A is the centroid of triangle IZS. Find 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": "2.29", "dataset": "MATH"}
{"messages": [{"content": "Let DJ = 7. Point T is the reflection of point J across point D. DB = 10. Let F be the reflection of point D across point T. Let X be the midpoint of segment BD. Let Z be the midpoint of segment BT. Point Q is the reflection of point Z across segment BD. Find the length of segment XQ. Express your 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": "ZK = 12. Point Y is obtained by rotating point Z by angle 3pi/4 about point K. Point L is the midpoint of segment ZK. Point J is the reflection of point Z across point Y. Let D be the line through point J parallel to segment LZ. Let W be the reflection of point Z across point J. Let R be the reflection of line D through point W. Point G is the reflection of point J across line R. Line U is the perpendicular bisector of segment LZ. Point X is the reflection of point W across line U. Find the measure of angle JGX, 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": "3968.11", "dataset": "MATH"}
{"messages": [{"content": "VH = 6. Point X is the reflection of point H across point V. Point U is obtained by rotating point H by angle pi/3 about point X. Let W be the rotation of point H by angle pi/2 around point U. Let J be the reflection of point X across point W. Line Z is the angle bisector of angle UHJ. Point C is obtained by rotating point V by angle 7pi/12 about point X. Let E be the line through point X parallel to line Z. Point Q is the reflection of point U across segment JX. Point K is the reflection of point U across point J. Point M is the midpoint of segment QC. Point F is the reflection of point K across line E. Find the circumradius of triangle MFW. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "42.69", "dataset": "MATH"}
{"messages": [{"content": "Let WB = 9. Point Z is the reflection of point W across point B. Point M is obtained by rotating point Z by angle pi/12 about point W. Point G is the midpoint of segment MW. Point Y is the midpoint of segment ZM. What is the circumradius of triangle GBY? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.54", "dataset": "MATH"}
{"messages": [{"content": "Let LX = 3. Let C be the rotation of point L by angle pi/3 around point X. Let H be the midpoint of segment CL. Point M is the circumcenter of triangle HXC. Circle Y is the incircle of triangle HXC. Let R be the inversion of point H in circle Y. What is the inradius of triangle MXR? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.14", "dataset": "MATH"}
{"messages": [{"content": "PC = 2. Let U be the rotation of point P by angle pi/3 around point C. Let E be the midpoint of segment PU. Let Z be the angle bisector of angle UEC. Point M is the reflection of point C across line Z. Let T be the incenter of triangle CPU. Let I be the midpoint of segment PT. Let V be the reflection of point I across point M. Find the length of segment VT. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.06", "dataset": "MATH"}
{"messages": [{"content": "CF = 2. Let S be the rotation of point F by angle 3pi/4 around point C. Point N is the reflection of point S across point C. Let R be the centroid of triangle NFC. Point D is the reflection of point R across segment CN. Find the length of segment DN. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.98", "dataset": "MATH"}
{"messages": [{"content": "Let IW = 4. Point C is the midpoint of segment WI. Point V is obtained by rotating point C by angle 7pi/12 about point I. Point X is the reflection of point V across segment IW. Let R be the incenter of triangle IWV. Let B be the midpoint of segment CR. Find the inradius of triangle XWB. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.65", "dataset": "MATH"}
{"messages": [{"content": "SE = 8. Point K is obtained by rotating point S by angle 2pi/3 about point E. Let B be the angle bisector of angle EKS. Point A is the midpoint of segment KE. Point N is the midpoint of segment AE. Point D is the midpoint of segment SA. Point Y is the reflection of point D across line B. Find the circumradius of triangle NYS. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.13", "dataset": "MATH"}
{"messages": [{"content": "Let WF = 4. Let S be the rotation of point F by angle 5pi/12 around point W. Point R is obtained by rotating point W by angle 3pi/4 about point S. Let D be the midpoint of segment WF. Point B is the reflection of point W across segment RD. Point T is the orthocenter of triangle RDS. Point C is the midpoint of segment FT. What is 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": "3.26", "dataset": "MATH"}
{"messages": [{"content": "JC = 9. Let Q be the rotation of point J by angle pi/6 around point C. Point D is the incenter of triangle CJQ. Point Y is the reflection of point D across point J. Point V is the reflection of point Y across segment CJ. Let O be the reflection of point V across point Q. What is the measure of angle OVC, 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": "37.05", "dataset": "MATH"}
{"messages": [{"content": "Let VE = 1. Point X is obtained by rotating point V by angle pi/6 about point E. Point A is the reflection of point X across point E. Point B is the midpoint of segment VX. Let S be the reflection of point V across point A. Line U passes through point B perpendicular to segment VX. Point Q is the reflection of point S across line U. What is the inradius of triangle BSQ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.24", "dataset": "MATH"}
{"messages": [{"content": "Let K be the orthocenter of triangle O. Point Q is the midpoint of segment IK. Point D is the reflection of point I across segment PW36. Let J be the midpoint of segment DQ. 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": "0.52", "dataset": "MATH"}
{"messages": [{"content": "TY = 9. Point C is obtained by rotating point T by angle 2pi/3 about point Y. Let J be the incenter of triangle CYT. Point U is the orthocenter of triangle CYT. Let Q be the midpoint of segment CU. Let I be the reflection of point Q across point J. 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": "4.67", "dataset": "MATH"}
{"messages": [{"content": "Let UW = 1. Let T be the midpoint of segment WU. Let I be the perpendicular bisector of segment TW. Let G be the reflection of line I through point W. Point X is on line G with |XT| = 11. Point K is the reflection of point X across line I. Let A be the midpoint of segment UT. Let F be the incenter of triangle KWU. What is the measure of angle AKF, 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 AR = 10. Let N be the rotation of point A by angle pi/12 around point R. Let T be the rotation of point A by angle pi/2 around point R. Let V be the perpendicular bisector of segment NR. Let O be the incircle of triangle RNA. Circle W is the reflection of circle O across line V. Let H be the midpoint of segment NT. Let S be the inversion of point A in circle W. Find the measure of angle TSH, 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": "22.8", "dataset": "MATH"}
{"messages": [{"content": "NH = 1. Point W is obtained by rotating point H by angle 5pi/12 about point N. Let P be the rotation of point N by angle 5pi/12 around point H. Let J be the incenter of triangle PNW. Let B be the incircle of triangle PNW. Point U is the inversion of point H with respect to circle B. Let R be the midpoint of segment JU. Point G is the midpoint of segment PN. Let Z be the midpoint of segment NR. Find the measure of angle ZJG, 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": "0.03", "dataset": "MATH"}
{"messages": [{"content": "Let VM = 12. Let D be the rotation of point V by angle pi/6 around point M. Point S is the reflection of point D across point V. Point N is the orthocenter of triangle MSV. Point B is the midpoint of segment SM. Point Y is the midpoint of segment BV. Find the measure of angle YBN, 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": "3.53", "dataset": "MATH"}
{"messages": [{"content": "Let MQ = 11. Point Z is obtained by rotating point Q by angle pi/6 about point M. Let T be the circumcenter of triangle ZQM. Let G be the angle bisector of angle TQZ. Point S is the reflection of point M across segment ZQ. Point O is the reflection of point S across line G. 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": "15.08", "dataset": "MATH"}
{"messages": [{"content": "LZ = 3. Point M is obtained by rotating point Z by angle 7pi/12 about point L. Let C be the orthocenter of triangle ZLM. Let Q be the vector from point Z to point C. Point S is the translation of point M by vector Q. Point F is the midpoint of segment SM. Let J be the incenter of triangle ZLM. Circle E is the incircle of triangle ZLM. Let K be the reflection of circle E through point F. Let A be the inversion of point S in circle E. Point G is the inversion of point J with respect to circle K. Find the distance between points G 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": "7.08", "dataset": "MATH"}
{"messages": [{"content": "IL = 11. Line V passes through point I perpendicular to segment LI. Point Q is the reflection of point L across line V. Point C lies on line V at distance 5 from point L. Point B is the reflection of point C across point Q. Point W is the reflection of point C across point I. Let Y be the midpoint of segment WL. 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": "6.04", "dataset": "MATH"}
{"messages": [{"content": "Let WN = 11. Let K be the rotation of point N by angle 5pi/12 around point W. Let S be the line through points N and K. Let F be the line through point W perpendicular to line S. Let A be the midpoint of segment NK. Line R is the reflection of line F across point K. Point M is the reflection of point N across point W. Point O is the reflection of point W across line R. Let H be the perpendicular bisector of segment MN. Point X is the reflection of point A across line H. What is the measure of angle WOX, 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": "30.25", "dataset": "MATH"}
{"messages": [{"content": "HC = 11. Let S be the rotation of point H by angle 3pi/4 around point C. Point J is obtained by rotating point H by angle 5pi/12 about point S. Point M is the reflection of point H across segment CS. Let K be the circumcenter of triangle CJH. Let A be the midpoint of segment CJ. Point L is the midpoint of segment AK. 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": "21.29", "dataset": "MATH"}
{"messages": [{"content": "Let TP = 7. Line C is the perpendicular bisector of segment PT. Let L be the reflection of line C through point P. Point D is on line L with |DT| = 7. Point M is the reflection of point D across point P. Point V is the reflection of point P across segment TM. Point Y is the midpoint of segment DT. What is 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": "13.12", "dataset": "MATH"}
{"messages": [{"content": "LV = 7. Point G is obtained by rotating point L by angle pi/12 about point V. Point X is obtained by rotating point L by angle pi/3 about point V. Let P be the reflection of point X across point L. Let F be the line through point G parallel to segment VX. Line Q passes through point X perpendicular to line F. Point U is the reflection of point V across line Q. Point J is the reflection of point P across segment VX. Let S be the midpoint of segment XG. Find the circumradius of triangle JUS. Express your 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 TK = 9. Let S be the reflection of point K across point T. Let X be the vector from point T to point K. Point B is obtained by rotating point K by angle 11pi/12 about point T. Point G is obtained by rotating point K by angle 5pi/12 about point S. Point W is the translation of point G by vector X. Let Z be the midpoint of segment BG. Point P is the reflection of point T across point S. Point L is the midpoint of segment PW. Point A is the circumcenter of triangle PWS. Let U be the midpoint of segment ZL. Find the measure of angle UKA, 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": "31.09", "dataset": "MATH"}
{"messages": [{"content": "DJ = 9. Let A be the rotation of point J by angle pi/12 around point D. Point I is the reflection of point D across point J. Point M lies on segment JD. Point P is the reflection of point M across segment DA. Find the measure of angle IDP, 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": "ZB = 7. Point L is obtained by rotating point B by angle 11pi/12 about point Z. Let X be the circumcircle of triangle ZBL. Let Q be the incenter of triangle ZBL. Let Y be the centroid of triangle ZBL. Point J is the reflection of point Y across segment ZB. Let G be the inversion of point J in circle X. What is the length of segment GQ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.19", "dataset": "MATH"}
{"messages": [{"content": "QW = 7. Let O be the perpendicular bisector of segment WQ. Line P is the reflection of line O across point W. Point J is the reflection of point Q across line P. Point T lies on line P at distance 1 from point J. Point A is the circumcenter of triangle QTW. Point G is the incenter of triangle QTW. What is the circumradius of triangle AQG? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "58.33", "dataset": "MATH"}
{"messages": [{"content": "Let ZR = 2. Let O be the midpoint of segment ZR. Point Y is the reflection of point O across point R. Let W be the rotation of point Y by angle pi/4 around point Z. Point L is obtained by rotating point O by angle pi/3 about point Z. Point K is the midpoint of segment WY. Let P be the reflection of point O across point L. Let I be the midpoint of segment KP. What is the inradius of triangle OIW? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.2", "dataset": "MATH"}
