image imagewidth (px) 139 1.6k | problem stringlengths 13 234 | answer stringlengths 1 34 | text_logic_form listlengths 1 11 | diagram_logic_form listlengths 0 456 | line_instances listlengths 1 110 | point_positions stringlengths 2 551 | circle_instances listlengths 0 4 |
|---|---|---|---|---|---|---|---|
<image>Find $y$ and so that each quadrilateral is a parallelogram. | \frac { 4 } { 3 } | [
"Parallelogram($)",
"Find(y)"
] | [
"Equals(LengthOf(Line(D, E)), 3y+4)",
"Equals(LengthOf(Line(C, E)), 4y)",
"Equals(LengthOf(Line(B, E)), \\frac{2}{3}x)",
"Equals(LengthOf(Line(A, E)), x)",
"PointLiesOnLine(E, Line(A, D))",
"PointLiesOnLine(E, Line(B, C))"
] | [
"AB",
"AC",
"AD",
"AE",
"BC",
"BD",
"BE",
"CD",
"CE",
"DE"
] | {"A": [0.0, 121.0], "B": [298.0, 120.0], "C": [74.0, 2.0], "D": [370.0, 1.0], "E": [187.0, 60.0]} | [
""
] | |
<image>In $\triangle A B C, \overline{B D}$ is a median. If $A D=3 x+5$ and $C D=5 x-1$, find AC . | 28 | [
"Triangle(A,B,C)",
"IsMedianOf(Line(B,D),Triangle(A,B,C))",
"Equals(LengthOf(Line(A,D)),3x+5)",
"Equals(LengthOf(Line(C,D)),5x-1)",
"Find(LengthOf(Line(A,C)))"
] | [
"PointLiesOnLine(D, Line(A, C))"
] | [
"BD",
"BA",
"BC",
"DA",
"DC",
"AC"
] | {"B": [55.49622296050203, 1.5103868586193983], "D": [79.9203821656051, 156.5], "A": [0.12121212121212466, 156.33333333333331], "C": [157.11677282377917, 156.33333333333331]} | [] | |
<image>Find y | 32.6 | [
"Find(y)"
] | [
"Equals(LengthOf(Line(C, A)), x)",
"Equals(LengthOf(Line(C, B)), 24)",
"Equals(MeasureOf(Angle(C, A, B)), 32)",
"PointLiesOnLine(D, Line(A, B))",
"Perpendicular(Line(C, A), Line(C, B))",
"Perpendicular(Line(C, D), Line(A, D))"
] | [
"CA",
"CB",
"CD",
"AB",
"AD",
"BD"
] | {"C": [224.50829926313358, 0.8024177098426151], "A": [0.5419664268585223, 139.33333333333331], "B": [311.12619047619046, 139.33333333333331], "D": [226.9892857142857, 139.5]} | [] | |
<image>Find the area of the figure. Round to the nearest tenth if necessary. | 261 | [
"Find(AreaOf(Shape($)))"
] | [
"Equals(LengthOf(Line(C, H)), 6)",
"Equals(LengthOf(Line(E, S)), 28)",
"Equals(LengthOf(Line(A, S)), 17)",
"Equals(LengthOf(Line(A, G)), 15)",
"PointLiesOnLine(H, Line(A, S))",
"Perpendicular(Line(C, H), Line(A, H))",
"PointLiesOnLine(G, Line(S, E))",
"Perpendicular(Line(A, G), Line(E, G))"
] | [
"AC",
"EA",
"ES",
"SC",
"AS",
"CG",
"AG",
"SG",
"AH",
"SH",
"EH"
] | {"A": [126.0, 2.0], "C": [32.0, 2.0], "E": [161.0, 160.0], "S": [1.0, 95.0]} | [
""
] | |
<image>Find $m \angle B$. | 30 | [
"Find(MeasureOf(Angle(B)))"
] | [
"Equals(MeasureOf(Angle(A, D, C)), 60)",
"PointLiesOnCircle(B, Circle(D, radius_3_0))",
"PointLiesOnCircle(A, Circle(D, radius_3_0))",
"PointLiesOnCircle(C, Circle(D, radius_3_0))"
] | [
"BA",
"BC"
] | {"A": [277.0, 52.0], "B": [4.0, 151.0], "C": [312.0, 204.0], "D": [162.0, 160.0]} | [
"D"
] | |
<image>Find $m \angle 2$. | 52 | [
"Find(MeasureOf(Angle(2)))"
] | [
"Equals(MeasureOf(Angle(F, E, A)), MeasureOf(angle 3))",
"Equals(MeasureOf(Angle(E, A, B)), MeasureOf(angle 2))",
"Equals(MeasureOf(Angle(A, C, D)), 50)",
"Equals(MeasureOf(Angle(G, A, D)), MeasureOf(angle 1))",
"Equals(MeasureOf(Angle(F, G, A)), 120)",
"Equals(MeasureOf(Angle(H, G, F)), MeasureOf(angle 4... | [
"AB",
"AC",
"AD",
"AE",
"AG",
"AH",
"BC",
"BF",
"BG",
"BH",
"CD",
"CG",
"CH",
"DE",
"EF",
"EG",
"GF",
"GH"
] | {"A": [241.97619116041153, 152.3584832461636], "B": [603.7846342718385, 152.44708834706938], "C": [2.668304281417761, 152.0019777347379], "D": [134.0, 309.0], "E": [347.0, 0.0], "F": [525.0866788714857, 315.4867257182659], "G": [433.2547647177885, 152.64185891069303], "H": [675.0, 153.0]} | [] | |
<image>The triangle is inscribed into the circle. Find the exact circumference of the circle. | 17 \pi | [
"InscribedIn(Triangle($),Circle($))",
"Find(CircumferenceOf(Circle($)))"
] | [
"Equals(LengthOf(Line(B, D)), 15)",
"Equals(LengthOf(Line(C, D)), 8)",
"PointLiesOnLine(A, Line(B, C))",
"PointLiesOnCircle(B, Circle(A, radius_0_0))",
"PointLiesOnCircle(C, Circle(A, radius_0_0))",
"PointLiesOnCircle(D, Circle(A, radius_0_0))",
"Perpendicular(Line(D, B), Line(D, C))"
] | [
"AB",
"AC",
"BC",
"BD",
"CD"
] | {"A": [151.0, 152.0], "B": [219.0, 20.0], "C": [83.0, 284.0], "D": [221.0, 284.0]} | [
"D"
] | |
<image>Find the area of the triangle. Round to the nearest tenth if necessary. | 137.5 | [
"Find(AreaOf(Triangle($)))"
] | [
"Equals(LengthOf(Line(B, O)), 35)",
"Equals(LengthOf(Line(O, A)), 25)",
