id stringlengths 4 8 | image_url stringlengths 77 81 | query stringlengths 7 1.32k | answer stringlengths 1 148 | choice stringlengths 4 597 | question_type stringclasses 2
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geo509 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/509.png | As shown in the figure, AB is the diameter of circle O, and chord CD is perpendicular to AB. Given that ∠CAB = 40°, connect BD and OD. What is the sum of ∠AOD and ∠ABD? | D | ['100°', '110°', '120°', '150°'] | multi_choice |
geo510 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/510.png | As shown in the figure, AB is a chord of circle O, and radius OC is perpendicular to AB at point D. Given that ∠A = 36°, and point P is on the circumference, what is the measure of ∠P? | A | ['27°', '30°', '36°', '40°'] | multi_choice |
geo511 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/511.png | As shown in the figure, AB is the diameter of circle O, C and D are points on circle O, and OC is parallel to BD. If ∠A = 30°, then ∠CBD = () | C | ['10°', '15°', '30°', '45°'] | multi_choice |
geo512 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/512.png | As shown in the figure, in the inscribed pentagon ABCDE of circle O, ∠CAD = 42°, then the sum of ∠B and ∠E is () | B | ['220°', '222°', '225°', '228°'] | multi_choice |
geo513 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/513.png | As shown in the figure, points A, B, C, and D are four consecutive points on circle O. OA is perpendicular to OB, and AE is perpendicular to OD at E. When ∠C = 70°, what is the measure of ∠A? | C | ['30°', '35°', '40°', '45°'] | multi_choice |
geo514 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/514.png | As shown in the figure, points A, B, and C are on circle O. If ∠A = 36° and ∠B = 64°, then what is the degree of ∠C? | A | ['28°', '32°', '44°', '52°'] | multi_choice |
geo515 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/515.png | As shown in the figure, A, B, and C are three points on the circle O, and ∠B = 75°. What is the degree measure of ∠AOC? | D | ['120°', '130°', '140°', '150°'] | multi_choice |
geo516 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/516.png | As shown in the figure, AB is the diameter of circle O, and ∠BAC=25°. What is ∠ADC? | D | ['25', '30°', '45°', '65°'] | multi_choice |
geo517 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/517.png | As shown in the figure, in circle O, AB is the diameter, and CD is a chord. If ∠ABD = 55°, then ∠C equals () | B | ['25°', '35°', '45°', '55°'] | multi_choice |
geo518 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/518.png | As shown in the figure, AB is the diameter of circle O, and point C is on circle O. If ∠B = 50°, then the measure of ∠A is () | D | ['80°', '60°', '50°', '40°'] | multi_choice |
geo519 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/519.png | AB is the diameter of circle O, and points C and D are on circle O. If ∠ABD = 42°, what is the measure of ∠BCD? | B | ['122°', '132°', '128°', '138°'] | multi_choice |
geo520 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/520.png | As shown in the figure, place the vertex of the 60° angle of a right triangle at the center of the circle O. The hypotenuse and one of the right-angle sides intersect the circle O at points A and B, respectively. P is any point on the major arc AB (not coinciding with A or B). What is the measure of ∠APB? | A | ['30°', '45°', '50°', '60°'] | multi_choice |
geo521 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/521.png | As shown in the figure, it is known that AB and CD are chords of circle O, and AB = CD. If ∠AOB = 70°, what is the measure of ∠CED? | D | ['70°', '60°', '45°', '35°'] | multi_choice |
geo522 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/522.png | As shown in the figure, AB is a chord of circle O, and C is a point on circle O. Given that ∠ABO = 40°, what is the measure of ∠ACB? | D | ['40°', '30°', '45°', '50°'] | multi_choice |
geo523 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/523.png | In a cylindrical oil tank with a diameter of 20 cm, some oil is added. The cross-section is as shown in the figure. If the width of the oil surface AB is 16 cm, what is the maximum depth of the oil? | A | ['4cm', '6cm', '8cm', '10cm'] | multi_choice |