{"messages": [{"content": "Let VJ = 4. Let B be the reflection of point J across point V. Let Z be the rotation of point B by angle 5pi/6 around point J. Let Q be the line through points B and Z. Point G is the reflection of point V across point J. Line W passes through point V parallel to line Q. Let X be the midpoint of segment GJ. Point T is the reflection of point V across line Q. Point C is the reflection of point G across line W. What is the circumradius of triangle CXT? Express your 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": "Let OZ = 8. Point Q is the midpoint of segment OZ. Point W is obtained by rotating point Z by angle 5pi/6 about point O. Let K be the midpoint of segment WQ. Line A is the perpendicular bisector of segment KQ. Line F is the perpendicular bisector of segment ZQ. Let J be the rotation of point Z by angle 3pi/4 around point K. Line H passes through point J parallel to line A. Line C passes through point J perpendicular to line H. Line E is the reflection of line F across line C. Point L is the reflection of point K across line E. What is the measure of angle OWL, 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": "0.93", "dataset": "MATH"}
{"messages": [{"content": "XI = 12. Let T be the midpoint of segment IX. Point D is obtained by rotating point T by angle 11pi/12 about point X. Point H is the midpoint of segment DT. Line J is the angle bisector of angle IHX. Point M is the reflection of point T across line J. Let Y be the centroid of triangle HMI. Find the length of segment YD. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.19", "dataset": "MATH"}
{"messages": [{"content": "UO = 6. Let A be the rotation of point O by angle pi/2 around point U. Let E be the rotation of point A by angle 5pi/12 around point U. Let J be the midpoint of segment OU. Point X is the incenter of triangle UAO. Point K is the reflection of point X across segment EA. Find the measure of angle KJX, 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": "0.37", "dataset": "MATH"}
{"messages": [{"content": "Let PT = 7. Let S be the midpoint of segment TP. Point X is the midpoint of segment PS. Point K is the reflection of point S across point P. Point Q is obtained by rotating point K by angle 3pi/4 about point X. Let Z be the reflection of point Q across point X. Point A is the reflection of point P across point Z. Point G is the orthocenter of triangle QZS. Find the distance between points A 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": "22.41", "dataset": "MATH"}
{"messages": [{"content": "TV = 8. Point R is obtained by rotating point V by angle pi/2 about point T. Point Y is the midpoint of segment TR. Point C is the reflection of point V across point Y. Point A is the reflection of point C across segment RV. What is the measure of angle TAC, 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 XO = 10. Point P is obtained by rotating point X by angle pi/2 about point O. Let W be the line through points O and P. Point G is the reflection of point X across line W. Let L be the midpoint of segment OG. Let T be the reflection of point G across point P. What is the measure of angle PLT, 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": "25.0", "dataset": "MATH"}
{"messages": [{"content": "Let ON = 7. Let E be the rotation of point O by angle 3pi/4 around point N. Point K is the midpoint of segment NE. Let T be the rotation of point K by angle pi/3 around point N. Point Q is the reflection of point T across point O. Line H is the perpendicular bisector of segment TQ. Point W is the reflection of point E across line H. Find 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": "12.93", "dataset": "MATH"}
{"messages": [{"content": "YW = 4. YP = 4. Let U be the perpendicular bisector of segment PW. Let S be the reflection of point Y across point W. Point N is the reflection of point S across line U. Let G be the midpoint of segment PY. 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": "6.0", "dataset": "MATH"}
{"messages": [{"content": "YU = 3. Point O is obtained by rotating point Y by angle pi/4 about point U. Let H be the midpoint of segment YO. Point J is the orthocenter of triangle OUY. Let G be the midpoint of segment YH. What is 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": "0.75", "dataset": "MATH"}
{"messages": [{"content": "WF = 7. Let J be the rotation of point F by angle 11pi/12 around point W. Point Z is the reflection of point J across segment WF. Point S is the circumcenter of triangle WJZ. Let O be the midpoint of segment ZS. Let U be the incenter of triangle WJZ. 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": "0.94", "dataset": "MATH"}
{"messages": [{"content": "Let CT = 8. Point S is obtained by rotating point T by angle pi/4 about point C. Point Y is the midpoint of segment CS. Point V is the reflection of point T across segment CS. Let G be the reflection of point V across point Y. Let I be the circumcenter of triangle TVG. Find 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": "7.62", "dataset": "MATH"}
{"messages": [{"content": "YH = 9. Point A is the midpoint of segment HY. Point X is obtained by rotating point A by angle 5pi/6 about point H. Point T is the reflection of point X across point A. Line I is the angle bisector of angle ATH. Point N is the reflection of point Y across line I. Point W is the midpoint of segment NH. 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": "8.86", "dataset": "MATH"}
{"messages": [{"content": "YF = 1. Point J is the midpoint of segment FY. Let W be the rotation of point Y by angle 2pi/3 around point F. Point L is the reflection of point F across point W. Let B be the midpoint of segment JF. Let K be the reflection of point Y across point B. Find the circumradius of triangle LKJ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.19", "dataset": "MATH"}
{"messages": [{"content": "YO = 7. Point B is obtained by rotating point O by angle 5pi/12 about point Y. Point V is the midpoint of segment BY. Let K be the reflection of point B across point O. Find the circumradius of triangle KOV. Express your 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": "Let ZI = 3. Let U be the reflection of point Z across point I. Let M be the rotation of point U by angle 11pi/12 around point I. Let N be the incenter of triangle IMZ. Let J be the circumcenter of triangle IMZ. Point Q is the reflection of point N across segment MZ. Point G is the reflection of point Q across segment IM. What is 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": "1.49", "dataset": "MATH"}
{"messages": [{"content": "Let BW = 3. Line F passes through points B and W. Point Q is obtained by rotating point B by angle 3pi/4 about point W. Let R be the rotation of point B by angle 5pi/12 around point Q. Point A is the reflection of point R across line F. Point P is the reflection of point Q across point R. Line M is the angle bisector of angle WPQ. Point U is the reflection of point B across line M. Point Z is the reflection of point R across line M. Let K be the reflection of point W across point Z. Find the measure of angle KUA, 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": "67.84", "dataset": "MATH"}
{"messages": [{"content": "Let UI = 12. Let G be the rotation of point U by angle 11pi/12 around point I. Let F be the line through points G and I. Let Q be the reflection of point G across point I. Point W is the reflection of point U across line F. What is the inradius of triangle WUQ? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.2", "dataset": "MATH"}
{"messages": [{"content": "Let XK = 2. Point Q is obtained by rotating point X by angle 11pi/12 about point K. Point E is the midpoint of segment QK. Let V be the centroid of triangle XQK. Point G is the reflection of point E across segment KX. 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": "1.08", "dataset": "MATH"}
{"messages": [{"content": "Let RO = 2. Point P is obtained by rotating point R by angle pi/2 about point O. Let H be the incenter of triangle RPO. Circle L is the incircle of triangle RPO. Point W is the inversion of point R with respect to circle L. Let J be the midpoint of segment HP. Find 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": "0.94", "dataset": "MATH"}
{"messages": [{"content": "OV = 3. Point G is obtained by rotating point O by angle pi/2 about point V. Let F be the reflection of point G across point V. Let T be the circumcenter of triangle GOF. Let W be the line through points F and T. Let C be the circumcircle of triangle GOF. Let N be the rotation of point T by angle 11pi/12 around point O. Circle Y is the reflection of circle C across line W. Point M is the reflection of point T across segment FG. Let H be the reflection of circle Y through point M. Let D be the reflection of point G across point T. Point K is the inversion of point N with respect to circle H. What is the distance between points D 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": "5.91", "dataset": "MATH"}
{"messages": [{"content": "ZW = 9. Let E be the rotation of point W by angle 3pi/4 around point Z. Let T be the perpendicular bisector of segment WE. Point G is the midpoint of segment EZ. Point Q is the reflection of point G across line T. Point N is the orthocenter of triangle WEZ. Find 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": "18.82", "dataset": "MATH"}
{"messages": [{"content": "Let JU = 7. Let C be the rotation of point U by angle pi/2 around point J. Let T be the rotation of point J by angle pi/12 around point U. Let L be the midpoint of segment UT. Point V is the reflection of point L across segment TC. Let O be the centroid of triangle TCU. What is the distance between points O 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": "6.27", "dataset": "MATH"}
{"messages": [{"content": "GO = 5. Line J is the perpendicular bisector of segment OG. Let Z be the line through point O parallel to line J. Let P be the reflection of line J across line Z. Point C is obtained by rotating point O by angle pi/3 about point G. Let Q be the rotation of point O by angle 5pi/12 around point G. Let D be the reflection of line Z across line P. Point X is the orthocenter of triangle GCO. Let R be the line through points Q and X. Point U is the intersection of lines D and R. 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": "25.09", "dataset": "MATH"}
{"messages": [{"content": "Let PV = 4. Let Z be the rotation of point P by angle pi/12 around point V. Line J is the perpendicular bisector of segment ZV. Let B be the line through point Z parallel to line J. Line G passes through point V parallel to line B. Point C is the reflection of point P across line G. Let R be the orthocenter of triangle CZP. Point D is the midpoint of segment CZ. What is the measure of angle RDP, 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": "1.93", "dataset": "MATH"}
{"messages": [{"content": "Let CX = 3. Point J is obtained by rotating point C by angle pi/2 about point X. Point H is the midpoint of segment XC. Point A is obtained by rotating point H by angle 5pi/6 about point C. Let F be the midpoint of segment XJ. Let G be the reflection of point F across point A. Let Z be the circumcenter of triangle GCA. Find the circumradius of triangle XGZ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "17.98", "dataset": "MATH"}
{"messages": [{"content": "Let YA = 4. Point E is obtained by rotating point A by angle pi/6 about point Y. Let V be the rotation of point Y by angle 5pi/12 around point E. Let G be the rotation of point V by angle pi/12 around point Y. Let I be the midpoint of segment YG. Let N be the circumcenter of triangle YIV. Point R is the reflection of point N across segment VY. 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": "2.42", "dataset": "MATH"}
{"messages": [{"content": "Let TC = 5. Point V is obtained by rotating point T by angle pi/3 about point C. Point K is the midpoint of segment CV. Let H be the circumcenter of triangle KTV. Let B be the midpoint of segment HV. What is 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": "1.1", "dataset": "MATH"}
{"messages": [{"content": "FJ = 8. Let K be the reflection of point F across point J. Point O is obtained by rotating point F by angle pi/4 about point K. Let W be the midpoint of segment JK. Let Y be the midpoint of segment WK. Point E is obtained by rotating point F by angle pi/3 about point K. Point A is the reflection of point E across point W. Let P be the reflection of point E across point Y. Let T be the incenter of triangle EOK. Point Z is the midpoint of segment TO. Find the measure of angle ZPA, 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": "50.16", "dataset": "MATH"}
{"messages": [{"content": "UM = 2. Let H be the perpendicular bisector of segment MU. Point N is obtained by rotating point U by angle 5pi/12 about point M. Point O is the reflection of point N across line H. Point W is the incenter of triangle MNO. Let J be the circumcenter of triangle MNO. Find the measure of angle WOJ, 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": "0.08", "dataset": "MATH"}
{"messages": [{"content": "Let OH = 8. Let U be the perpendicular bisector of segment OH. Point G is on line U with |GH| = 3. Let W be the reflection of point O across point G. Let C be the orthocenter of triangle HGO. Let N be the reflection of point H across point C. Find the measure of angle NCW, 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": "12.0", "dataset": "MATH"}
{"messages": [{"content": "Point E is the reflection of point B across segment YY40. Point V is obtained by rotating point B by angle pi about point E. Let Q be the angle bisector of angle DBE. Circle J is the circumcircle of triangle D. Circle X is the reflection of circle J across line Q. Point C is the inversion of point V with respect to circle X. What is 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": "1.92", "dataset": "MATH"}
{"messages": [{"content": "SC = 12. Point A is obtained by rotating point C by angle 2pi/3 about point S. Point P is the centroid of triangle ASC. Line B passes through point S perpendicular to segment SC. Point G is the reflection of point P across line B. 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": "4.0", "dataset": "MATH"}
{"messages": [{"content": "Let BH = 9. Line J is the perpendicular bisector of segment HB. Let O be the line through point B perpendicular to line J. Point A is the midpoint of segment HB. Let X be the midpoint of segment HA. Let I be the rotation of point A by angle pi/6 around point X. Point U is obtained by rotating point X by angle pi/3 about point H. Point E is obtained by rotating point A by angle 5pi/6 about point U. Point S is the reflection of point E across line O. Line K is the angle bisector of angle HSB. Point T is the reflection of point U across line K. Point V is the centroid of triangle EIH. What is 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": "5.31", "dataset": "MATH"}