"Equals(LengthOf(Line(C, A)), 23)",
"Equals(LengthOf(Line(B, C)), 11)",
"PointLiesOnLine(C, Line(A, B))",
"Perpendicular(Line(O, A), Line(A, C))"
] | [
"AB",
"AC",
"CB",
"OA",
"OB",
"OC"
] | {"A": [1.0, 172.0], "B": [236.0, 172.0], "C": [161.0, 174.0], "O": [1.0, 2.0]} | [
""
] | |
<image>Use parallelogram MNPR to find $m \angle MQN$ | 97 | [
"Parallelogram(M,N,P,R)",
"Find(MeasureOf(Angle(M,Q,N)))"
] | [
"Equals(LengthOf(Line(P, Q)), 11.1)",
"Equals(MeasureOf(Angle(N, Q, P)), 83)",
"Equals(LengthOf(Line(R, P)), 20)",
"Equals(LengthOf(Line(M, N)), 3x-4)",
"Equals(LengthOf(Line(P, N)), 2y+5)",
"Equals(LengthOf(Line(N, Q)), 15.4)",
"Equals(MeasureOf(Angle(Q, N, M)), 33)",
"Equals(LengthOf(Line(M, N)), 3x... | [
"MN",
"MQ",
"NQ",
"PM",
"PN",
"PQ",
"RM",
"RN",
"RP",
"RQ"
] | {"M": [65.0, 0.0], "N": [260.0, 1.0], "P": [197.0, 166.0], "Q": [131.0, 83.0], "R": [0.0, 166.0]} | [
""
] | |
<image>Find x. | 20 \sqrt { 2 } | [
"Find(x)"
] | [
"Equals(LengthOf(Line(B, C)), 20)",
"Equals(LengthOf(Line(A, B)), x)",
"Equals(MeasureOf(Angle(C, A, B)), 45)",
"Equals(LengthOf(Line(A, C)), 20)",
"Perpendicular(Line(B, C), Line(A, C))"
] | [
"AB",
"AC",
"BC"
] | {"A": [166.98575647038388, 244.3505297898211], "B": [80.0, 0.0], "C": [0.735928458705942, 166.5202524986849]} | [] | |
<image>If $a=4,$ find $b$ | 4 \sqrt { 3 } | [
"Equals(a,4)",
"Find(b)"
] | [
"Equals(MeasureOf(Angle(A, B, C)), 60)",
"Equals(LengthOf(Line(B, A)), LengthOf(c))",
"Equals(LengthOf(Line(B, C)), LengthOf(a))",
"Equals(MeasureOf(Angle(C, A, B)), 30)",
"Equals(LengthOf(Line(A, C)), LengthOf(b))",
"Perpendicular(Line(B, C), Line(A, C))"
] | [
"BA",
"BC",
"AC"
] | {"B": [1.9851485148514882, 2.5643564356435604], "A": [199.0, 116.0], "C": [0.035069169089780416, 113.33403824125475]} | [] | |
<image>In the figure, $m ∠1 = 53$. Find the measure of $∠6$. | 53 | [
"Equals(MeasureOf(Angle(1)),53)",
"Find(MeasureOf(Angle(6)))"
] | [
"Equals(MeasureOf(Angle(L, W, N)), MeasureOf(angle 1))",
"Equals(MeasureOf(Angle(G, X, E)), MeasureOf(angle 2))",
"Equals(MeasureOf(Angle(I, Y, L)), MeasureOf(angle 7))",
"Equals(MeasureOf(Angle(L, Y, Z)), MeasureOf(angle 6))",
"Equals(MeasureOf(Angle(K, Y, M)), MeasureOf(angle 5))",
"Equals(MeasureOf(Ang... | [
"GN",
"GW",
"GX",
"HE",
"HZ",
"HZ",
"IK",
"IY",
"IZ",
"ML",
"MW",
"MY",
"WL",
"WN",
"WX",
"XE",
"XN",
"YK",
"YL",
"YW",
"YZ",
"ZE",
"ZK",
"ZX"
] | {"E": [331.0, 74.0], "G": [57.0, 80.0], "H": [165.0, 365.0], "I": [1.0, 198.0], "K": [295.0, 360.0], "L": [184.0, 6.0], "M": [23.0, 289.0], "N": [358.0, 248.0], "W": [122.0, 116.0], "X": [261.0, 194.0], "Y": [58.0, 228.0], "Z": [199.0, 305.0]} | [
""
] | |
<image>Find the area of the quadrilateral. | 240 | [
"Find(AreaOf(Quadrilateral($)))"
] | [
"Equals(LengthOf(Line(B, C)), 20)",
"Equals(LengthOf(Line(D, E)), 24)",
"PointLiesOnLine(E, Line(A, C))",
"PointLiesOnLine(E, Line(B, D))",
"Perpendicular(Line(B, E), Line(C, E))",
"Equals(LengthOf(Line(A, D)), LengthOf(Line(A, B)))",
"Equals(LengthOf(Line(A, D)), LengthOf(Line(C, D)))",
"Equals(Lengt... | [
"AB",
"AC",
"AD",
"AE",
"BC",
"BD",
"BE",
"CE",
"DC",
"DE"
] | {"A": [1.0, 89.0], "B": [62.0, 1.0], "C": [123.0, 88.0], "D": [62.0, 187.0], "E": [62.0, 89.0]} | [
""
] | |
<image>Find $JK$. | 14 | [
"Find(LengthOf(Line(K,L)))"
] | [
"Equals(LengthOf(Line(K, L)), 11x-8)",
"Equals(LengthOf(Line(L, J)), x+12)",
"Equals(LengthOf(Line(K, J)), 7x)",
"Equals(LengthOf(Line(K, L)), LengthOf(Line(K, J)))",
"Equals(LengthOf(Line(K, J)), LengthOf(Line(L, J)))"
] | [
"JK",
"LJ",
"LK"
] | {"J": [2.0, 182.0], "K": [124.0, 2.0], "L": [274.0, 184.0]} | [
""
] | |
<image>Find $AB$. | 29 | [
"Find(LengthOf(Line(A,B)))"
] | [
"Equals(LengthOf(Line(D, A)), 3x+5)",
"Equals(LengthOf(Line(B, A)), 5x-11)",
"Equals(LengthOf(Line(D, F)), 15)",
"Equals(LengthOf(Line(B, F)), 15)",
"PointLiesOnLine(F, Line(B, D))",
"PointLiesOnLine(F, Line(A, N))",
"PointLiesOnLine(A, Line(F, G))",
"PointLiesOnLine(F, Line(E, N))",
"PointLiesOnLin... | [
"BD",
"BA",
"BF",
"DA",
"DF",
"AE",
"AN",
"AF",
"AG",
"AH",
"FE",
"FN",
"FG",
"FH",
"EN",
"EG",
"EH",
"NG",
"NH",
"GH"
] | {"B": [68.67450490596683, 335.4192404360631], "E": [22.0, 21.5], "D": [289.8988179885927, 175.98178815541996], "N": [205.0, 300.0], "A": [44.06492905218602, 62.9841617716574], "F": [176.25347886588972, 257.49184640807096], "G": [3.4266666666666623, 4.933333333333351], "H": [11.0, 32.0]} | [] | |
<image>Find x. Assume that segments that appear to be tangent are tangent. | 145 | [