geo524 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/524.png | As shown in the figure, ABCD is a cyclic quadrilateral inscribed in circle O. If ∠D = 65°, then ∠B = () | D | ['85°', '95°', '105°', '115°'] | multi_choice |
geo525 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/525.png | As shown in the figure, quadrilateral ABCD is inscribed in circle O. E is a point on the extension of BC. If ∠BAD = 100°, then the measure of ∠DCE is () | C | ['115°', '105°', '100°', '95°'] | multi_choice |
geo526 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/526.png | As shown in the figure, the diameter AB of circle O is 2. Points C and D are on circle O, and ∠ADC = 30°. What is the length of BC? | B | ['√{2}', '√{3}', '2', '1'] | multi_choice |
geo527 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/527.png | Given the diagram, ∠A=32°, ∠B=45°, ∠C=38°, what is ∠DFE? | B | ['120°', '115°', '110°', '105°'] | multi_choice |
geo528 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/528.png | As shown in the figure, quadrilateral ABCD is inscribed in circle O. F is a point on the circle such that DF = BC. Connect CF and extend it to intersect the extension of AD at point E. Connect AC. If ∠ABC = 110°, ∠BAC = 20°, then the measure of ∠E is () | C | ['60°', '55°', '50°', '45°'] | multi_choice |
geo529 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/529.png | As shown in the figure, quadrilateral ABCD is inscribed in circle O. F is a point on CD, and DF = BC. Connect CF and extend it to intersect the extension line of AD at point E. Connect AC. If ∠ABC = 110° and ∠BAC = 25°, then the degree of ∠E is () | A | ['45°', '50°', '55°', '60°'] | multi_choice |
geo530 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/530.png | As shown in the figure, quadrilateral ABCD is inscribed in circle O. If ∠A = 110°, what is the measure of ∠BOD? | D | ['70°', '110°', '120°', '140°'] | multi_choice |
geo531 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/531.png | As shown in the figure, quadrilateral ABCD is an inscribed quadrilateral in circle O. If ∠AOC = 120°, what is the measure of ∠ABC? | B | ['100°', '120°', '140°', '110°'] | multi_choice |
geo532 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/532.png | As shown in the figure, the extensions of the two pairs of opposite sides of the inscribed quadrilateral ABCD in circle O intersect at points E and F, respectively. If ∠E = ∠F = 35°, what is the degree measure of ∠A? | B | ['35°', '55°', '60°', '65°'] | multi_choice |
geo533 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/533.png | As shown in the figure, △ABC is inscribed in circle O, and ∠OBC = 42°. What is the measure of ∠A? | D | ['84°', '96°', '116°', '132°'] | multi_choice |
geo534 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/534.png | As shown in the figure, in the cyclic quadrilateral ABCD, ∠C = 110°, what is the measure of ∠BOD? | A | ['140°', '70°', '80°', '60°'] | multi_choice |
geo535 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/535.png | As shown in the figure, quadrilateral ABCD is inscribed in circle O. If one of its exterior angles ∠DCE is 72°, then ∠BOD is equal to () | A | ['144°', '70°', '110°', '140°'] | multi_choice |
geo536 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/536.png | As shown in the figure, quadrilateral ABCD is inscribed in circle O. If the exterior angle ∠DCE of quadrilateral ABCD is 70°, then the measure of ∠BAD is () | D | ['140°', '110°', '220°', '70°'] | multi_choice |
geo537 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/537.png | As shown in the figure, AB is the diameter of circle O, and points C and D are on circle O. If ∠AOD = 30°, what is the degree measure of ∠BCD? | C | ['75°', '95°', '105°', '115°'] | multi_choice |
geo538 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/538.png | As shown in the figure, ∠E + ∠D = 45°, then the measure of ∠EFC is () | D | ['30°', '60°', '90°', '45°'] | multi_choice |