{"messages": [{"content": "DB = 6. Let S be the rotation of point B by angle 3pi/4 around point D. Point W is the reflection of point S across point D. Let H be the reflection of point W across point S. Let C be the centroid of triangle SBD. What is the distance between points H 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": "17.47", "dataset": "MATH"}
{"messages": [{"content": "JH = 12. JA = 1. Let D be the rotation of point A by angle 3pi/4 around point H. Point O lies on segment HA. Let L be the line through point O parallel to segment DH. Let W be the incenter of triangle ADH. Let Z be the line through point W parallel to line L. Segment HA and line Z intersect at point G. Find the measure of angle WGA, 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": "LQ = 8. Let H be the vector from point Q to point L. Point E is the translation of point L by vector H. Let I be the line through points E and Q. Line O passes through point E perpendicular to line I. Point V is on line O with |VQ| = 10. Let U be the perpendicular bisector of segment QV. Point Y is the reflection of point L across line U. Point X is the reflection of point L across point Q. Point J is the reflection of point Y across line O. Point N is the reflection of point J across point V. What is the measure of angle XNJ, 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": "207.64", "dataset": "MATH"}
{"messages": [{"content": "Let QL = 11. Point M is obtained by rotating point L by angle pi/2 about point Q. Let C be the rotation of point Q by angle pi/6 around point M. Point K is obtained by rotating point M by angle 5pi/12 about point L. Let R be the incircle of triangle QKM. Point G is the midpoint of segment MQ. Let W be the reflection of point M across point C. Let O be the inversion of point G in circle R. Point Y is the reflection of point C across point W. What is the measure of angle QYO, 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": "4.49", "dataset": "MATH"}
{"messages": [{"content": "GB = 7. Point I is obtained by rotating point G by angle 2pi/3 about point B. Let U be the rotation of point B by angle pi/2 around point G. Let C be the midpoint of segment GU. Point L is the incenter of triangle GUI. Line K passes through points L and G. Point E is the intersection of segment UI and line K. What is the length of segment CE? Express your 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 HN = 6. Point S is the midpoint of segment NH. Point P is the midpoint of segment SN. Point A is the reflection of point S across point N. Point L is the midpoint of segment AN. Point W is obtained by rotating point H by angle 2pi/3 about point S. Let V be the reflection of point S across point L. Let J be the reflection of point P across point W. Point T is the reflection of point V across segment HJ. Point K is the midpoint of segment HW. Find the measure of angle HKT, 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": "14.47", "dataset": "MATH"}
{"messages": [{"content": "OE = 9. Let A be the perpendicular bisector of segment OE. Let Q be the reflection of line A through point O. Let R be the rotation of point E by angle 11pi/12 around point O. Let T be the reflection of line Q across line A. Let Y be the rotation of point R by angle pi/6 around point E. Point P is the reflection of point O across line T. Line M is the angle bisector of angle ROE. Line C is the reflection of line M across point R. Point V is the reflection of point Y across segment EO. Point N is the intersection of lines C and T. Point J is the reflection of point N across segment EO. Point K is the reflection of point V across point P. What is the measure of angle KJE, 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": "6263.89", "dataset": "MATH"}
{"messages": [{"content": "FE = 1. Point A is obtained by rotating point E by angle pi/2 about point F. Point I is obtained by rotating point A by angle pi/6 about point F. Line V is the angle bisector of angle FAI. Point M is the midpoint of segment EF. Let L be the perpendicular bisector of segment FI. Let G be the circumcircle of triangle EFA. Point X is the reflection of point E across line L. Let T be the reflection of circle G across line V. Let K be the inversion of point E in circle T. Let N be the reflection of point M across point K. Point R is the midpoint of segment XE. 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": "2.41", "dataset": "MATH"}
{"messages": [{"content": "Let FR = 6. Let Y be the rotation of point F by angle 2pi/3 around point R. Let W be the reflection of point F across point R. Point C is the reflection of point F across segment RY. Let V be the centroid of triangle RYF. What is the circumradius of triangle VCW? Express your 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": "Let VU = 9. Point J is the midpoint of segment VU. Point O is the reflection of point V across point U. Let D be the rotation of point V by angle pi/6 around point J. Let A be the perpendicular bisector of segment OJ. Point X is the reflection of point V across segment OD. Point T is the reflection of point U across segment OD. Point H is the reflection of point U across line A. What is the inradius of triangle XTH? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.37", "dataset": "MATH"}
{"messages": [{"content": "Let LW = 9. Point D is obtained by rotating point W by angle 5pi/12 about point L. Let R be the angle bisector of angle DLW. Let K be the line through point W parallel to line R. Point Z is the reflection of point L across line K. Let G be the orthocenter of triangle LDW. What is the measure of angle LZG, 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": "YK = 10. Line C is the perpendicular bisector of segment YK. Point G lies on line C at distance 12 from point K. Point J is the midpoint of segment KG. Let T be the midpoint of segment JY. Circle B is the circumcircle of triangle GJY. Point A is the midpoint of segment YT. Let F be the inversion of point A in circle B. Let P be the centroid of triangle GJY. 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": "7.84", "dataset": "MATH"}
{"messages": [{"content": "Let AI = 6. Point J is the reflection of point I across point A. Let B be the line through points A and J. Point D is obtained by rotating point I by angle pi/3 about point J. Point T is the reflection of point D across point I. Line R is the reflection of line B across point T. Line K is the reflection of line R across line B. Point H is the reflection of point I across line K. Point O is the reflection of point H across line R. What is the measure of angle DHO, 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": "374.12", "dataset": "MATH"}
{"messages": [{"content": "LH = 6. Line R is the perpendicular bisector of segment LH. Let Y be the vector from point H to point L. Point X is the reflection of point L across point H. Point G is on line R with |GX| = 8. Line S passes through points X and L. Let M be point G translated by vector Y. Point D lies on line S at distance 6 from point M. Let J be the midpoint of segment MX. Let C be the midpoint of segment GL. Let W be the reflection of point D across point X. Find the measure of angle CWJ, 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": "6.0", "dataset": "MATH"}
{"messages": [{"content": "XJ = 5. Point K is obtained by rotating point J by angle 3pi/4 about point X. Line A is the angle bisector of angle XKJ. Point N is on line A. Point R is the reflection of point K across segment XN. Find 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": "5.0", "dataset": "MATH"}
{"messages": [{"content": "Let UO = 1. Point B is obtained by rotating point U by angle pi/6 about point O. Let K be the reflection of point B across point U. Let F be the orthocenter of triangle OBK. Let G be the midpoint of segment OF. Let P be the incenter of triangle OBK. Find the distance between points G 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": "0.31", "dataset": "MATH"}
{"messages": [{"content": "FY = 5. Point S is obtained by rotating point F by angle pi/6 about point Y. Point A is obtained by rotating point Y by angle 3pi/4 about point S. Let M be the midpoint of segment FA. Point Q is the midpoint of segment AS. Point N is the midpoint of segment MQ. Find 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": "2.88", "dataset": "MATH"}
{"messages": [{"content": "WT = 3. Point X is obtained by rotating point T by angle pi/4 about point W. Let K be the incenter of triangle WTX. Line V passes through points K and W. Point M is the intersection of segment TX and line V. 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": "2.77", "dataset": "MATH"}
{"messages": [{"content": "Let AS = 2. Let W be the midpoint of segment SA. Let H be the line through points W and A. Let R be the line through point A perpendicular to line H. Line G passes through point W parallel to line R. Point V is on line G with |VS| = 4. Point T is the midpoint of segment VS. Let C be the reflection of line R across line G. Let U be the midpoint of segment SW. Point O is the reflection of point U across line R. Point Q is the reflection of point T across line C. Find the measure of angle OVQ, 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": "5.5", "dataset": "MATH"}
{"messages": [{"content": "Let LP = 11. Let M be the rotation of point L by angle 7pi/12 around point P. Point B is the midpoint of segment ML. Let F be the rotation of point B by angle pi/2 around point L. Point Q is the incenter of triangle LMF. Point T is the midpoint of segment FL. Let I be the centroid of triangle LMF. Find the measure of angle TQI, 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": "0.57", "dataset": "MATH"}
{"messages": [{"content": "Let FZ = 11. Let R be the midpoint of segment FZ. Point W is obtained by rotating point R by angle 5pi/6 about point Z. Let H be the midpoint of segment RW. Let K be the rotation of point F by angle pi/4 around point W. Line Y is the angle bisector of angle ZKF. Point E is the reflection of point R across segment FH. Circle A is the incircle of triangle KFH. Let G be the reflection of circle A across line Y. Point D is the reflection of point Z across line Y. Let I be the inversion of point H in circle G. Find the measure of angle EDI, 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": "5.08", "dataset": "MATH"}
{"messages": [{"content": "Let LU = 6. Let O be the rotation of point U by angle 7pi/12 around point L. Let P be the orthocenter of triangle LOU. Let Y be the reflection of point L across point P. Point J is the reflection of point Y across segment LO. Find the distance between points J 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": "5.1", "dataset": "MATH"}
{"messages": [{"content": "QR = 10. Let N be the reflection of point R across point Q. Let D be the rotation of point Q by angle pi/4 around point R. Point O is the reflection of point D across point N. Point X is the midpoint of segment QN. Let G be the midpoint of segment XD. What is the distance between points G 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": "24.33", "dataset": "MATH"}
{"messages": [{"content": "DN = 5. Let W be the reflection of point D across point N. Let G be the angle bisector of angle DNW. Point F lies on line G at distance 3 from point D. Let K be the incenter of triangle WFD. Point T is the midpoint of segment WF. Point M is the midpoint of segment TK. What is the length of segment KM? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.25", "dataset": "MATH"}
{"messages": [{"content": "Let QK = 7. Point S is the midpoint of segment KQ. Let L be the perpendicular bisector of segment SQ. Let F be the reflection of point S across point Q. I is constructed so that SI = 12. Point P lies on line L. Let N be the reflection of point S across point I. Let X be the midpoint of segment KP. Point R is the midpoint of segment PF. What is the length of segment XR? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.25", "dataset": "MATH"}
{"messages": [{"content": "Let KZ = 7. Let V be the rotation of point Z by angle 5pi/12 around point K. Let X be the rotation of point K by angle 11pi/12 around point Z. Point N is the reflection of point V across segment KZ. Line T passes through point Z perpendicular to segment KZ. Point O is the reflection of point X across line T. Find the circumradius of triangle ONZ. Express your 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": "Let SC = 10. Point F is obtained by rotating point C by angle 2pi/3 about point S. Line V is the angle bisector of angle FCS. Line Z is the reflection of line V across point F. Point P is the reflection of point F across point C. Let Q be the orthocenter of triangle PSF. Point U is the reflection of point F across line Z. What is the length of segment QU? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "54.77", "dataset": "MATH"}
{"messages": [{"content": "Let NW = 8. Let K be the midpoint of segment NW. Let M be the perpendicular bisector of segment KN. Point Y lies on line M at distance 2 from point W. Point E is the midpoint of segment KY. Let A be the orthocenter of triangle WKY. Find the inradius of triangle EWA. Express your 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": "RQ = 10. Point D is obtained by rotating point R by angle 5pi/6 about point Q. Let L be the incenter of triangle DRQ. Let F be the circumcircle of triangle DRQ. Point X is the circumcenter of triangle DRQ. Point U is the circumcenter of triangle RLX. Let Y be the inversion of point U in circle F. What is the distance between points Y 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": "4.92", "dataset": "MATH"}
{"messages": [{"content": "Let AG = 10. Line F passes through points A and G. Point N is obtained by rotating point G by angle pi/3 about point A. Let W be the midpoint of segment NG. Let D be the midpoint of segment WA. Point L is the reflection of point W across line F. Let T be the circumcenter of triangle LDN. Point H is the intersection of line F and segment LD. Find 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": "2.2", "dataset": "MATH"}