"Tangent(Line($),Circle($))",
"Find(x)"
] | [
"Equals(MeasureOf(Angle(A, B, C)), 35)",
"Equals(MeasureOf(Arc(A, C)), x)",
"PointLiesOnCircle(A, Circle(D, radius_3_0))",
"PointLiesOnCircle(C, Circle(D, radius_3_0))"
] | [
"AB",
"BC"
] | {"A": [70.0, 113.0], "B": [170.0, 1.0], "C": [43.0, 31.0], "D": [53.0, 74.0]} | [
"D"
] | |
<image>Find $x$. | 2 | [
"Find(x)"
] | [
"Equals(LengthOf(Line(R, S)), 3x+2)",
"Equals(LengthOf(Line(T, R)), 2x+4)",
"Equals(LengthOf(Line(S, R)), LengthOf(Line(R, T)))"
] | [
"RS",
"TR",
"TS"
] | {"R": [3.0, 0.0], "S": [387.0, 0.0], "T": [330.0, 192.0]} | [
""
] | |
<image>Find $z$. | 20 | [
"Find(z)"
] | [
"Equals(LengthOf(Line(B, E)), z)",
"Equals(LengthOf(Line(B, C)), 32)",
"Equals(LengthOf(Line(B, D)), 15)",
"Equals(LengthOf(Line(C, D)), 9)",
"PointLiesOnLine(E, Line(B, C))"
] | [
"BC",
"BD",
"BE",
"CD",
"CE",
"DE"
] | {"B": [174.0, 36.0], "C": [288.0, 323.0], "D": [2.0, 317.0], "E": [233.0, 179.0]} | [
""
] | |
<image>$\overline{PM}$ is a diameter of $\odot R$. Find $m \widehat {MQ}$. | 65 | [
"IsDiameterOf(Line(P,M),Circle(R))",
"Find(MeasureOf(Arc(M,Q)))"
] | [
"Equals(MeasureOf(Angle(P, R, Q)), 115)",
"PointLiesOnLine(R, Line(M, P))",
"PointLiesOnCircle(Q, Circle(R, radius_1_0))",
"PointLiesOnCircle(M, Circle(R, radius_1_0))",
"PointLiesOnCircle(P, Circle(R, radius_1_0))",
"PointLiesOnCircle(N, Circle(R, radius_1_0))",
"Perpendicular(Line(R, P), Line(R, N))"
... | [
"MP",
"QR",
"RM",
"RN",
"RP"
] | {"M": [134.0, 345.0], "N": [4.0, 139.0], "P": [215.0, 6.0], "Q": [312.0, 283.0], "R": [173.0, 172.0]} | [
"R"
] | |
<image>Find y. | 30 | [
"Find(y)"
] | [
"Equals(LengthOf(Line(A, B)), 16)",
"Equals(LengthOf(Line(A, D)), 8)",
"Equals(MeasureOf(Angle(A, B, D)), y)",
"Equals(LengthOf(Line(B, D)), x)",
"Perpendicular(Line(A, D), Line(B, D))"
] | [
"AB",
"AC",
"AD",
"BC",
"BD"
] | {"A": [180.3636484252907, 2.974985424859071], "B": [1.9747926606719943, 90.35133195612872], "C": [0.0, 0.0], "D": [181.3296296296296, 91.32962962962964]} | [] | |
<image>Find the area of the parallelogram. Round to the nearest tenth if necessary. | 1448.0 | [
"Find(AreaOf(Parallelogram($)))"
] | [
"Equals(LengthOf(Line(C, B)), 44)",
"Equals(MeasureOf(Angle(E, C, A)), 30)",
"Equals(LengthOf(Line(A, E)), 19)",
"PointLiesOnLine(E, Line(A, D))",
"Perpendicular(Line(C, E), Line(E, D))"
] | [
"AC",
"AD",
"AE",
"CB",
"CE",
"DB",
"ED"
] | {"A": [235.0, 120.0], "B": [2.0, 2.0], "C": [165.0, 1.0], "D": [67.0, 118.0]} | [
""
] | |
<image>Assume that the segment is tangent, find the value of $x$. | 3 | [
"Tangent(Line($),Circle($))",
"Find(x)"
] | [
"Equals(LengthOf(Line(S, Q)), 4)",
"Equals(LengthOf(Line(S, B)), 2)",
"Equals(LengthOf(Line(B, R)), x)",
"PointLiesOnLine(B, Line(S, R))",
"PointLiesOnCircle(B, Circle(R, radius_2_0))",
"PointLiesOnCircle(Q, Circle(R, radius_2_0))",
"PointLiesOnCircle(C, Circle(R, radius_2_0))"
] | [
"BR",
"QR",
"SB",
"SQ",
"SR"
] | {"B": [122.0, 228.0], "Q": [128.0, 75.0], "R": [257.0, 157.0], "S": [0.0, 294.0]} | [
"R"
] | |
<image>Express the ratio of $ \cos C $ as a decimal to the nearest hundredth. | 0.80 | [
"Find(RatioOf(SinOf(Angle(A))))"
] | [
"Equals(LengthOf(Line(A, C)), 25)",
"Equals(LengthOf(Line(A, B)), 15)",
"Equals(LengthOf(Line(C, B)), 20)",
"Perpendicular(Line(A, B), Line(C, B))"
] | [
"AB",
"AC",
"CB"
] | {"A": [238.0, 1.0], "B": [236.0, 131.0], "C": [3.0, 132.0]} | [
""
] | |
<image>Find the area of the triangle. Round to the nearest hundredth. | 6.79 | [
"Find(AreaOf(Triangle($)))"
] | [
"Equals(LengthOf(Line(B, C)), 4)",
"Equals(MeasureOf(Angle(C, B, A)), 29)",
"Equals(LengthOf(Line(A, B)), 7)",
"PointLiesOnLine(D, Line(C,D))",
"Perpendicular(Line(C,D), Line(A,B))"
] | [
"AB",
"AC",
"BC",
"CD"
] | {"A": [323.0, 92.0], "B": [2.0, 92.0], "C": [166.0, 1.0], "D": [167.0, 71.0], "N": [188.0, 70.0]} | [
""
] | |
<image>Find the perimeter of the parallelogram. Round to the nearest tenth if necessary. | 38.4 | [
"Find(PerimeterOf(Parallelogram($)))"
] | [
"Equals(LengthOf(Line(A, E)), 10)",
"Equals(LengthOf(Line(N, B)), 7)",
"Equals(LengthOf(Line(N, C)), 12.2)",
"PointLiesOnLine(E, Line(B, N))",
"Perpendicular(Line(A, E), Line(E, N))"
] | [
"AB",
"AC",
"BN",
"CN"
] | {"A": [38.0, -0.0], "B": [0.0, 131.0], "C": [142.0, 1.0], "N": [107.0, 132.0]} | [
""
] | |
<image>Find x. | 6 | [
"Find(x)"
] | [
"Equals(LengthOf(Line(A, B)), y)",
"Equals(LengthOf(Line(B, C)), 6\\sqrt{3})",