geo539 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/539.png | As shown in the figure, quadrilateral ABCD is a cyclic quadrilateral, ∠BAD=108°, E is a point on the extension of BC, if CF bisects ∠DCE, then the measure of ∠DCF is () | B | ['52°', '54°', '56°', '60°'] | multi_choice |
geo540 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/540.png | As shown in the figure, points A, B, and C are on circle O. If ∠AOB = 130°, then ∠ACB equals () | B | ['105°', '115°', '125°', '135°'] | multi_choice |
geo541 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/541.png | As shown in the figure, A, B, C, and D are four points on circle O. If ∠AOC = 80°, then what are ∠ADC and ∠ABC equal to? | A | ['140°', '110°', '50°', '40°'] | multi_choice |
geo542 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/542.png | As shown in the figure, the exterior angle ∠ABE of the cyclic quadrilateral ABCD is 85°. What is the measure of ∠ADC? | C | ['120°', '95°', '85°', '42.5°'] | multi_choice |
geo543 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/543.png | As shown in the figure, △ABC is an inscribed triangle in circle O, and ∠BOC = 120°. Then, ∠A equals () | B | ['50°', '60°', '55°', '65°'] | multi_choice |
geo544 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/544.png | As shown in the figure, △ABC is inscribed in circle O, C is the midpoint of arc AB, and D is a point on circle O. Given that ∠ACB = 100°, what is the measure of ∠ADC? | A | ['40°', '39°', '38°', '36°'] | multi_choice |
geo545 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/545.png | As shown in the figure, △ABC is inscribed in circle O. Connect OA and OB. Given that ∠C = 40°, what is the measure of ∠AOB? | A | ['80°', '50°', '45°', '40°'] | multi_choice |
geo546 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/546.png | As shown in the figure, in the plane Cartesian coordinate system, given points A(0,1), B(0,1+t), and C(0,1-t) (t > 0), point P moves on a circle centered at D(3,5) with a radius of 1, and always satisfies ∠BPC=90°. What is the minimum value of t? | B | ['3', '4', '5', '6'] | multi_choice |
geo547 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/547.png | As shown in the figure, ⊙O is the circumcircle of △ABC. If ∠A = 58°, then the measure of ∠BOC is () | B | ['124°', '116°', '62°', '58°'] | multi_choice |
geo548 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/548.png | As shown in the figure, △ABC is an inscribed triangle in circle O, AB=2, ∠C=30°, then the radius of circle O is () | B | ['1', '2', '3', '4'] | multi_choice |
geo549 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/549.png | As shown in the figure, △ABC is inscribed in circle O. If ∠OBA = 40°, then ∠ACB = () | B | ['40°', '50°', '60°', '80°'] | multi_choice |
geo550 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/550.png | As shown in the figure, in a grid composed of unit squares with a side length of 1, the vertices of △ABC are all on the grid intersections. What is the radius R of the circumcircle of △ABC? | A | ['√{5}', '√{6}', '√{7}', '2√{2}'] | multi_choice |
geo551 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/551.png | As shown in the figure, the radius of circle O is 1, and △ABC is an inscribed triangle of circle O. Connect OB and OC. If ∠BAC and ∠BOC are supplementary, then the length of chord BC is () | A | ['√{3}', '2√{3}', '3√{3}', '1.5√{3}'] | multi_choice |
geo552 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/552.png | As shown in the figure, △ABC is inscribed in circle O, ∠C=30°, AB=4. What is the radius of circle O? | B | ['2√{2}', '4', '2√{3}', '5'] | multi_choice |
geo553 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/553.png | As shown in the figure, ⊙O is the circumcircle of △ABC, the radius of ⊙O is 2, and ∠A = 30°. What is the length of BC? | C | ['√{2}', '2√{2}', '2', '\\frac{1}{3}π'] | multi_choice |
geo554 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/554.png | As shown in the figure, △ABC is inscribed in circle O. If ∠ACB = 50°, what is the measure of ∠AOB? | A | ['100°', '90°', '80°', '130°'] | multi_choice |
geo555 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/555.png | As shown in the figure, D is a point on the circumcircle of the equilateral triangle △ABC, and ∠DAC=20°. What is the measure of ∠ACD? | C | ['20°', '30°', '40°', '45°'] | multi_choice |