{"messages": [{"content": "Let IP = 12. Let R be the midpoint of segment PI. Point A is obtained by rotating point P by angle 11pi/12 about point I. Point U is the midpoint of segment AI. Let M be the circumcenter of triangle RAI. Point H lies on segment AI. Let N be the angle bisector of angle HIM. Point G is the reflection of point U across line N. Point S is the midpoint of segment GP. Point E is the midpoint of segment SG. What is the length of segment SE? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.37", "dataset": "MATH"}
{"messages": [{"content": "Let HB = 2. Let Y be the rotation of point B by angle pi/12 around point H. Line V is the perpendicular bisector of segment HB. Line C passes through point Y perpendicular to segment HB. Point W is obtained by rotating point Y by angle 5pi/12 about point B. Point O is obtained by rotating point W by angle pi/4 about point Y. Point S is the reflection of point O across line V. Segments SY and HB intersect at point F. Point J is the reflection of point S across line C. Find 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": "3.38", "dataset": "MATH"}
{"messages": [{"content": "Let SI = 8. E is constructed so that IE = 1. Let Z be the midpoint of segment SE. Point F is the reflection of point Z across point I. Line J passes through point I perpendicular to segment EI. Let Y be the midpoint of segment EI. Point L is the reflection of point Y across line J. What is the distance between points L 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": "4.0", "dataset": "MATH"}
{"messages": [{"content": "US = 2. Point B is obtained by rotating point U by angle 5pi/12 about point S. Point N is the midpoint of segment BS. Let Q be the midpoint of segment UN. Let P be the orthocenter of triangle SUN. Point W is the centroid of triangle SUN. Find the measure of angle QWP, 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": "0.07", "dataset": "MATH"}
{"messages": [{"content": "WQ = 1. Point L is obtained by rotating point Q by angle pi about point W. Let Y be the perpendicular bisector of segment WL. Point J is obtained by rotating point W by angle 5pi/6 about point Q. Point Z is the reflection of point J across line Y. Point M is the orthocenter of triangle ZJQ. Let H be the midpoint of segment JQ. 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": "6.46", "dataset": "MATH"}
{"messages": [{"content": "NB = 7. Let E be the reflection of point B across point N. Point R is obtained by rotating point N by angle pi/12 about point E. Let I be the rotation of point R by angle pi around point E. Let V be the midpoint of segment RN. Let U be the incenter of triangle NBI. Find the circumradius of triangle UVR. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "65.14", "dataset": "MATH"}
{"messages": [{"content": "JE = 3. Let F be the midpoint of segment EJ. Let V be the rotation of point J by angle pi/2 around point F. Let D be the line through point J perpendicular to segment EV. Point X is the midpoint of segment FE. Point Z is the reflection of point E across line D. Point G is the reflection of point X across segment EV. Point I is the reflection of point F across line D. Find the measure of angle IZG, 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": "2.25", "dataset": "MATH"}
{"messages": [{"content": "LE = 3. Point Q is obtained by rotating point E by angle 3pi/4 about point L. Let U be the reflection of point Q across point E. Point J is the reflection of point U across segment LE. Let X be the midpoint of segment LE. 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": "6.95", "dataset": "MATH"}
{"messages": [{"content": "CO = 3. Point K is obtained by rotating point O by angle pi/6 about point C. Let L be the rotation of point C by angle pi/3 around point O. Point N is the midpoint of segment KO. Let H be the midpoint of segment OL. Point G is the reflection of point N across point C. Find the inradius of triangle GKH. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.04", "dataset": "MATH"}
{"messages": [{"content": "Let FX = 1. Let L be the rotation of point X by angle pi/2 around point F. Point P is obtained by rotating point F by angle 11pi/12 about point X. Circle G is the incircle of triangle PXL. Let C be the orthocenter of triangle PXL. Let V be the circumcenter of triangle PXL. Let W be the inversion of point L in circle G. Find the measure of angle CVW, 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": "0.25", "dataset": "MATH"}
{"messages": [{"content": "CL = 10. Let H be the line through points L and C. Point M lies on line H at distance 4 from point L. Point B is obtained by rotating point L by angle pi/12 about point M. Point N is the reflection of point B across line H. Let Q be the reflection of point L across point B. Line R passes through points M and B. Line Z is the perpendicular bisector of segment LN. Let V be the midpoint of segment BQ. Let G be the reflection of line Z across line R. Point D is the reflection of point Q across line G. Let U be the midpoint of segment VQ. Find the circumradius of triangle DNU. Express your 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": "GT = 4. J is constructed so that TJ = 5. Let U be the reflection of point J across point T. Let X be the midpoint of segment GU. Let O be the midpoint of segment JG. What is the distance between points O 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": "5.0", "dataset": "MATH"}
{"messages": [{"content": "SL = 3. Point I is obtained by rotating point L by angle 3pi/4 about point S. Point V is the midpoint of segment LI. Let Q be the midpoint of segment VL. Let N be the line through points L and S. Point Z is the midpoint of segment QS. Point X is the orthocenter of triangle QZI. Point P is the reflection of point V across line N. Point A is the reflection of point X across line N. 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": "4.24", "dataset": "MATH"}
{"messages": [{"content": "EK = 9. Point Y is obtained by rotating point K by angle pi/4 about point E. Point T is the midpoint of segment YK. Point B is the reflection of point K across point Y. Point N is the midpoint of segment BT. Point X is the reflection of point T across segment NE. Point U is the incenter of triangle YNE. Point W is the midpoint of segment XT. 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": "1.63", "dataset": "MATH"}
{"messages": [{"content": "Let LV = 3. Point G is obtained by rotating point L by angle 5pi/6 about point V. Line E is the perpendicular bisector of segment GL. Line Y is the reflection of line E across point G. Line S is the reflection of line Y across line E. Let A be the midpoint of segment LV. Point M is the circumcenter of triangle VLG. Let D be the incenter of triangle VLG. Point B is the reflection of point M across line S. Find the circumradius of triangle ABD. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "333.06", "dataset": "MATH"}
{"messages": [{"content": "RX = 9. Point W is obtained by rotating point R by angle 11pi/12 about point X. Point N is the midpoint of segment XR. Let Z be the midpoint of segment WR. Point H is obtained by rotating point W by angle 5pi/12 about point Z. Let Y be the reflection of point W across point R. Let B be the angle bisector of angle YZN. Point K is the reflection of point X across line B. What is the circumradius of triangle KRH? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "7.13", "dataset": "MATH"}
{"messages": [{"content": "UF = 4. Point D is the midpoint of segment FU. Point T is obtained by rotating point U by angle 11pi/12 about point F. Let Q be the midpoint of segment FT. Point C is the reflection of point U across segment TD. Line J is the angle bisector of angle CQU. Point I is the reflection of point U across line J. What is the measure of angle CDI, 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": "0.0", "dataset": "MATH"}
{"messages": [{"content": "Let LN = 7. Let B be the rotation of point L by angle 7pi/12 around point N. Point Q is the reflection of point N across segment BL. Point G is the reflection of point L across point B. Point S is the reflection of point Q across point B. What is the inradius of triangle GSN? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.34", "dataset": "MATH"}
{"messages": [{"content": "Let WL = 4. Let T be the reflection of point L across point W. Let R be the rotation of point W by angle 11pi/12 around point L. Point Z is the orthocenter of triangle TWR. Point F is the midpoint of segment RW. Point J is the orthocenter of triangle RLZ. Find 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": "20.22", "dataset": "MATH"}
{"messages": [{"content": "RS = 10. Let D be the midpoint of segment SR. Point Y is obtained by rotating point S by angle pi/12 about point R. Let Q be the line through points Y and D. Point E is the reflection of point S across line Q. Let T be the midpoint of segment YE. 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": "5.0", "dataset": "MATH"}
{"messages": [{"content": "Let KU = 3. Point H is the reflection of point U across point K. Let R be the line through points K and U. Let G be the rotation of point U by angle 2pi/3 around point H. Let F be the line through point G parallel to line R. Let V be the reflection of point G across point U. Let L be the rotation of point G by angle pi/12 around point V. Point S is the reflection of point H across line F. Let E be the reflection of point H across point S. Point X is the reflection of point H across point U. What is the circumradius of triangle XEL? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.33", "dataset": "MATH"}
{"messages": [{"content": "ZO = 7. Let B be the midpoint of segment ZO. Let V be the rotation of point B by angle pi/4 around point O. Let X be the midpoint of segment BZ. Let J be the midpoint of segment ZX. Line W passes through point V perpendicular to segment XZ. Point L is the reflection of point J across line W. Let T be the circumcircle of triangle VXZ. Let G be the inversion of point L in circle T. Point S is the midpoint of segment GJ. What is 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": "1.68", "dataset": "MATH"}
{"messages": [{"content": "SN = 8. Let X be the midpoint of segment NS. Let K be the rotation of point N by angle pi/12 around point X. Let H be the line through points K and S. Line T passes through point N parallel to line H. Point Q is the reflection of point N across point K. Point L is the reflection of point Q across line T. Find the inradius of triangle LQS. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.6", "dataset": "MATH"}
{"messages": [{"content": "OP = 5. Let Y be the line through points O and P. Let H be the perpendicular bisector of segment OP. Let J be the midpoint of segment PO. Point X is the reflection of point J across point P. Line D is the reflection of line H across point P. Line K passes through point O perpendicular to line Y. Point S lies on line K at distance 9 from point J. Point B is the reflection of point S across line D. Let W be the midpoint of segment XS. What is the circumradius of triangle OWB? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "18.39", "dataset": "MATH"}
{"messages": [{"content": "DT = 2. Let C be the rotation of point D by angle 3pi/4 around point T. Let Y be the circumcenter of triangle TCD. Point R is the incenter of triangle TCD. Let Z be the line through point Y perpendicular to segment TC. Point Q is the reflection of point T across line Z. 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": "0.44", "dataset": "MATH"}
{"messages": [{"content": "Let EP = 6. Line V is the perpendicular bisector of segment PE. Point L lies on line V. Point F is obtained by rotating point E by angle 11pi/12 about point L. Let W be the midpoint of segment EP. Let G be the midpoint of segment FP. Let Z be the midpoint of segment PL. Find the measure of angle GZW, 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": "Let CN = 4. Let E be the rotation of point N by angle 5pi/12 around point C. Let L be the reflection of point C across point N. Point U is the midpoint of segment CE. Let F be the orthocenter of triangle LEN. Let K be the reflection of point N across point F. Point H is the circumcenter of triangle LEN. Let M be the incenter of triangle LEN. Let Z be the midpoint of segment KU. What is the length of segment ZM? Express your 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 SM = 4. Line X is the perpendicular bisector of segment MS. Point F is obtained by rotating point S by angle pi/4 about point M. Point R is the reflection of point F across line X. Let J be the angle bisector of angle FRM. Point C is the reflection of point R across point M. Point H is the reflection of point C across point R. Point I is the reflection of point C across line J. Find the measure of angle IHS, 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": "14.66", "dataset": "MATH"}
{"messages": [{"content": "RL = 9. Line U is the perpendicular bisector of segment LR. Let E be the line through point L parallel to line U. Let C be the midpoint of segment RL. Point P is obtained by rotating point L by angle 3pi/4 about point R. Line K passes through point P parallel to segment LR. Point A is the reflection of point C across line E. Point Q is the reflection of point C across line K. What is the circumradius of triangle QPA? Express your 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 MG = 3. Let E be the rotation of point M by angle 5pi/6 around point G. Point V is the midpoint of segment EG. Let D be the incenter of triangle EGM. Let S be the centroid of triangle EGM. What is the measure of angle VDS, 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": "0.09", "dataset": "MATH"}
{"messages": [{"content": "SG = 12. Point L is obtained by rotating point S by angle pi/2 about point G. Point Z is obtained by rotating point S by angle pi/2 about point L. Point A is the circumcenter of triangle ZSL. Line R is the angle bisector of angle ZAG. Point U is the reflection of point S across line R. What is the measure of angle USG, 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": "97.26", "dataset": "MATH"}