"Equals(MeasureOf(Angle(B, A, C)), 60)",
"Equals(LengthOf(Line(A, C)), x)",
"Perpendicular(Line(B, C), Line(A, C))"
] | [
"AB",
"AC",
"BC"
] | {"A": [0.0, 161.0], "B": [94.0, 1.0], "C": [92.0, 161.0]} | [
""
] | |
<image>If PR || WX, WX = 10, XY = 6, WY = 8, RY = 5, and PS = 3, find PQ | 6 | [
"Parallel(Line(P,R),Line(W,X))",
"Equals(LengthOf(Line(W,X)),10)",
"Equals(LengthOf(Line(X,Y)),6)",
"Equals(LengthOf(Line(W,Y)),8)",
"Equals(LengthOf(Line(R,Y)),5)",
"Equals(LengthOf(Line(P,S)),3)",
"Find(LengthOf(Line(P,Q)))"
] | [
"PointLiesOnLine(S, Line(W, Y))",
"PointLiesOnLine(S, Line(Q, P))",
"PointLiesOnLine(Q, Line(W, X))",
"PointLiesOnLine(B, Line(Y, X))",
"PointLiesOnLine(B, Line(Q, R))",
"PointLiesOnLine(B, Line(Q, R))",
"PointLiesOnLine(Y, Line(P, R))",
"Perpendicular(Line(C, P), Line(Y, S))",
"Perpendicular(Line(Y... | [
"PR",
"QB",
"QP",
"QR",
"QS",
"QX",
"RB",
"SP",
"WQ",
"WS",
"WX",
"WY",
"XB",
"YB",
"YP",
"YR",
"YS",
"YX"
] | {"B": [120.0, 130.0], "P": [290.0, 0.0], "Q": [291.0, 133.0], "R": [3.0, 131.0], "S": [291.0, 82.0], "W": [409.0, 81.0], "X": [121.0, 212.0], "Y": [119.0, 80.0]} | [
""
] | |
<image>Find y | \frac { 19 \sqrt { 2 } } { 2 } | [
"Find(y)"
] | [
"Equals(LengthOf(Line(A, B)), 19)",
"Equals(LengthOf(Line(A, C)), x)",
"Equals(MeasureOf(Angle(C, B, A)), 45)",
"Equals(LengthOf(Line(B, C)), y)",
"Perpendicular(Line(A, C), Line(B, C))"
] | [
"AB",
"AC",
"BC"
] | {"A": [0.6624067717165047, 1.9936170506086484], "B": [155.0, 156.0], "C": [0.006450807391374269, 154.0001248543366]} | [] | |
<image>Find $x$ so that the quadrilateral is a parallelogram. | 8 | [
"Parallelogram($)",
"Find(x)"
] | [
"Equals(LengthOf(Line(A, B)), 4x-17)",
"Equals(MeasureOf(Angle(C, B, A)), 3y+3)",
"Equals(LengthOf(Line(C, D)), 2x-1)",
"Equals(MeasureOf(Angle(B, C, D)), 4y-19)"
] | [
"AB",
"AC",
"BC",
"BD",
"CD"
] | {"A": [0.0, 0.0], "B": [232.0, 1.0], "C": [3.0, 140.0], "D": [233.0, 142.0]} | [
""
] | |
<image>Find the area of the shaded region. Round to the nearest tenth. | 7.9 | [
"Find(AreaOf(Shaded(Shape($))))"
] | [
"Equals(LengthOf(Line(C, F)), 3)",
"Equals(MeasureOf(Angle(E, A, D)), 130)",
"PointLiesOnLine(A, Line(B, D))",
"PointLiesOnLine(A, Line(C, E))",
"PointLiesOnCircle(B, Circle(A, radius_0_0))",
"Equals(MeasureOf(Angle(C, A, D)), MeasureOf(Angle(B, A, E)))",
"PointLiesOnCircle(C, Circle(A, radius_0_0))",
... | [
"AB",
"AC",
"AD",
"AE",
"BD",
"CD",
"CE"
] | {"A": [84.0, 91.0], "B": [167.0, 55.0], "C": [8.0, 52.0], "D": [7.0, 126.0], "E": [169.0, 128.0]} | [
"A"
] | |
<image>If $ABCDE \sim PQRST,$ find the perimeter of $PQRST$. | 22.5 | [
"Similar(Pentagon(A,B,C,D,E),Pentagon(P,Q,R,S,T))",
"Find(PerimeterOf(Pentagon(P,Q,R,S,T)))"
] | [
"Equals(LengthOf(Line(B, A)), 8)",
"Equals(LengthOf(Line(D, C)), 4)",
"Equals(LengthOf(Line(E, D)), 6)",
"Equals(LengthOf(Line(R, S)), 3)",
"Equals(LengthOf(Line(B, A)), LengthOf(Line(B, C)))",
"Equals(LengthOf(Line(E, A)), LengthOf(Line(D, C)))",
"Equals(LengthOf(Line(P, Q)), LengthOf(Line(Q, R)))",
... | [
"BA",
"CB",
"DC",
"DE",
"EA",
"PQ",
"QR",
"RS",
"ST",
"TP"
] | {"A": [1.0, 152.0], "B": [207.0, 2.0], "C": [414.0, 150.0], "D": [307.0, 296.0], "E": [104.0, 295.0], "P": [414.0, 550.0], "Q": [207.0, 402.0], "R": [1.0, 552.0], "S": [104.0, 695.0], "T": [307.0, 696.0]} | [
""
] | |
<image>Find the value of the variable $x$ in the figure. | 70 | [
"Find(x)"
] | [
"Equals(MeasureOf(Line(A, E,N)), 2x-15)",
"Equals(MeasureOf(Angle(B, G, E)), x+55)",
"PointLiesOnLine(E, Line(A, H))",
"PointLiesOnLine(G, Line(C, B))",
"Parallel(Line(B, C), Line(A, H))"
] | [
"AE",
"AH",
"HE",
"CG",
"GB",
"CB",
"NG",
"GE",
"EI",
"NE",
"GI",
"NI"
] | {"A": [76.0, 3.0], "B": [268.0, 272.0], "C": [7.0, 123.0], "E": [249.0, 100.0], "G": [202.0, 234.0], "H": [365.0, 166.0], "I": [276.0, 24.0], "N": [196.0, 255.0]} | [
""
] | |
<image>Each triangle in the figure is a $45^{\circ}-45^{\circ}-90^{\circ}$ triangle. Find $x$. | \frac { 3 } { 2 } | [
"Equals(Triangle($),Triangle($1,$2,$3))",
"Equals(MeasureOf(Angle($1)),45)",
"Equals(MeasureOf(Angle($2)),45)",
"Equals(MeasureOf(Angle($3)),90)",
"Find(x)"
] | [
"Equals(MeasureOf(Angle(E, A, D)), 45)",
"Equals(MeasureOf(Angle(B, E, D)), 45)",
"Equals(MeasureOf(Angle(D, B, F)), 45)",
"Equals(MeasureOf(Angle(A, C, D)), 45)",
"Equals(LengthOf(Line(D, F)), x)",
"Equals(LengthOf(Line(C, D)), 6)",
"PointLiesOnLine(D, Line(C, F))",
"Perpendicular(Line(A, E), Line(D,... | [
"AC",
"AD",
"AE",
"BF",
"CD",
"CF",