geo556 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/556.png | As shown in the figure, point P is the circumcenter of triangle ABC. Given that ∠A = 75°, what is the measure of ∠BPC? | A | ['150°', '165°', "142°30'", "127°30'"] | multi_choice |
geo557 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/557.png | As shown in the figure, AB is the diameter of circle O with a radius of 4. P is an arbitrary point on the circle other than A and B. The bisector of ∠APB intersects circle O at point C. Connect AC and BC. The line containing the midline of △ABC intersects circle O at points E and F. What is the length of EF? | A | ['4√{3}', '2√{3}', '6', '2√{5}'] | multi_choice |
geo558 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/558.png | As shown in the figure, AB is tangent to circle O at point A, BO intersects circle O at point C, and point D is on circle O. If ∠ADC = 32°, what is the measure of ∠ABO? | C | ['32⁰', '64⁰', '26⁰', '36⁰'] | multi_choice |
geo559 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/559.png | As shown in the figure, circle O is tangent to BC at point B, and chord AB is parallel to OC. If ∠C = 40°, what is the measure of ∠AOB? | C | ['60', '70°', '80°', '90°'] | multi_choice |
geo560 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/560.png | As shown in the figure, AB is the diameter of circle O, point C is on the extension line of AB, and CD is tangent to circle O at point D. If ∠CDA = 122°, then the degree of ∠C is () | B | ['22°', '26°', '28°', '30°'] | multi_choice |
geo561 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/561.png | As shown in the figure, ray BM is tangent to circle O at point B. If ∠MBA = 140°, what is the measure of ∠ACB? | A | ['40°', '50°', '60°', '70°'] | multi_choice |
geo562 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/562.png | As shown in the figure, it is known that AB is a tangent to circle O, point A is the point of tangency, and connecting OB intersects circle O at point C. ∠B = 38°, point D is a point on circle O, connecting CD and AD. Then ∠D equals () | D | ['76°', '38°', '30°', '26°'] | multi_choice |
geo563 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/563.png | As shown in the figure, AB is the diameter of circle O, CD is the tangent to circle O at point D, and CD intersects the extension of AB at point C. Given that ∠A = 30°, CD = 3, find the value of AB. | B | ['3', '2√{3}', '6', '√{3}'] | multi_choice |
geo564 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/564.png | As shown in the figure, in circle O, CD is a tangent line, and the point of tangency is D. The line CO intersects circle O at points B and A. If ∠A = 20°, then what is the measure of ∠C? | C | ['25°', '65°', '50°', '75°'] | multi_choice |
geo565 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/565.png | As shown in the figure, AB is the diameter of circle O, and the line PA is tangent to circle O at point A. PO intersects circle O at point C, and BC is connected. If ∠P = 50°, what is the measure of ∠ABC? | A | ['20°', '25°', '40°', '50°'] | multi_choice |
geo566 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/566.png | As shown in the figure, AB is the diameter of circle O, CA is tangent to circle O at point A, CO intersects circle O at point D, and BD is connected. If ∠C = 40°, then ∠B equals () | B | ['20°', '25°', '30°', '40°'] | multi_choice |
geo567 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/567.png | As shown in the figure, point P is a point outside circle O. PA is a tangent to circle O at point A. PO intersects circle O at point B. Given that ∠P = 30°, OB = 3, find the length of segment BP. | A | ['3', '3√{3}', '6', '9'] | multi_choice |
geo568 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/568.png | As shown in the figure, AB is tangent to circle O at point A, BO intersects circle O at point C, and point D is a point on arc CEA different from points C and A. If ∠ABO = 32°, what is the measure of ∠ADC? | D | ['32°', '16°', '28°', '29°'] | multi_choice |