{"messages": [{"content": "EJ = 6. Let C be the rotation of point J by angle 5pi/12 around point E. Point G is the reflection of point C across segment EJ. Point Y is the reflection of point J across point G. Point Z is the incenter of triangle EJC. Let H be the midpoint of segment JC. What is the measure of angle ZHY, 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": "6.7", "dataset": "MATH"}
{"messages": [{"content": "SB = 9. Let R be the rotation of point S by angle 7pi/12 around point B. Let Z be the reflection of point B across point R. Point D is the centroid of triangle BSR. Point M is the orthocenter of triangle BSR. Find the measure of angle ZDM, 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": "53.4", "dataset": "MATH"}
{"messages": [{"content": "SO = 10. Let T be the rotation of point O by angle 2pi/3 around point S. Let R be the reflection of point T across point O. Let U be the reflection of point S across point T. Let I be the reflection of point S across point O. What is the circumradius of triangle URI? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "43.59", "dataset": "MATH"}
{"messages": [{"content": "JG = 2. Let I be the rotation of point G by angle 7pi/12 around point J. Point N is the midpoint of segment GI. Point R is the centroid of triangle GIJ. Point Y is the reflection of point R across segment GI. Point B is the reflection of point N across segment JG. What is 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": "2.27", "dataset": "MATH"}
{"messages": [{"content": "Let XG = 6. Point B is obtained by rotating point X by angle pi about point G. Let Z be the rotation of point B by angle 7pi/12 around point G. Let U be the midpoint of segment ZG. Point V is on segment UX. Let W be the midpoint of segment UB. What is the measure of angle VWG, 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": "4.35", "dataset": "MATH"}
{"messages": [{"content": "VD = 11. Point U is obtained by rotating point V by angle 3pi/4 about point D. Line H is the angle bisector of angle VUD. Let R be the midpoint of segment VD. Point A is the reflection of point R across segment DU. Point P is the reflection of point V across line H. Find the distance between points A 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": "6.68", "dataset": "MATH"}
{"messages": [{"content": "NO = 1. Point Y is obtained by rotating point O by angle pi/12 about point N. Let R be the midpoint of segment ON. Point E is the reflection of point R across point N. Let K be the orthocenter of triangle YRE. Find the distance between points K 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": "2.9", "dataset": "MATH"}
{"messages": [{"content": "VU = 8. Let J be the rotation of point V by angle pi/4 around point U. Let B be the midpoint of segment VU. Point T is the orthocenter of triangle BVJ. Let K be the midpoint of segment VJ. Find the measure of angle UTK, 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": "5.66", "dataset": "MATH"}
{"messages": [{"content": "HQ = 6. Let Y be the midpoint of segment QH. Point A is the reflection of point Y across point H. Point B is obtained by rotating point A by angle 3pi/4 about point Y. Let K be the midpoint of segment BH. Let R be the line through point K perpendicular to segment AB. Point V is the reflection of point B across line R. 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": "4.2", "dataset": "MATH"}
{"messages": [{"content": "Let RY = 8. Let H be the vector from point R to point Y. Point O is obtained by rotating point Y by angle 3pi/4 about point R. Let L be point O translated by vector H. Point K is the midpoint of segment RY. Let D be the reflection of point K across point Y. Point M is the reflection of point K across segment RO. Point C is the reflection of point D across segment RO. Point E is the reflection of point D across segment MO. What is the inradius of triangle ELY? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.31", "dataset": "MATH"}
{"messages": [{"content": "Let BN = 12. Let G be the rotation of point B by angle 2pi/3 around point N. Point W is the reflection of point B across point N. Let I be the midpoint of segment NW. Line V is the perpendicular bisector of segment NG. Let K be the midpoint of segment IW. Point L is the reflection of point K across line V. Find the circumradius of triangle GNL. Express your 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": "Let RP = 9. Let S be the rotation of point P by angle 11pi/12 around point R. Let N be the perpendicular bisector of segment SP. Point O is the reflection of point S across point R. Line I is the perpendicular bisector of segment SR. Let T be the reflection of line I across line N. Point Z is the midpoint of segment SR. Line T and segment SO intersect at point H. 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": "9.16", "dataset": "MATH"}
{"messages": [{"content": "Let TL = 12. Point P is obtained by rotating point T by angle pi/4 about point L. Point M is the orthocenter of triangle TPL. Circle V is the incircle of triangle TPL. Point Y is the reflection of point M across segment PL. Let K be the inversion of point Y in circle V. 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": "4.67", "dataset": "MATH"}
{"messages": [{"content": "Let OH = 2. Let U be the rotation of point H by angle pi/6 around point O. Point V is obtained by rotating point H by angle 2pi/3 about point U. Point T is the midpoint of segment OV. Let K be the line through point U perpendicular to segment HO. Point R is the reflection of point V across line K. Let J be the reflection of point H across point V. What is the circumradius of triangle HJR? Express your 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": "DL = 10. Point B is the midpoint of segment LD. Let M be the perpendicular bisector of segment DL. Point R is on line M with |RD| = 4. Line X passes through point R perpendicular to segment DR. Point S is the midpoint of segment DR. Point C is the reflection of point B across line X. Find the length of segment CS. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.5", "dataset": "MATH"}
{"messages": [{"content": "EV = 2. Point G is the reflection of point E across point V. Let P be the reflection of point E across point G. Point Y is obtained by rotating point G by angle 7pi/12 about point V. Point Q is obtained by rotating point P by angle 5pi/12 about point Y. Point J is the midpoint of segment YE. Let W be the reflection of point P across point J. Let U be the line through point W parallel to segment YE. Point O is the reflection of point Q across line U. What is the measure of angle JGO, 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": "19.95", "dataset": "MATH"}
{"messages": [{"content": "TA = 12. Point S is obtained by rotating point T by angle 5pi/12 about point A. Let X be the reflection of point T across point A. Point U is the midpoint of segment AS. Point I is the midpoint of segment UX. What is the measure of angle STI, 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": "86.93", "dataset": "MATH"}
{"messages": [{"content": "OF = 5. Point Z is obtained by rotating point O by angle 2pi/3 about point F. Point Y is the incenter of triangle OZF. Let I be the centroid of triangle OZF. Let S be the incircle of triangle OZF. Circle B is the reflection of circle S across point I. Let D be the inversion of point I in circle B. Point W is the reflection of point Y across segment ZF. Find the distance between points W 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": "1.86", "dataset": "MATH"}
{"messages": [{"content": "UB = 3. Let E be the rotation of point B by angle 5pi/12 around point U. Let Z be the vector from point B to point U. Let H be point U translated by vector Z. Point O is the midpoint of segment EH. Point P is the reflection of point U across point H. Let Q be the midpoint of segment PE. Find the circumradius of triangle OQP. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "6.48", "dataset": "MATH"}
{"messages": [{"content": "NW = 8. Point Y is obtained by rotating point W by angle 3pi/4 about point N. Line P passes through point N perpendicular to segment NY. Let Q be the reflection of point Y across point N. Let X be the reflection of point Q across point W. Point D is the reflection of point X across line P. Point M is the centroid of triangle YWN. 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": "9.76", "dataset": "MATH"}
{"messages": [{"content": "Let IS = 11. Let Z be the rotation of point S by angle 3pi/4 around point I. Point F is obtained by rotating point Z by angle 2pi/3 about point S. Let T be the incenter of triangle ZFI. Point L is the reflection of point S across point I. Let A be the circumcenter of triangle ZFI. Find the measure of angle ATL, 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": "18.57", "dataset": "MATH"}
{"messages": [{"content": "ZC = 8. Let U be the reflection of point Z across point C. Point K is obtained by rotating point U by angle 11pi/12 about point Z. Point H is the reflection of point K across segment ZU. Let F be the centroid of triangle ZUK. Point X is the reflection of point Z across point H. 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": "32.56", "dataset": "MATH"}
{"messages": [{"content": "Let CJ = 8. Line P is the perpendicular bisector of segment JC. Let H be the rotation of point C by angle 11pi/12 around point J. Point M is the reflection of point H across line P. Let X be the centroid of triangle CMH. Point E is the reflection of point X across line P. Let L be the midpoint of segment HC. What is the distance between points E 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": "2.55", "dataset": "MATH"}
{"messages": [{"content": "QJ = 2. Let W be the rotation of point J by angle pi/4 around point Q. Let U be the rotation of point W by angle 2pi/3 around point Q. Let C be the rotation of point Q by angle 11pi/12 around point U. Point I is the midpoint of segment CU. Line A is the perpendicular bisector of segment CI. Let H be the circumcenter of triangle JWQ. Point K is the reflection of point H across line A. Find the inradius of triangle QHK. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.47", "dataset": "MATH"}
{"messages": [{"content": "Let UJ = 4. Point P is the reflection of point U across point J. Let F be the rotation of point J by angle pi/3 around point U. Point E is the midpoint of segment JF. Let I be the midpoint of segment EP. Point Y is the circumcenter of triangle EUF. Point H is the midpoint of segment UY. Find the measure of angle PHI, 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": "2.02", "dataset": "MATH"}
{"messages": [{"content": "LZ = 12. Let B be the midpoint of segment ZL. Let E be the rotation of point L by angle 7pi/12 around point Z. Point P is the reflection of point E across point Z. Let I be the midpoint of segment PZ. Let F be the orthocenter of triangle PIB. Find 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": "19.67", "dataset": "MATH"}
{"messages": [{"content": "Let VD = 5. Let K be the rotation of point V by angle pi/4 around point D. Let G be the rotation of point K by angle 2pi/3 around point V. Line W is the perpendicular bisector of segment DV. Let X be the reflection of line W through point D. Point P is the reflection of point V across point K. Point E is the reflection of point G across line X. Let S be the reflection of point P across point E. What is the inradius of triangle DES? Express your 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": "Let IB = 1. Let L be the vector from point B to point I. Let G be the rotation of point I by angle pi/3 around point B. Let Z be the midpoint of segment BG. Point M is the translation of point I by vector L. Line R passes through point M perpendicular to segment MZ. Point N is the midpoint of segment MZ. Point A is the reflection of point G across line R. Point C is the reflection of point M across segment NG. 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": "0.96", "dataset": "MATH"}
{"messages": [{"content": "YB = 12. Line T is the perpendicular bisector of segment YB. Line Q is the reflection of line T across point B. Point H is on line Q with |HB| = 3. Let L be the orthocenter of triangle YHB. Point O is the reflection of point L across segment YH. Point V is the reflection of point Y across line Q. Point W is the reflection of point O across point V. Find the length of segment YW. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "77.67", "dataset": "MATH"}
{"messages": [{"content": "ZQ = 5. Point D is obtained by rotating point Q by angle pi/6 about point Z. Point S is the incenter of triangle QDZ. Point V is the orthocenter of triangle QDZ. Point A is the reflection of point V across segment DZ. 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": "2.24", "dataset": "MATH"}
{"messages": [{"content": "Let BG = 4. K is constructed so that GK = 6. Let P be the midpoint of segment KB. Let N be the midpoint of segment KP. Point A is the midpoint of segment PG. Find the length of segment AN. Express your 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": "WK = 3. Let Q be the rotation of point W by angle 11pi/12 around point K. Let P be the reflection of point K across point Q. Point L is the midpoint of segment WP. Point E is the reflection of point K across point L. Point F is the reflection of point L across segment KW. What is the inradius of triangle EWF? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.86", "dataset": "MATH"}
{"messages": [{"content": "XY = 9. Let G be the rotation of point X by angle pi/2 around point Y. Let W be the line through point Y perpendicular to segment XY. Point R is on line W with |RY| = 8. Let Z be the midpoint of segment YG. Let J be the midpoint of segment XY. Point V is the reflection of point G across point J. Point E is the midpoint of segment RX. Let Q be the midpoint of segment VX. What is the measure of angle ZEQ, 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": "18.0", "dataset": "MATH"}
{"messages": [{"content": "Let RP = 7. Let C be the vector from point P to point R. Point I is obtained by rotating point R by angle 11pi/12 about point P. Let Y be point I translated by vector C. Let N be the midpoint of segment IY. Let X be the midpoint of segment RI. Point B is the incenter of triangle YRX. Find the circumradius of triangle IBN. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "26.71", "dataset": "MATH"}