"DB",
"DE",
"DF",
"EB"
] | {"A": [163.0, 2.0], "B": [414.0, 85.0], "C": [1.0, 170.0], "D": [333.0, 170.0], "E": [330.0, 2.0], "F": [416.0, 170.0]} | [
""
] | |
<image>Find $m\angle L$ | 52 | [
"Find(MeasureOf(Angle(L)))"
] | [
"Equals(MeasureOf(Angle(J, M, L)), 2x-8)",
"Equals(MeasureOf(Angle(M, J, K)), 3x-6)",
"Equals(MeasureOf(Angle(J, K, L)), x+10)",
"Equals(MeasureOf(Angle(K, L, M)), x)"
] | [
"KJ",
"LK",
"LM",
"MJ"
] | {"J": [121.0, 43.0], "K": [1.0, 206.0], "L": [412.0, 206.0], "M": [288.0, 1.0]} | [
""
] | |
<image>The segment is tangent to the circle. Find $x$. | 20 | [
"Tangent(Line($),Circle($))",
"Circle($)",
"Find(x)"
] | [
"Equals(LengthOf(Line(M, N)), 16)",
"Equals(LengthOf(Line(L, N)), 12)",
"Equals(LengthOf(Line(L, M)), x)",
"PointLiesOnLine(F, Line(M, L))",
"PointLiesOnCircle(F, Circle(L, radius_9_0))",
"PointLiesOnCircle(N, Circle(L, radius_9_0))"
] | [
"ML",
"MN",
"NL",
"FM",
"FL"
] | {"L": [208.0, 123.0], "M": [0.0, 124.0], "N": [160.0, 10.0], "F": [80.0, 123.0]} | [
"L"
] | |
<image>Find the area of the triangle. Round to the nearest tenth if necessary. | 20 | [
"Find(AreaOf(Triangle($)))"
] | [
"Equals(LengthOf(Line(A, B)), 5)",
"Equals(LengthOf(Line(O, B)), 21.5)",
"Equals(LengthOf(Line(O, C)), 8)",
"Equals(LengthOf(Line(A, C)), 15)",
"PointLiesOnLine(A, Line(C, B))",
"Perpendicular(Line(O, C), Line(C, A))"
] | [
"AB",
"AC",
"AO",
"BC",
"OB",
"OC"
] | {"A": [227.0, 121.0], "B": [300.0, 121.0], "C": [2.0, 120.0], "O": [2.0, 2.0]} | [
""
] | |
<image>Find the area of the parallelogram. | 396 | [
"Find(AreaOf(Parallelogram($)))"
] | [
"Equals(LengthOf(Line(A, B)), 36)",
"Equals(LengthOf(Line(A, D)), 22)",
"Equals(MeasureOf(Angle(B, C, E)), 30)",
"Perpendicular(Line(B, E), Line(C, E))",
"PointLiesOnLine(E, Line(C, D))"
] | [
"AB",
"AD",
"BC",
"BE",
"CE",
"DC",
"DE"
] | {"A": [339.0, 1.0], "B": [131.0, 2.0], "C": [2.0, 74.0], "D": [215.0, 74.0]} | [
""
] | |
<image>Find $x$ in parallelogram $ABCD$. | 7.2 | [
"Parallelogram(A,B,C,D)",
"Find(x)"
] | [
"Equals(LengthOf(Line(D, E)), x)",
"Equals(LengthOf(Line(A, B)), 15)",
"Equals(LengthOf(Line(D, A)), 9)",
"Equals(LengthOf(Line(D, B)), 12)",
"PointLiesOnLine(E, Line(A, B))",
"Perpendicular(Line(D, E), Line(E, B))",
"Perpendicular(Line(D, B), Line(B, C))"
] | [
"AB",
"CB",
"DA",
"DC"
] | {"A": [0.0, 0.0], "C": [305.0, 118.0], "D": [66.0, 116.0], "B": [240.0, 1.0]} | [
""
] | |
<image>If QRST is a parallelogram, Find $z$ | 11 | [
"Parallelogram(Q,R,S,T)",
"Find(z)"
] | [
"Equals(MeasureOf(Angle(R, Q, S)), 33)",
"Equals(LengthOf(Line(R, P)), y+4)",
"Equals(MeasureOf(Angle(S, Q, T)), 95)",
"Equals(LengthOf(Line(T, Q)), 5x)",
"Equals(LengthOf(Line(R, S)), 27)",
"Equals(LengthOf(Line(T, P)), 2y-5)",
"Equals(MeasureOf(Angle(T, S, Q)), 3z)",
"PointLiesOnLine(P, Line(R, T))"... | [
"PQ",
"RP",
"RQ",
"RS",
"RT",
"SP",
"SQ",
"TP",
"TQ",
"TS"
] | {"P": [322.0, 106.0], "Q": [162.0, 3.0], "R": [644.0, 1.0], "S": [481.0, 208.0], "T": [2.0, 209.0]} | [
""
] | |
<image>Refer to the figure at the right. Find the value of $m ∠ ABD $ if $ p \parallel q $. | 75 | [
"Parallel(Line(p),Line(q))",
"Find(MeasureOf(Angle(A,B,D)))"
] | [
"Equals(LengthOf(Line(P, O)), p)",
"Equals(LengthOf(Line(M, N)), q)",
"Equals(MeasureOf(Angle(A, B, D)), 3x-60)",
"Equals(MeasureOf(Angle(D, B, C)), y)",
"Equals(MeasureOf(Angle(E, C, F)), 2x+15)",
"PointLiesOnLine(A, Line(B, I))",
"PointLiesOnLine(A, Line(C, I))",
"PointLiesOnLine(A, Line(F, I))",
... | [
"AF",
"BA",
"BD",
"BF",
"BG",
"BO",
"BP",
"CA",
"CB",
"CE",
"CF",
"CI",
"CL",
"CM",
"CN",
"CT",
"DO",
"DP",
"ET",
"GD",
"GO",
"GP",
"IA",
"IB",
"IF",
"IL",
"LA",
"LB",
"LF",
"ME",
"MN",
"MT",
"NE",
"NT",
"OP"
] | {"A": [195.0, 31.0], "B": [192.0, 79.0], "C": [181.0, 162.0], "D": [341.0, 49.0], "E": [358.0, 129.0], "F": [178.0, 203.0], "G": [380.0, 45.0], "I": [202.0, 1.0], "L": [172.0, 236.0], "M": [425.0, 116.0], "N": [25.0, 191.0], "O": [1.0, 116.0], "P": [403.0, 39.0], "T": [405.0, 119.0]} | [
""
] | |
<image>Find $KL$. | 24 | [
"Find(LengthOf(Line(J,K)))"
] | [
"Equals(LengthOf(Line(L, K)), 4x)",
"Equals(LengthOf(Line(J, K)), 3x+6)",
"Equals(LengthOf(Line(J, L)), 5x-6)",
"Equals(LengthOf(Line(J, K)), LengthOf(Line(L, K)))",
"Equals(LengthOf(Line(L, K)), LengthOf(Line(J, L)))"
] | [
"JL",
"JK",
"LK"
] | {"J": [1.1529790660225387, 206.66666666666669], "L": [300.0, 206.0], "K": [150.50242718446603, 1.6821031312287005]} | [] | |