geo569 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/569.png | As shown in the figure, PA and PB are tangent to circle O at points A and B, respectively. AC is the diameter of circle O, and ∠P = 40°. What is the measure of ∠ACB? | C | ['50°', '60°', '70°', '80°'] | multi_choice |
geo570 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/570.png | As shown in the figure, circle C is tangent to both sides of angle AOB, with side OA tangent to circle C at point P. If ∠AOB = 90° and OP = 6, what is the length of OC? | C | ['12', '12√{2}', '6√{2}', '6√{3}'] | multi_choice |
geo571 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/571.png | As shown in the figure, PA and PB are two tangents to circle O, with A and B being the points of tangency. AC is the diameter of circle O, and ∠P = 70°. What is ∠PBC? | D | ['110°', '120°', '135°', '145°'] | multi_choice |
geo572 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/572.png | As shown in the figure, it is known that AB is a tangent to circle O, and B is the point of tangency. AO intersects circle O at point C. If ∠BAO = 40°, then what is the measure of ∠OCB? | D | ['40°', '50°', '60°', '65°'] | multi_choice |
geo573 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/573.png | As shown in the figure, PA and PB are tangents to circle O, with A and B being the points of tangency. If ∠P = 50°, what is the measure of ∠PAB? | C | ['50°', '60°', '65°', '70°'] | multi_choice |
geo574 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/574.png | As shown in the figure, in △ABC, ∠ACB=90°, AC=6, AB=10. A circle is drawn with B as the center and tangent to AC. What is the radius of the circle? | D | ['5', '4', '10', '8'] | multi_choice |
geo575 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/575.png | As shown in the figure, △ABC is inscribed in circle O, ∠B=60°, CD is the diameter of circle O, the tangent line passing through point A intersects the extension line of CD at point P. If the radius of circle O is 1, then the length of PA is equal to () | B | ['√{2}', '√{3}', '√{5}', '2'] | multi_choice |
geo576 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/576.png | As shown in the figure, AB and AC are two chords of circle O, ∠A = 31°. The tangent line passing through point C intersects the extension of OB at point D. What is the degree of ∠D? | A | ['28°', '29°', '30°', '31°'] | multi_choice |
geo577 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/577.png | As shown in the figure, ∠AOB = 45°, point C is on OB, OC = 4. If a circle with center at point C and radius r is tangent to OA, then r equals () | C | ['2', '8', '2√{2}', '2√{3}'] | multi_choice |
geo578 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/578.png | As shown in the figure, points A, B, and C are on circle O, ∠ABC=29°. A tangent to circle O is drawn through point C and intersects the extension of OA at point D. What is the measure of ∠D? | B | ['29°', '32°', '42°', '58°'] | multi_choice |
geo579 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/579.png | As shown in the figure, PA is tangent to circle O at point A. PO intersects circle O at point C. Point B is on the major arc CBA. If ∠ABC = 32°, then ∠P = () degrees. | B | ['16', '26', '36', '46'] | multi_choice |
geo580 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/580.png | As shown in the figure, AB is a chord of circle O. The extension of AO intersects the tangent to circle O at point B at point C. If ∠C = 40°, what is the measure of ∠ABO? | C | ['50°', '40°', '25°', '20度'] | multi_choice |
geo581 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/581.png | As shown in the figure, AB and AC are two chords of circle O, ∠A = 30°. The tangent line passing through point C intersects the extension of OB at point D. What is the degree of ∠D? | A | ['30°', '35°', '40°', '45°'] | multi_choice |
geo582 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/582.png | As shown in the figure, the radius of circle O is 2, the distance from point O to line l is 3, and point P is a moving point on line l. PQ is tangent to circle O at point Q. What is the minimum value of PQ? | B | ['√{13}', '√{5}', '3', '5'] | multi_choice |