{"messages": [{"content": "IF = 2. Point O is the reflection of point I across point F. Point A is obtained by rotating point I by angle pi/6 about point F. Let C be the midpoint of segment IF. Point J is the midpoint of segment CO. Let V be the reflection of point C across point A. Point L is the midpoint of segment JV. 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": "1.79", "dataset": "MATH"}
{"messages": [{"content": "Let ZQ = 1. Let D be the rotation of point Q by angle pi/4 around point Z. Point E is the reflection of point Q across point Z. Point K is the midpoint of segment QZ. Let Y be the incircle of triangle KZD. Let O be the inversion of point Q in circle Y. Point J is the incenter of triangle KZD. What is the measure of angle OJE, 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": "0.01", "dataset": "MATH"}
{"messages": [{"content": "TB = 1. Let U be the rotation of point B by angle pi/2 around point T. Point O is the incenter of triangle UTB. Let K be the line through points T and O. Point N is the intersection of segment BU and line K. Point M is the midpoint of segment BO. 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": "0.38", "dataset": "MATH"}
{"messages": [{"content": "SY = 10. Let X be the rotation of point S by angle pi/2 around point Y. Let N be the midpoint of segment XY. Line A is the perpendicular bisector of segment NX. Let B be the reflection of point X across point S. Line J is the perpendicular bisector of segment SN. Point D is the reflection of point B across line J. Point K is the reflection of point D across line A. Let T be the reflection of point N across point K. Find the inradius of triangle KTD. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.47", "dataset": "MATH"}
{"messages": [{"content": "Let EA = 11. Let T be the reflection of point A across point E. Let K be the rotation of point A by angle 11pi/12 around point E. Point D is the orthocenter of triangle EKA. Point P is the midpoint of segment TK. Point M is the reflection of point T across point A. Let R be the circumcenter of triangle MPD. Find the measure of angle RAD, 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": "2.61", "dataset": "MATH"}
{"messages": [{"content": "Let QN = 6. Point C is the reflection of point Q across point N. Let K be the reflection of point N across point C. Line Z passes through points Q and C. Let S be the line through point C perpendicular to line Z. Let M be the rotation of point K by angle pi/2 around point N. Let V be the reflection of point M across point Q. Let I be the angle bisector of angle QVK. Point G is obtained by rotating point V by angle 7pi/12 about point Q. Line U is the reflection of line S across line I. Line D is the reflection of line I across point C. Let P be the midpoint of segment CG. Lines D and U intersect at point J. What is the measure of angle JQP, 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": "53.85", "dataset": "MATH"}
{"messages": [{"content": "HO = 7. Point S is the reflection of point H across point O. Let G be the midpoint of segment OH. Let U be the rotation of point G by angle pi/4 around point S. Point X is the reflection of point G across point U. Let M be the midpoint of segment OU. Point Y is the midpoint of segment XM. What is the measure of angle YUS, 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": "12.99", "dataset": "MATH"}
{"messages": [{"content": "Let FI = 2. Let D be the rotation of point I by angle pi/4 around point F. Point O is the midpoint of segment FD. Let Y be the reflection of point O across point F. Point N is the midpoint of segment YF. Point E is the midpoint of segment IF. Point U is the reflection of point D across segment IN. What is the measure of angle YFU, 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": "1.19", "dataset": "MATH"}
{"messages": [{"content": "Let UH = 10. Let Z be the midpoint of segment UH. Point T is obtained by rotating point U by angle pi/6 about point Z. Point G is the reflection of point T across point H. Point J is the orthocenter of triangle ZUT. What is the inradius of triangle JHG? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.33", "dataset": "MATH"}
{"messages": [{"content": "Let PZ = 12. Line X is the perpendicular bisector of segment ZP. Point L is the reflection of point Z across point P. Let O be the rotation of point P by angle pi/3 around point L. Point A is the reflection of point O across line X. Let N be the centroid of triangle LAO. Point T is the orthocenter of triangle LAO. 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": "24.0", "dataset": "MATH"}
{"messages": [{"content": "Let QR = 12. O is placed 12 units away from R. Let I be the rotation of point R by angle 5pi/6 around point O. Point M is the midpoint of segment OR. Let C be the midpoint of segment RQ. Let W be the reflection of point M across point C. Point E is the reflection of point R across segment WI. What is 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": "23.18", "dataset": "MATH"}
{"messages": [{"content": "SJ = 5. Let P be the rotation of point S by angle 11pi/12 around point J. Let L be the perpendicular bisector of segment PJ. Point H is the reflection of point P across point S. Point G is the orthocenter of triangle HJP. Point N is the reflection of point H across line L. 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": "118.99", "dataset": "MATH"}
{"messages": [{"content": "HS = 6. Let X be the rotation of point H by angle pi/12 around point S. Let A be the midpoint of segment XH. Point P is obtained by rotating point X by angle pi/2 about point S. Let K be the reflection of point A across point H. Let O be the midpoint of segment PA. Let U be the reflection of point S across point A. Find the circumradius of triangle KOU. Express your 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": "Let NS = 10. O is constructed so that SO = 5. Let A be the centroid of triangle SON. Let M be the line through point A parallel to segment NS. Point J is the intersection of line M and segment ON. 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": "3.33", "dataset": "MATH"}
{"messages": [{"content": "Let HY = 8. Let I be the midpoint of segment YH. Let A be the reflection of point I across point H. Let Z be the midpoint of segment YA. Line R is the perpendicular bisector of segment IH. Let M be the midpoint of segment YZ. Let V be the line through point M parallel to line R. Let C be the line through point A parallel to line R. Let U be the rotation of point H by angle pi/3 around point Z. Point G is the reflection of point A across point U. Let F be the midpoint of segment GU. Line P is the angle bisector of angle FMZ. Let L be the reflection of line P across line C. Point S is the intersection of lines L and V. Find the measure of angle HSF, 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": "11.69", "dataset": "MATH"}
{"messages": [{"content": "JZ = 9. Let C be the reflection of point J across point Z. Let Y be the rotation of point Z by angle pi/12 around point C. Let F be the midpoint of segment JY. Let H be the incircle of triangle JYC. Let I be the incenter of triangle JYC. Point T is the inversion of point Y with respect to circle H. Point P is the reflection of point T across point F. Find the measure of angle CIP, 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": "9.99", "dataset": "MATH"}
{"messages": [{"content": "MK = 8. Point A is obtained by rotating point K by angle pi/4 about point M. Point N is the midpoint of segment MK. Point O is the reflection of point N across segment MA. Let Y be the midpoint of segment AK. What is the circumradius of triangle ONY? Express your 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 BE = 3. Let L be the reflection of point E across point B. Let A be the rotation of point E by angle 11pi/12 around point L. Let Z be the midpoint of segment AB. Point K is the reflection of point B across point A. Point F is the reflection of point L across point E. What is the inradius of triangle FKZ? Express your 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": "IS = 8. Let R be the reflection of point S across point I. Line B passes through points R and S. Point L is obtained by rotating point S by angle pi/12 about point R. Point Q is the reflection of point L across line B. X is constructed so that RX = 3. Point V is the midpoint of segment IQ. Point J is the reflection of point V across segment LR. Point E is the midpoint of segment LR. 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": "5.76", "dataset": "MATH"}
{"messages": [{"content": "Let PC = 8. Let W be the vector from point P to point C. Let I be the rotation of point P by angle 5pi/12 around point C. Point N is the translation of point I by vector W. Point D is the incenter of triangle NCI. Line U is the perpendicular bisector of segment IN. Line J is the reflection of line U across point C. Line J and segment IN intersect at point Q. Point E is the reflection of point D across segment NC. Find the length of segment QE. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.64", "dataset": "MATH"}
{"messages": [{"content": "CT = 4. Point S is obtained by rotating point T by angle pi/6 about point C. Point G is the reflection of point C across segment ST. Point E is the reflection of point G across point T. Point L is the reflection of point G across point E. What is the circumradius of triangle ELS? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "15.73", "dataset": "MATH"}
{"messages": [{"content": "Let VC = 10. Point Q is obtained by rotating point V by angle pi/4 about point C. Point W is the reflection of point Q across point V. Point I is the midpoint of segment CQ. Let M be the midpoint of segment WC. Point G is the reflection of point M across segment CQ. Find the measure of angle IMG, 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": "20.71", "dataset": "MATH"}
{"messages": [{"content": "NK = 10. Point E is obtained by rotating point K by angle pi/6 about point N. Let L be the centroid of triangle NEK. Let Q be the orthocenter of triangle NEK. Let C be the midpoint of segment QK. What is the length of segment CL? Express your 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 PB = 8. Let O be the rotation of point P by angle pi/4 around point B. Point S is the midpoint of segment BP. Point K is the midpoint of segment SP. Point G is the reflection of point K across point P. What is the circumradius of triangle KGO? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.57", "dataset": "MATH"}
{"messages": [{"content": "Let MJ = 12. Let B be the midpoint of segment MJ. Let Q be the reflection of point B across point M. Line I is the perpendicular bisector of segment BJ. Let E be the reflection of point M across point Q. Let T be the rotation of point M by angle 2pi/3 around point E. Point W is the midpoint of segment TB. Point Y is the reflection of point M across line I. Point O is the centroid of triangle TWE. 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": "30.45", "dataset": "MATH"}
{"messages": [{"content": "GU = 3. Line X is the perpendicular bisector of segment UG. Let Z be the reflection of line X through point U. Point B is the reflection of point U across point G. Let Y be the rotation of point U by angle pi/3 around point B. Let R be the midpoint of segment UG. Point P is the midpoint of segment YR. Let T be the reflection of line Z through point P. Point D is the reflection of point P across line T. Find the measure of angle BDP, 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": "9.74", "dataset": "MATH"}
{"messages": [{"content": "EH = 2. Point R is the midpoint of segment HE. Point T is the reflection of point R across point H. Let B be the reflection of point H across point T. Point L is obtained by rotating point B by angle 3pi/4 about point R. Let P be the rotation of point T by angle pi/6 around point E. Line D is the perpendicular bisector of segment RT. Point X is the reflection of point P across line D. Point K is the reflection of point R across segment XL. 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": "1.55", "dataset": "MATH"}
{"messages": [{"content": "Let IE = 9. Let U be the reflection of point I across point E. Let T be the rotation of point U by angle 2pi/3 around point I. Point C is the midpoint of segment ET. Let W be the midpoint of segment TC. Let Q be the incenter of triangle ETU. Line G is the perpendicular bisector of segment IQ. Point N is the circumcenter of triangle ETU. Point D is the reflection of point W across line G. Find the inradius of triangle NDT. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.74", "dataset": "MATH"}
{"messages": [{"content": "Let DC = 9. Point G is obtained by rotating point D by angle 3pi/4 about point C. Let A be the perpendicular bisector of segment CG. Let J be the midpoint of segment DC. Line F passes through point D parallel to line A. Point T is the reflection of point J across line F. Find 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": "6.36", "dataset": "MATH"}
{"messages": [{"content": "Let OC = 3. Point A is the reflection of point C across point O. Let N be the rotation of point C by angle 7pi/12 around point O. Line B is the perpendicular bisector of segment NO. Line V passes through point A parallel to line B. Point J is the reflection of point O across line V. Point U is the reflection of point N across segment OA. Point T is the midpoint of segment OA. Segments OA and JU intersect at point H. Find 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": "0.97", "dataset": "MATH"}
{"messages": [{"content": "MC = 2. Let E be the rotation of point C by angle 7pi/12 around point M. Point O is the centroid of triangle CME. Let V be the incenter of triangle CME. Circle A is the circumcircle of triangle CME. Let G be the inversion of point V in circle A. Let Y be the midpoint of segment OC. Let T be the incircle of triangle CME. Line P passes through point G perpendicular to segment OC. Let W be the reflection of circle T across line P. Let N be the inversion of point V in circle W. Find the circumradius of triangle NYM. Express your 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": "Let SK = 7. Let N be the rotation of point S by angle pi/12 around point K. Point U is the reflection of point S across segment NK. Point T is the reflection of point N across point U. Let O be the midpoint of segment SN. Let Y be the reflection of point O across point K. What is 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": "13.68", "dataset": "MATH"}