<image>In $\odot D, \overline{D E} \cong \overline{E C}, m \widehat{C F}=60,$ and
$\overline{D E} \perp \overline{E C} .$ Find $m \angle 5$ | 30 | [
"Circle(D)",
"Equals(LengthOf(Line(D,E)),Line(E,C))",
"Equals(MeasureOf(Arc(C,F)),60)",
"Perpendicular(Line(D,E),Line(E,C))",
"Find(MeasureOf(Angle(5)))"
] | [
"Equals(MeasureOf(Angle(C, B, A)), MeasureOf(angle 3))",
"Equals(MeasureOf(Angle(B, A, C)), MeasureOf(angle 4))",
"Equals(MeasureOf(Angle(A, C, B)), MeasureOf(angle 1))",
"Equals(MeasureOf(Angle(E, D, C)), MeasureOf(angle 2))",
"Equals(MeasureOf(Angle(F, C, D)), MeasureOf(angle 6))",
"Equals(MeasureOf(Ang... | [
"AB",
"AD",
"AF",
"BG",
"CA",
"CB",
"CD",
"CE",
"CF",
"CG",
"DE",
"DG",
"EB",
"EG"
] | {"A": [0.0, 80.0], "B": [86.0, 3.0], "C": [163.0, 81.0], "D": [81.0, 82.0], "E": [144.0, 21.0], "F": [122.0, 153.0], "G": [124.0, 40.0]} | [
"D"
] | |
<image>For trapezoid $D E G H, X$ and $Y$ are midpoints of the legs. Find $D E$ | 8 | [
"Trapezoid(D,E,G,H)",
"IsMidpointOf(Point(X),LegOf(Trapezoid(D,E,G,H)))",
"IsMidpointOf(Point(Y),LegOf(Trapezoid(D,E,G,H)))",
"Find(LengthOf(Line(D,E)))"
] | [
"Equals(LengthOf(Line(X, Y)), 20)",
"Equals(LengthOf(Line(H, G)), 32)",
"PointLiesOnLine(X, Line(H, D))",
"PointLiesOnLine(Y, Line(G, E))"
] | [
"HD",
"HX",
"HG",
"DX",
"DE",
"XY",
"GY",
"GE",
"YE"
] | {"H": [0.0, 94.0], "Y": [291.0851063829787, 48.0], "D": [150.40866331218552, 0.6966382265004043], "X": [73.93617021276596, 48.0], "G": [344.0, 94.0], "E": [237.47384710615907, 1.9746208604147384]} | [] | |
<image>Point $S$ is the incenter of $\triangle JPL$. Find $m \angle MPQ$. | 56 | [
"IsIncenterOf(Point(S),Triangle(J,P,L))",
"Find(MeasureOf(Angle(M,P,Q)))"
] | [
"Equals(LengthOf(Line(J, K)), 8)",
"Equals(LengthOf(Line(J, S)), 10)",
"Equals(MeasureOf(Angle(K, L, S)), 24.5)",
"Equals(MeasureOf(Angle(K, P, L)), 28)",
"PointLiesOnLine(K, Line(L, J))",
"PointLiesOnLine(M, Line(L, P))",
"PointLiesOnLine(S, Line(L, Q))",
"PointLiesOnLine(Q, Line(J, P))",
"PointLie... | [
"JK",
"JM",
"JP",
"JQ",
"JS",
"LJ",
"LK",
"LM",
"LP",
"LQ",
"LS",
"PK",
"PM",
"PQ",
"SK",
"SM",
"SP",
"SQ"
] | {"J": [274.0, 1.0], "K": [350.0, 149.0], "L": [444.0, 336.0], "M": [252.0, 324.0], "P": [1.0, 307.0], "Q": [169.0, 117.0], "S": [262.0, 190.0]} | [
""
] | |
<image>Find x. | 40 | [
"Find(x)"
] | [
"Equals(LengthOf(Line(B, D)), x)",
"Equals(LengthOf(Line(A, B)), z)",
"Equals(LengthOf(Line(A, D)), 20)",
"Equals(LengthOf(Line(C, D)), 10)",
"Equals(LengthOf(Line(A, C)), y)",
"PointLiesOnLine(D, Line(B, C))",
"Perpendicular(Line(B, D), Line(A, D))",
"Perpendicular(Line(A, B), Line(A, C))"
] | [
"AB",
"AC",
"AD",
"BC",
"BD",
"CD"
] | {"A": [240.39999999999998, 263.0217265154588], "B": [240.0, -2.842170943040401e-14], "C": [0.9660575502768722, 263.3359587450219], "D": [108.45373850965032, 144.40256499352432]} | [] | |
<image>Find $m \angle 8$. | 28.5 | [
"Find(MeasureOf(Angle(8)))"
] | [
"Equals(MeasureOf(Angle(Z, Y, W)), 58)",
"Equals(MeasureOf(Angle(Y, X, Z)), MeasureOf(angle 7))",
"Equals(MeasureOf(Angle(X, Z, Y)), 65)",
"Equals(MeasureOf(Angle(Z, V, W)), MeasureOf(angle 4))",
"Equals(MeasureOf(Angle(W, X, V)), MeasureOf(angle 5))",
"Equals(MeasureOf(Angle(Z, X, W)), MeasureOf(angle 6)... | [
"VX",
"WV",
"WX",
"WY",
"WZ",
"YX",
"ZV",
"ZX",
"ZY"
] | {"V": [2.0, 206.0], "W": [87.0, 336.0], "X": [172.0, 205.0], "Y": [306.0, 0.0], "Z": [402.0, 205.0]} | [
""
] | |
<image>Find the height of the parallelogram given its area with 100 square units. | 5 | [
"Equals(AreaOf(Parallelogram($)),100)",
"Find(HeightOf(Parallelogram($)))"
] | [
"Equals(LengthOf(Line(A, B)), x+15)",
"Equals(LengthOf(Line(B, E)), x)",
"PointLiesOnLine(E, Line(D, C))",
"Perpendicular(Line(B, E), Line(C, E))"
] | [
"AB",
"AD",
"BC",
"BE",
"CE",
"DC",
"DE"
] | {"A": [483.0, 0.0], "B": [141.0, 2.0], "C": [0.0, 90.0], "D": [345.0, 89.0], "E": [142.0, 90.0]} | [
""
] | |
<image>Find y | 16 | [
"Find(y)"
] | [
"Equals(LengthOf(Line(B, C)), y)",
"Equals(LengthOf(Line(A, C)), 8)",
"Equals(MeasureOf(Angle(C, B, A)), 30)",
"Equals(LengthOf(Line(A, B)), x)",
"Perpendicular(Line(A, C), Line(A, B))"
] | [
"AB",
"AC",
"BC"
] | {"A": [239.0, 138.0], "B": [1.3429131550758768, 139.9943410335698], "C": [238.3323417870322, 0.5308248234127575]} | [] | |
<image>The height of a triangle is 5 centimeters more than its base. The area of the triangle is 52 square centimeters. Find the height. | 13 | [
"Equals(HeightOf(Triangle($)),Add(BaseOf(Polygon($)),5))",
"Equals(AreaOf(Triangle($)),52)",