geo583 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/583.png | As shown in the figure, PA and PB are tangents to circle O at points A and B, respectively. CD is a tangent to circle O, intersecting PA and PB at points C and D. The perimeter of triangle PCD is 36. What is the length of AP? | B | ['12', '18', '24', '9'] | multi_choice |
geo584 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/584.png | As shown in the figure, PA and PB are tangent to circle O at points A and B respectively, and PA = 8. CD is tangent to circle O at point E and intersects PA and PB at points C and D. What is the perimeter of triangle PCD? | B | ['8', '16', '24', '32'] | multi_choice |
geo585 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/585.png | As shown in the figure, AB is a chord of circle O, and BC is tangent to circle O at point B. Connect OA. If ∠ABC = 70°, then ∠A equals () | C | ['10°', '15°', '20°', '30°'] | multi_choice |
geo586 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/586.png | As shown in the figure, AB is a chord of circle O, AC is a tangent to circle O at point A, and BC passes through the center of the circle. If ∠C = 40°, what is the measure of ∠B? | B | ['20°', '25°', '40°', '50°'] | multi_choice |
geo587 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/587.png | As shown in the figure, PA and PB are tangents to circle O at points A and B respectively, ∠APB=60°, PA=8. What is the radius OA of circle O? | D | ['4', '8', '\\frac{4}{3}√{3}', '\\frac{8}{3}√{3}'] | multi_choice |
geo588 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/588.png | As shown in the figure, AB is the diameter of circle O, and C is a point outside circle O. A tangent to circle O is drawn through point C, with the point of tangency being B. Line segment AC intersects circle O at point D, and ∠C = 38°. Point E moves on the semicircle to the right of AB (not coinciding with A or B). What is the measure of ∠AED? | C | ['62°', '52°', '38°', '28°'] | multi_choice |
geo589 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/589.png | As shown in the figure, AB is the diameter of circle O, point C is a point on circle O, the tangent line passing through point C intersects the extension line of AB at point D, connect AC and CO. If ∠A = 35°, then what is the degree of ∠ADC? | A | ['20°', '30°', '35°', '55°'] | multi_choice |
geo590 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/590.png | As shown in the figure, the side length of the rhombus ABCD is 10. Circle O is tangent to AB and AD at points E and F, respectively, and is tangent to BG at point G. If AO = 5 and the radius of circle O is 3, what is the length of BG? | C | ['4', '5', '6', '7'] | multi_choice |
geo591 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/591.png | As shown in the figure, the three sides of △ABC are tangent to circle O at points D, E, and F respectively. If ∠A = 50°, then what is ∠DEF? | A | ['65°', '50°', '130°', '80°'] | multi_choice |
geo592 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/592.png | As shown in the figure, the endpoints of a line segment AB of length 10 slide on the positive half-axes of the x-axis and y-axis respectively (the length of the line segment AB remains unchanged). Circle O is tangent to the line segment AB. What is the maximum area of Circle O? | B | ['100π', '25π', '22π', '20π'] | multi_choice |
geo593 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/593.png | As shown in the figure, AB is the diameter of circle O, P is a point on the extension line of AB, and PT is tangent to circle O at T. If PT = 6 and PB = 2, what is the diameter of circle O? | C | ['8', '10', '16', '18'] | multi_choice |
geo594 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/594.png | In circle O, the radius of the circle is 6, ∠B=30°, AC is the tangent to circle O, then the minimum value of CD is () | B | ['1', '3', '√{3}', '2'] | multi_choice |
geo595 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/595.png | As shown in the figure, two concentric circles have center O. The chord AB of the larger circle is a tangent to the smaller circle. If AB = 10, what is the area of the shaded region in the figure? | D | ['100π', '75π', '50π', '25π'] | multi_choice |