{"messages": [{"content": "Let VT = 6. Let X be the reflection of point V across point T. Point U is obtained by rotating point T by angle pi/4 about point V. Point G is the midpoint of segment UT. Let Z be the midpoint of segment UG. Let H be the orthocenter of triangle UXZ. Point C is the midpoint of segment ZH. 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": "5.37", "dataset": "MATH"}
{"messages": [{"content": "Let WB = 11. Let X be the reflection of point W across point B. Let H be the rotation of point W by angle pi/12 around point X. Let D be the incenter of triangle HWX. Point P is the reflection of point B across segment HD. What is the distance between points P 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": "20.78", "dataset": "MATH"}
{"messages": [{"content": "DH = 1. Let O be the rotation of point H by angle 5pi/6 around point D. Point R is obtained by rotating point O by angle 5pi/12 about point D. Point V is the reflection of point R across point O. Point Y is the reflection of point V across segment DH. Let J be the perpendicular bisector of segment HY. Let Z be the reflection of point H across point V. Point X is the reflection of point O across line J. What is the measure of angle XVZ, 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": "1.66", "dataset": "MATH"}
{"messages": [{"content": "CK = 1. Let I be the line through points K and C. Let Q be the line through point K perpendicular to line I. Point V is the reflection of point K across point C. Point G is on line Q with |GC| = 5. Point U is the reflection of point V across point G. Point M is the midpoint of segment GC. Let J be the midpoint of segment UK. Find the circumradius of triangle JMC. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "8.01", "dataset": "MATH"}
{"messages": [{"content": "Let DH = 11. Let T be the reflection of point D across point H. Point Y is obtained by rotating point D by angle 3pi/4 about point T. Let P be the reflection of point D across point Y. Let V be the line through points H and P. Point C is the reflection of point T across line V. Line K is the angle bisector of angle HTC. Point L is the reflection of point P across line K. Find the measure of angle CLP, 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": "1999.1", "dataset": "MATH"}
{"messages": [{"content": "Let KQ = 2. Let L be the rotation of point K by angle 3pi/4 around point Q. Let F be the incenter of triangle LQK. Let O be the circumcenter of triangle LQK. Point H is the midpoint of segment KF. Line X passes through points O and H. Segment LQ and line X intersect at point Z. What is 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": "1.04", "dataset": "MATH"}
{"messages": [{"content": "Let BX = 9. Point R is obtained by rotating point X by angle 7pi/12 about point B. Let U be the perpendicular bisector of segment RX. Let W be the reflection of line U through point X. Let G be the line through points R and X. Point E is the midpoint of segment RB. Lines G and W intersect at point J. Find the circumradius of triangle BJE. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "10.59", "dataset": "MATH"}
{"messages": [{"content": "DK = 1. Point O is the reflection of point D across point K. Let Y be the rotation of point D by angle 5pi/6 around point O. Line I is the perpendicular bisector of segment DY. Point L is the reflection of point K across line I. Point U is the reflection of point L across point D. What is the inradius of triangle UYL? Express your 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": "Let DU = 12. Let J be the line through point D perpendicular to segment UD. Point Z is obtained by rotating point D by angle pi/4 about point U. Point Y is the midpoint of segment ZU. Let I be the rotation of point D by angle pi/6 around point U. Let G be the angle bisector of angle UYD. Point B is the midpoint of segment IZ. Point X is the reflection of point U across line G. Point C is the reflection of point X across line J. Find the circumradius of triangle BCU. Express your 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": "Let PU = 2. Point S is obtained by rotating point P by angle pi/12 about point U. Point Z is the midpoint of segment SP. Let A be the line through point P perpendicular to segment US. Point M is the reflection of point Z across line A. Let R be the reflection of point Z across point M. Let Y be the reflection of point U across point R. Point Q is the midpoint of segment US. 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": "2.71", "dataset": "MATH"}
{"messages": [{"content": "HR = 8. Point A is obtained by rotating point R by angle 5pi/12 about point H. Let B be the midpoint of segment RH. Point L is the reflection of point B across point R. Let J be the reflection of point L across point B. 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": "8.0", "dataset": "MATH"}
{"messages": [{"content": "ZJ = 9. Let X be the rotation of point Z by angle 2pi/3 around point J. Point V is obtained by rotating point J by angle 5pi/6 about point Z. Point C is the reflection of point X across segment ZJ. Let T be the reflection of point Z across point J. Let N be the midpoint of segment ZC. Find the circumradius of triangle TNV. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "45.47", "dataset": "MATH"}
{"messages": [{"content": "ME = 4. Point L is obtained by rotating point E by angle pi/2 about point M. Let A be the midpoint of segment LM. Let N be the line through points M and L. Point D is on line N with |DL| = 7. Point Z is the reflection of point D across segment ME. Let Y be the reflection of point L across point D. Let P be the midpoint of segment YA. Let X be the midpoint of segment EZ. Let S be the midpoint of segment XP. Point I is the reflection of point S across line N. What is the measure of angle IXM, 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": "0.5", "dataset": "MATH"}
{"messages": [{"content": "Let DJ = 1. Point V is the reflection of point J across point D. Point Z is obtained by rotating point D by angle 3pi/4 about point J. Let Q be the midpoint of segment JZ. Let O be the reflection of point V across point Q. Let I be the incenter of triangle JZD. Find 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": "2.83", "dataset": "MATH"}
{"messages": [{"content": "Let HS = 8. Point M is obtained by rotating point S by angle 3pi/4 about point H. Point E is the midpoint of segment SH. Point T is the reflection of point E across point S. Let R be the midpoint of segment ST. Let D be the line through point M parallel to segment HE. Point P is the reflection of point R across line D. What is the inradius of triangle MPE? Express your 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": "JT = 1. Let V be the vector from point J to point T. Let K be the perpendicular bisector of segment TJ. Point E is the translation of point T by vector V. Let A be the line through point J parallel to line K. Line Y passes through point J perpendicular to line A. Let L be the line through point E perpendicular to line Y. Point W is obtained by rotating point T by angle 5pi/12 about point J. Point P is the reflection of point W across line L. Point Z is the reflection of point E across line K. Let H be the reflection of point Z across point W. Point X is the midpoint of segment TH. Find the circumradius of triangle TPX. Express your 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": "EY = 2. Let M be the perpendicular bisector of segment YE. Point L lies on line M at distance 10 from point Y. Let S be the orthocenter of triangle YLE. Let I be the line through point E perpendicular to segment EY. Point Q is the reflection of point Y across line I. Find 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": "3.0", "dataset": "MATH"}
{"messages": [{"content": "KF = 8. Point D is obtained by rotating point K by angle pi/4 about point F. Point E is the midpoint of segment FK. Point S is the reflection of point E across point F. Point Q is the reflection of point S across segment KD. Point H is the midpoint of segment KD. Find the measure of angle EQH, 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": "5.66", "dataset": "MATH"}
{"messages": [{"content": "BD = 1. Point Q is obtained by rotating point B by angle 2pi/3 about point D. Let Y be the midpoint of segment BD. Point U is the centroid of triangle DQB. Point V is the midpoint of segment UB. 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": "0.17", "dataset": "MATH"}
{"messages": [{"content": "PL = 11. Let F be the rotation of point L by angle 7pi/12 around point P. Let Z be the rotation of point P by angle 2pi/3 around point L. Point C is obtained by rotating point F by angle 3pi/4 about point P. Point S is the reflection of point P across point Z. Point R is the reflection of point S across point P. Let U be the centroid of triangle LZC. What is the measure of angle ZRU, 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": "52.39", "dataset": "MATH"}
{"messages": [{"content": "Let QE = 9. Point P is obtained by rotating point E by angle 11pi/12 about point Q. Let N be the midpoint of segment QP. Let W be the centroid of triangle QEN. Let J be the midpoint of segment NW. 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": "2.97", "dataset": "MATH"}
{"messages": [{"content": "Let FI = 9. Let B be the rotation of point F by angle pi/6 around point I. Let R be the reflection of point I across point F. Point E is obtained by rotating point B by angle pi/2 about point F. Let X be the line through point E parallel to segment BR. Circle G is the incircle of triangle EBR. Let H be the centroid of triangle EBR. Let D be the midpoint of segment FH. Let Q be the reflection of circle G across line X. Let L be the inversion of point B in circle Q. What is the measure of angle BDL, 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": "13.54", "dataset": "MATH"}
{"messages": [{"content": "Point T is the centroid of triangle B. Let U be the midpoint of segment BR. Let V be the reflection of point T across point U. Let J be the midpoint of segment VQ. What is 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": "0.29", "dataset": "MATH"}
{"messages": [{"content": "Let XM = 11. Let E be the rotation of point X by angle 5pi/12 around point M. Let T be the incenter of triangle XME. Point Y is the midpoint of segment XT. Point V is the midpoint of segment XY. Point U is the reflection of point V across segment XM. 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": "2.99", "dataset": "MATH"}
{"messages": [{"content": "Let YZ = 6. Point Q is the reflection of point Z across point Y. Let N be the rotation of point Q by angle 3pi/4 around point Z. Point A is the reflection of point N across point Y. Let V be the midpoint of segment QA. Point F is the midpoint of segment YZ. Let S be the midpoint of segment VN. Let I be the reflection of point F across point A. What is the measure of angle NIS, 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": "9.55", "dataset": "MATH"}
{"messages": [{"content": "Let ZT = 8. Let V be the reflection of point T across point Z. Point L is obtained by rotating point T by angle 7pi/12 about point V. Point N is the reflection of point T across point V. Let O be the rotation of point T by angle pi/6 around point L. Point I is the midpoint of segment VN. Let E be the orthocenter of triangle NVO. Let A be the reflection of point I across point E. Find the distance between points A 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": "140.8", "dataset": "MATH"}
{"messages": [{"content": "NF = 11. Let E be the rotation of point N by angle pi/4 around point F. Let R be the circumcircle of triangle EFN. Let Y be the incenter of triangle EFN. Let T be the inversion of point Y in circle R. 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": "23.98", "dataset": "MATH"}
{"messages": [{"content": "RJ = 8. Point K is obtained by rotating point R by angle pi/12 about point J. Let O be the reflection of point J across point K. Let G be the rotation of point K by angle 11pi/12 around point O. Point T is the reflection of point O across point J. Circle Q is the incircle of triangle RGJ. Let I be the rotation of point O by angle 11pi/12 around point T. Let D be the inversion of point I in circle Q. Point E is the midpoint of segment RD. Let M be the inversion of point T in circle Q. What is the measure of angle MEK, 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": "0.01", "dataset": "MATH"}
{"messages": [{"content": "Let ZI = 11. Let K be the rotation of point Z by angle 7pi/12 around point I. Let S be the midpoint of segment ZK. Point F is the reflection of point I across segment SZ. Point U is the incenter of triangle ISZ. 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": "9.18", "dataset": "MATH"}
{"messages": [{"content": "IZ = 2. Point P is obtained by rotating point Z by angle 5pi/6 about point I. Point K is the reflection of point P across point Z. Point B is the centroid of triangle ZIP. Point O is the midpoint of segment IB. What is the inradius of triangle OKI? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.08", "dataset": "MATH"}
{"messages": [{"content": "Let BQ = 2. Point T is the midpoint of segment QB. Let A be the vector from point T to point B. Let E be the midpoint of segment BT. Let Z be point B translated by vector A. Let L be the line through point Z perpendicular to segment QB. Let D be the line through point T perpendicular to segment QB. Point K lies on line L at distance 10 from point Z. Let J be the midpoint of segment BE. Line S is the reflection of line L across line D. Point O is the reflection of point B across line S. Point N is the reflection of point K across point O. Find the measure of angle KJN, 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": "57.5", "dataset": "MATH"}