"Find(HeightOf(Polygon($)))"
] | [
"Equals(LengthOf(Line(B, C)), b+5)",
"Equals(LengthOf(Line(A, E)), b)",
"PointLiesOnLine(B, Line(A, E))",
"Perpendicular(Line(C, B), Line(B, E))"
] | [
"AB",
"AC",
"AE",
"CB",
"CE",
"EB"
] | {"A": [207.0, 284.0], "C": [71.0, 6.0], "E": [1.0, 282.0]} | [
""
] | |
<image>Find the diameter of circle M. | 38.0 | [
"Find(DiameterOf(Circle(M)))"
] | [
"Equals(LengthOf(Line(W, Y)), 10.5)",
"Equals(LengthOf(Line(V, W)), 17)",
"Equals(LengthOf(Line(X, W)), 17)",
"PointLiesOnLine(W, Line(X, V))",
"PointLiesOnLine(W, Line(Z, Y))",
"PointLiesOnLine(W, Line(M, Y))",
"PointLiesOnCircle(X, Circle(M, radius_2_0))",
"PointLiesOnCircle(Z, Circle(M, radius_2_0)... | [
"MY",
"VW",
"WM",
"WY",
"WZ",
"XV",
"XW",
"ZM",
"ZY"
] | {"M": [181.0, 180.0], "V": [18.0, 105.0], "W": [184.0, 104.0], "X": [347.0, 104.0], "Y": [184.0, 1.0], "Z": [185.0, 363.0]} | [
"M"
] | |
<image>Find $RT$. | 8 | [
"Find(LengthOf(Line(R,T)))"
] | [
"Equals(LengthOf(Line(R, S)), 3x+2)",
"Equals(LengthOf(Line(T, R)), 2x+4)",
"Equals(LengthOf(Line(S, R)), LengthOf(Line(R, T)))"
] | [
"RS",
"TR",
"TS"
] | {"R": [3.0, 0.0], "S": [387.0, 0.0], "T": [330.0, 192.0]} | [
""
] | |
<image>Find $x$. | 20 | [
"Find(x)"
] | [
"Equals(LengthOf(Line(F, N)), \\frac{1}{4}x+6)",
"Equals(LengthOf(Line(P, I)), 12-3y)",
"Equals(LengthOf(Line(P, D)), 16-5y)",
"Equals(LengthOf(Line(N, A)), 2x-29)",
"PointLiesOnLine(N, Line(A, F))",
"PointLiesOnLine(D, Line(A, G))",
"PointLiesOnLine(F, Line(A, K))",
"PointLiesOnLine(N, Line(A, K))",
... | [
"AD",
"AF",
"AG",
"AJ",
"AK",
"AL",
"AN",
"BC",
"BI",
"CI",
"DE",
"DG",
"DI",
"DL",
"DM",
"DP",
"EM",
"EP",
"FB",
"FC",
"FI",
"FJ",
"FK",
"FN",
"GL",
"HO",
"IE",
"IM",
"IP",
"JK",
"JN",
"KN",
"MP",
"NH",
"NO",
"NP",
"PH",
"PO"
] | {"A": [279.0, 276.0], "B": [0.0, 93.0], "C": [399.0, 95.0], "D": [132.0, 275.0], "E": [133.0, 0.0], "F": [302.0, 95.0], "G": [0.0, 275.0], "H": [398.0, 184.0], "I": [133.0, 94.0], "J": [267.0, 368.0], "K": [314.0, 4.0], "L": [398.0, 276.0], "M": [132.0, 367.0], "N": [291.0, 183.0], "O": [1.0, 182.0], "P": [133.0, 183.0... | [
""
] | |
<image>$\overline{E C}$ and $\overline{A B}$ are diameters of $\odot O$. Find its measure of $\widehat{A C B}$. | 180 | [
"IsDiameterOf(Line(E,C),Circle(O))",
"IsDiameterOf(Line(A,B),Circle(O))",
"Find(MeasureOf(Arc(A,C,B)))"
] | [
"Equals(MeasureOf(Angle(F, O, E)), 45)",
"PointLiesOnLine(O, Line(A, B))",
"PointLiesOnLine(O, Line(E, C))",
"PointLiesOnCircle(A, Circle(O, radius_1_0))",
"PointLiesOnCircle(C, Circle(O, radius_1_0))",
"PointLiesOnCircle(F, Circle(O, radius_1_0))",
"PointLiesOnCircle(E, Circle(O, radius_1_0))",
"Poin... | [
"AB",
"AO",
"CE",
"OB",
"OC",
"OE",
"OF"
] | {"A": [32.0, 180.0], "B": [181.0, 31.0], "C": [179.0, 180.0], "E": [31.0, 31.0], "F": [1.0, 109.0], "O": [105.0, 104.0]} | [
"O"
] | |
<image>Use parallelogram $JKLM$ to find $m \angle JML$. | 80 | [
"Parallelogram(J,K,L,M)",
"Find(MeasureOf(Angle(J,M,L)))"
] | [
"Equals(LengthOf(Line(J, K)), 2b+3)",
"Equals(LengthOf(Line(J, M)), 3)",
"Equals(LengthOf(Line(L, K)), 21)",
"Equals(MeasureOf(Angle(R, L, K)), 70)",
"Equals(MeasureOf(Angle(M, L, J)), 30)",
"Equals(LengthOf(Line(L, M)), 45)",
"PointLiesOnLine(R, Line(L, J))",
"PointLiesOnLine(R, Line(K, M))"
] | [
"JK",
"JM",
"JR",
"KM",
"KR",
"LJ",
"LK",
"LM",
"LR",
"MR"
] | {"J": [39.0, 0.0], "K": [417.0, 1.0], "L": [378.0, 204.0], "M": [3.0, 203.0], "R": [209.0, 102.0]} | [
""
] | |
<image>Find $m\angle C$ | 127 | [
"Find(MeasureOf(Angle(C)))"
] | [
"Equals(MeasureOf(Angle(A, B, C)), x-8)",
"Equals(MeasureOf(Angle(F, A, B)), x+2)",
"Equals(MeasureOf(Angle(B, C, D)), x+7)",
"Equals(MeasureOf(Angle(A, F, E)), x-4)",
"Equals(MeasureOf(Angle(F, E, D)), x+6)",
"Equals(MeasureOf(Angle(C, E, D)), x-3)"
] | [
"AB",
"BC",
"CD",
"DE",
"EF",
"FA"
] | {"A": [85.0, 2.0], "B": [255.0, 2.0], "C": [337.0, 112.0], "D": [255.0, 232.0], "E": [90.0, 234.0], "F": [1.0, 117.0]} | [
""
] | |
<image>If QRST is a kite, find $m\angle RST$. | 78 | [
"Kite(Q,R,S,T)",
"Find(MeasureOf(Angle(R,S,T)))"
] | [
"Equals(MeasureOf(Angle(S, R, Q)), 136)",
"Equals(MeasureOf(Angle(Q, T, S)), 68)",
"Equals(LengthOf(Line(Q, R)), LengthOf(Line(R, S)))",
"Equals(LengthOf(Line(Q, T)), LengthOf(Line(T, S)))"
] | [
"QR",
"QT",
"RS",
"TS"
] | {"Q": [1.0, 91.0], "R": [134.0, 2.0], "S": [264.0, 89.0], "T": [135.0, 298.0]} | [
""
] | |
<image>In $\odot P, m \widehat{E N}=66$ and $m \angle G P M=89$ .