geo596 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/596.png | As shown in the figure, PA and PB are tangent to circle O at points A and B respectively. If AP = 5, then BP = () | D | ['4', '10', '3', '5'] | multi_choice |
geo597 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/597.png | As shown in the figure, the side AC of triangle △ABC intersects the circle ⊙O at points C and D, and passes through the center O. The side AB is tangent to the circle ⊙O at point B. Given that ∠A = 30°, what is the measure of ∠C? | A | ['30°', '45°', '60°', '40°'] | multi_choice |
geo598 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/598.png | As shown in the figure, given that the radius of circle O is 5 cm, a straight line is tangent to circle O at point C. If this straight line is translated 2 cm along the direction of CO and intersects circle O at points A and B, what is the length of chord AB? | D | ['2', '3', '5', '8'] | multi_choice |
geo599 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/599.png | As shown in the figure, AB is the diameter of circle O, PA is tangent to circle O at point A, PO intersects circle O at point C, and BC is connected. If ∠P = 36°, then ∠B equals () | A | ['27°', '30°', '36°', '54°'] | multi_choice |
geo600 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/600.png | As shown in the figure, with O as the center of two concentric circles, the chord AB of the larger circle is tangent to the smaller circle at point C. Given that AB = 12 and AO = 8, find the length of OC. | D | ['5', '4', '2√{5}', '2√{7}'] | multi_choice |
geo601 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/601.png | As shown in the figure, AB is the diameter of circle O, EF is tangent to circle O at point D, and BH is perpendicular to EF at point H, intersecting circle O at point C. Connect BD. If ∠ABH = 50°, what is the measure of ∠ABD? | D | ['50°', '40°', '30°', '25°'] | multi_choice |
geo602 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/602.png | As shown in the figure, AB is tangent to circle O at point B. The extension of AO intersects circle O at point C. Connect BC. If ∠ABC = 120° and OC = 2, then the length of chord BC is () | C | ['2', '√{3}', '2√{3}', '4'] | multi_choice |
geo603 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/603.png | As shown in the figure, P is a point outside circle O. PA and PB are tangent to circle O at points A and B, respectively. CD is tangent to circle O at point E and intersects PA and PB at points C and D, respectively. If PA = 5, what is the perimeter of triangle PCD? | C | ['5', '8', '10', '12'] | multi_choice |
geo604 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/604.png | As shown in the figure, two concentric circles have the same center O. The chord AB of the larger circle is a tangent to the smaller circle. If the radius of the larger circle is 10 cm and the radius of the smaller circle is 6 cm, what is the length of the chord AB? | D | ['8cm', '10cm', '12cm', '16cm'] | multi_choice |
geo605 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/605.png | As shown in the figure, two concentric circles have the same center O. The chord AB of the larger circle is tangent to the smaller circle at point C. If the radius of the larger circle is 10 cm and the radius of the smaller circle is 6 cm, what is the length of the chord AB? | D | ['2cm', '4cm', '8cm', '16cm'] | multi_choice |
geo606 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/606.png | AB is the diameter of circle O, and chord CD is tangent to circle O, with AB ∥ CD. If chord CD = 16 cm, what is the area of the shaded region? | B | ['144πcm²', '64πcm²', '79πcm²', '81πcm²'] | multi_choice |
geo607 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/607.png | As shown in the figure, AB is the diameter of the semicircle, with AB = 4. The semicircle rotates 45° clockwise around point B, and point A rotates to position A'. What is the area of the shaded region in the figure? | B | ['π', '2π', '\\frac{π}{2}', '4π'] | multi_choice |
geo608 | /home/yz979/scratch/chengye/math/data/original_set/geoset/images/images/608.png | As shown in the figure, in △ABC, DE∥BC, AD=5, BD=10, DE=6, what is the value of BC? | C | ['6', '12', '18', '24'] | multi_choice |
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