{"messages": [{"content": "ID = 5. Point R is obtained by rotating point D by angle 2pi/3 about point I. Point L is the midpoint of segment IR. Let H be the angle bisector of angle ILR. Point A lies on line H at distance 8 from point I. Point S is the midpoint of segment RA. Let E be the reflection of point S across point A. Let O be the centroid of triangle ISE. Find the distance between points O 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": "5.4", "dataset": "MATH"}
{"messages": [{"content": "Let OK = 4. Let Q be the midpoint of segment KO. Point F is obtained by rotating point O by angle 11pi/12 about point K. Let L be the reflection of point F across point O. Point X is obtained by rotating point F by angle pi/6 about point O. Let A be the reflection of point O across point X. Let R be the perpendicular bisector of segment LX. Let P be the line through point L perpendicular to line R. Let J be the perpendicular bisector of segment FO. Point B is the reflection of point Q across line J. Point E is the reflection of point A across line P. What is the circumradius of triangle EBX? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "4.94", "dataset": "MATH"}
{"messages": [{"content": "OW = 9. Point R is obtained by rotating point W by angle 5pi/6 about point O. Point D is the midpoint of segment OW. Let G be the midpoint of segment DW. Point L is the reflection of point G across segment RD. What is the circumradius of triangle LRW? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "11.25", "dataset": "MATH"}
{"messages": [{"content": "Let NH = 4. Let K be the rotation of point N by angle 2pi/3 around point H. Point M is the midpoint of segment KN. Let I be the centroid of triangle MHN. Let O be the incenter of triangle MHN. 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": "0.43", "dataset": "MATH"}
{"messages": [{"content": "FT = 3. Let N be the rotation of point T by angle 3pi/4 around point F. Let V be the rotation of point N by angle pi/6 around point T. Let Z be the angle bisector of angle VFT. Point J is the reflection of point F across point N. Point O is the reflection of point T across line Z. Point P is the incenter of triangle TJN. Find the inradius of triangle PNO. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.06", "dataset": "MATH"}
{"messages": [{"content": "PF = 9. Point B is the reflection of point F across point P. Point Q is the reflection of point P across point B. Point U is obtained by rotating point F by angle pi/3 about point B. Line C is the angle bisector of angle BUQ. Let K be the midpoint of segment BU. Let R be the reflection of line C through point Q. Let H be the line through point K perpendicular to segment UQ. Line W is the reflection of line R across line H. Let X be the incenter of triangle UQP. Point M is the reflection of point P across line W. Point Y is the reflection of point M across line C. Let V be the reflection of point Q across point Y. What is the length of segment XV? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "61.45", "dataset": "MATH"}
{"messages": [{"content": "KB = 10. Point P is the midpoint of segment KB. Line H is the angle bisector of angle BKP. Let V be the reflection of point K across point B. Point Z is on line H with |ZP| = 9. Point G is obtained by rotating point Z by angle 7pi/12 about point P. Let F be the perpendicular bisector of segment PV. Point A is the midpoint of segment GZ. Point Q is the reflection of point K across line F. Let R be the midpoint of segment QA. What is the measure of angle ABR, 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": "16.3", "dataset": "MATH"}
{"messages": [{"content": "GI = 5. Point Y is obtained by rotating point I by angle pi/3 about point G. Let R be the rotation of point Y by angle pi/4 around point I. Line B passes through point G perpendicular to segment IR. Let F be the midpoint of segment RY. Point C is the reflection of point F across line B. Find the measure of angle FCG, 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": "1.72", "dataset": "MATH"}
{"messages": [{"content": "RY = 11. Point V is the midpoint of segment RY. Let X be the rotation of point R by angle pi/2 around point Y. Point O is the centroid of triangle VXY. Let G be the perpendicular bisector of segment VX. Line L is the reflection of line G across point R. Point K is the reflection of point O across line L. Point C is the midpoint of segment RX. 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": "32.24", "dataset": "MATH"}
{"messages": [{"content": "Let IF = 9. Let R be the rotation of point I by angle 5pi/6 around point F. Let H be the reflection of point R across point I. Let C be the circumcenter of triangle FIH. Let E be the midpoint of segment FI. Point U is the reflection of point C across point R. 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": "33.38", "dataset": "MATH"}
{"messages": [{"content": "Let LA = 7. Point K is the midpoint of segment LA. Point E is obtained by rotating point K by angle pi/2 about point L. Point I is the midpoint of segment KL. Point D is the centroid of triangle LIE. Let T be the midpoint of segment EL. Let B be the reflection of point T across point D. Point C is the midpoint of segment IE. What is the length of segment BC? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "1.2", "dataset": "MATH"}
{"messages": [{"content": "GV = 9. Point D is obtained by rotating point G by angle 7pi/12 about point V. Point E is the reflection of point V across point D. Point N is obtained by rotating point D by angle 11pi/12 about point E. Point K is the orthocenter of triangle NED. Circle S is the circumcircle of triangle NED. Let P be the angle bisector of angle DKV. Let B be the inversion of point V in circle S. Point T is the reflection of point V across line P. Let O be the reflection of point B across point E. Find 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": "27.01", "dataset": "MATH"}
{"messages": [{"content": "Let UY = 7. Let N be the midpoint of segment YU. Point P is obtained by rotating point N by angle 7pi/12 about point U. Circle M is the incircle of triangle NUP. Let J be the reflection of circle M through point Y. Let V be the rotation of point Y by angle 5pi/6 around point U. Let C be the midpoint of segment PU. Let L be the circumcenter of triangle NUP. Point D is the inversion of point V with respect to circle J. Let W be the midpoint of segment PN. Let R be the line through point U parallel to segment CD. Point O is the reflection of point L across line R. 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": "3.19", "dataset": "MATH"}
{"messages": [{"content": "OC = 9. Point P is obtained by rotating point O by angle 3pi/4 about point C. Point K is obtained by rotating point P by angle 11pi/12 about point O. Let W be the midpoint of segment CK. Let F be the centroid of triangle WOP. Point L is the reflection of point F across segment OP. Find the measure of angle FLW, 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": "10.49", "dataset": "MATH"}
{"messages": [{"content": "MG = 10. Point P is the reflection of point G across point M. Let K be the rotation of point M by angle pi/6 around point G. Point I is the reflection of point G across segment PK. Line V passes through point P perpendicular to segment PK. Let O be the line through point I parallel to line V. Point C is the midpoint of segment PK. Point E is the midpoint of segment CG. Let F be the reflection of point E across point K. Point A is the reflection of point M across line O. What is the measure of angle FPA, 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": "80.67", "dataset": "MATH"}
{"messages": [{"content": "KF = 3. Line E is the perpendicular bisector of segment KF. Point A lies on line E at distance 6 from point K. Point L is the incenter of triangle KFA. Point B is the orthocenter of triangle KFA. Line I passes through point L parallel to segment KF. Segment FA and line I intersect at point W. Find the distance between points W 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": "YK = 6. Point U is obtained by rotating point K by angle 2pi/3 about point Y. Point A is obtained by rotating point K by angle 11pi/12 about point U. Point J is the reflection of point A across point U. Let Q be the centroid of triangle KYJ. Let Z be the line through point U perpendicular to segment JK. Point W is the reflection of point Q across line Z. Point F is the midpoint of segment YW. Find the inradius of triangle FAW. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.77", "dataset": "MATH"}
{"messages": [{"content": "Let FA = 6. Let U be the rotation of point A by angle pi/12 around point F. Let P be the rotation of point F by angle 2pi/3 around point A. Let G be the reflection of point A across point P. Point K is the midpoint of segment PG. Let Q be the midpoint of segment KF. Point Z is the reflection of point G across segment KF. Find the measure of angle ZUQ, 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": "23.52", "dataset": "MATH"}
{"messages": [{"content": "ZC = 4. Let J be the midpoint of segment ZC. Point L is obtained by rotating point Z by angle 7pi/12 about point J. Point R is obtained by rotating point J by angle 3pi/4 about point C. Let Y be the perpendicular bisector of segment JL. Point Q is the reflection of point Z across line Y. Point P is the circumcenter of triangle JLR. Find the inradius of triangle PQZ. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "0.92", "dataset": "MATH"}
{"messages": [{"content": "Let US = 10. Point V is obtained by rotating point U by angle pi/12 about point S. Point O is the reflection of point U across point S. Let B be the midpoint of segment SO. Let N be the centroid of triangle OVU. Point J is the reflection of point N across segment VU. 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": "21.48", "dataset": "MATH"}
{"messages": [{"content": "Let GT = 12. Let A be the rotation of point T by angle pi/4 around point G. Point O is the orthocenter of triangle GTA. Let V be the incenter of triangle GTA. Let F be the midpoint of segment TA. Point C is the midpoint of segment VF. Point R is the midpoint of segment CT. Point L is on segment TA. Let P be the midpoint of segment RO. Find 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": "1.17", "dataset": "MATH"}
{"messages": [{"content": "NZ = 3. Let L be the midpoint of segment NZ. Line E is the perpendicular bisector of segment ZN. Line X passes through point N parallel to line E. Let M be the reflection of line X across line E. Point D lies on line M at distance 5 from point N. Point P lies on line X. Let T be the reflection of point L across point P. Point V is the centroid of triangle DNT. Find the measure of angle VTL, 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": "2.5", "dataset": "MATH"}
{"messages": [{"content": "Let IM = 11. Let H be the midpoint of segment IM. Point P is obtained by rotating point H by angle 11pi/12 about point I. Let O be the reflection of point P across point M. Point V is the orthocenter of triangle IOP. Point R is the reflection of point O across segment VP. 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": "32.75", "dataset": "MATH"}
{"messages": [{"content": "YG = 4. Let N be the vector from point G to point Y. Point U is the translation of point Y by vector N. Let T be the reflection of point G across point U. Let V be the rotation of point T by angle 5pi/12 around point Y. Let B be the rotation of point U by angle pi/12 around point V. Line X is the angle bisector of angle TBG. Let L be the line through point Y parallel to line X. Line W passes through point T parallel to line L. Point F is the reflection of point B across line W. Find the measure of angle VTF, 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": "30.62", "dataset": "MATH"}
{"messages": [{"content": "Let GZ = 10. Point X is obtained by rotating point Z by angle pi/12 about point G. Point Q is the midpoint of segment GX. Let M be the reflection of point Q across point Z. Circle R is the circumcircle of triangle GXZ. Point C is the incenter of triangle GXZ. Point D is the reflection of point M across segment XZ. Point B is the inversion of point Q with respect to circle R. Find the inradius of triangle DBC. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "2.03", "dataset": "MATH"}
{"messages": [{"content": "Let QG = 11. Let B be the rotation of point G by angle 7pi/12 around point Q. Let H be the midpoint of segment GQ. Line E passes through point H perpendicular to segment GQ. Let R be the midpoint of segment HB. Point W is the reflection of point B across line E. What is the inradius of triangle GRW? Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "3.31", "dataset": "MATH"}
{"messages": [{"content": "DM = 3. Let U be the reflection of point D across point M. Point C is the reflection of point M across point U. Point N is obtained by rotating point C by angle pi/4 about point U. Let Y be the reflection of point N across point C. Point A is the reflection of point N across point U. Point R is the midpoint of segment DM. Find the circumradius of triangle AYR. Express your 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": "XY = 12. Let A be the rotation of point Y by angle pi/4 around point X. Let U be the midpoint of segment YA. Let L be the incenter of triangle XYA. Point S is the midpoint of segment XY. Find the measure of angle LUS, 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": "3.52", "dataset": "MATH"}
{"messages": [{"content": "YA = 4. Let D be the rotation of point Y by angle pi around point A. Point X is the midpoint of segment AD. Let H be the rotation of point A by angle pi/4 around point Y. Point Z is the reflection of point H across point A. Point B is the reflection of point H across point D. Let T be the orthocenter of triangle ZXH. Find the length of segment TB. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "5.97", "dataset": "MATH"}
{"messages": [{"content": "Let RX = 8. Let M be the rotation of point X by angle pi/6 around point R. Let E be the reflection of point R across point X. Point N is the reflection of point M across segment RX. Point B is the reflection of point N across point M. Let C be the reflection of point R across point B. Find the circumradius of triangle ECM. Express your answer as a decimal number rounded to 2 decimal places.\n\nPresent the answer in LaTex format: \\boxed{Your answer}", "role": "user"}], "ground_truth": "12.09", "dataset": "MATH"}