Find $m \angle G M E$ | 57 | [
"Circle(P)",
"Equals(MeasureOf(Arc(E,N)),66)",
"Equals(MeasureOf(Angle(G,P,M)),89)",
"Find(MeasureOf(Angle(G,M,E)))"
] | [
"PointLiesOnLine(B, Line(M, E))",
"PointLiesOnLine(B, Line(N, P))",
"PointLiesOnLine(P, Line(N, G))",
"PointLiesOnLine(B, Line(N, G))",
"PointLiesOnLine(P, Line(G, B))",
"PointLiesOnCircle(M, Circle(P, radius_4_0))",
"PointLiesOnCircle(E, Circle(P, radius_4_0))",
"PointLiesOnCircle(N, Circle(P, radius... | [
"EB",
"EG",
"EN",
"GB",
"MB",
"ME",
"MG",
"MN",
"MP",
"NB",
"NG",
"NP",
"PB",
"PG"
] | {"B": [117.0, 110.0], "E": [186.0, 45.0], "G": [18.0, 46.0], "M": [41.0, 183.0], "N": [186.0, 155.0], "P": [103.0, 100.0]} | [
"P"
] | |
<image>Find $x$ if $AC=x-3$, $BE=20, AB=16,$ and $CD=x+5$ | 35 | [
"Equals(LengthOf(Line(A,C)),x-3)",
"Equals(LengthOf(Line(B,E)),20)",
"Equals(LengthOf(Line(A,B)),16)",
"Equals(LengthOf(Line(C,D)),x+5)",
"Find(x)"
] | [
"PointLiesOnLine(B, Line(E, A))",
"PointLiesOnLine(C, Line(A, D))",
"Parallel(Line(C, B), Line(E, D))"
] | [
"AB",
"AC",
"AD",
"BC",
"DC",
"EA",
"EB",
"ED"
] | {"A": [1.0, 82.0], "B": [90.0, 124.0], "C": [87.0, 43.0], "D": [180.0, 1.0], "E": [181.0, 165.0]} | [
""
] | |
<image>Find $ED$ if $AE=3, AB=2$, $BC=6,$ and $ED=2x-3$ | 9 | [
"Equals(LengthOf(Line(A,E)),3)",
"Equals(LengthOf(Line(A,B)),2)",
"Equals(LengthOf(Line(B,C)),6)",
"Equals(LengthOf(Line(E,D)),2x-3)",
"Find(LengthOf(Line(E,D)))"
] | [
"PointLiesOnLine(E, Line(A, D))",
"PointLiesOnLine(B, Line(A, C))",
"Parallel(Line(D, C), Line(E, B))"
] | [
"AB",
"AC",
"AD",
"AE",
"CB",
"DC",
"DE",
"EB"
] | {"A": [247.0, 102.0], "B": [215.0, 118.0], "C": [123.0, 164.0], "D": [0.0, 0.0], "E": [178.0, 73.0]} | [
""
] | |
<image>Use a Pythagorean Triple to find x. | 34 | [
"UseTheorem(Pythagorean_Triple)",
"Find(x)"
] | [
"Equals(LengthOf(Line(A, C)), 16)",
"Equals(LengthOf(Line(B, C)), 30)",
"Equals(LengthOf(Line(A, B)), x)",
"Perpendicular(Line(A, C), Line(B, C))"
] | [
"AB",
"AC",
"BC"
] | {"A": [1.3404255319148888, 0.014184397163120366], "B": [94.5, 183.5], "C": [93.01058749064964, 1.312503596294384]} | [] | |
<image>Find $x$. | 5 | [
"Find(x)"
] | [
"Equals(LengthOf(Line(J, K)), x+7)",
"Equals(LengthOf(Line(J, L)), 4x-8)",
"Equals(LengthOf(Line(J, K)), LengthOf(Line(J, L)))",
"Equals(LengthOf(Line(J, K)), LengthOf(Line(K, L)))",
"Equals(LengthOf(Line(L, K)), LengthOf(Line(J, L)))"
] | [
"JK",
"JL",
"KL"
] | {"J": [0.0, 278.0], "K": [158.0, 1.0], "L": [315.0, 276.0]} | [
""
] | |
<image>Find EC | 4.8 | [
"Find(LengthOf(Line(E,C)))"
] | [
"Equals(LengthOf(Line(B, D)), x-1)",
"Equals(LengthOf(Line(C, E)), x+2)",
"Equals(LengthOf(Line(E, F)), 8)",
"Equals(LengthOf(Line(B, A)), 3)",
"PointLiesOnLine(C, Line(D, B))",
"PointLiesOnLine(G, Line(D, A))",
"PointLiesOnLine(D, Line(B, E))",
"PointLiesOnLine(C, Line(B, E))",
"PointLiesOnLine(D, ... | [
"AG",
"BA",
"BC",
"BE",
"CF",
"CG",
"DA",
"DB",
"DC",
"DE",
"DG",
"EC",
"EF",
"GF"
] | {"A": [1.0, 129.0], "B": [2.0, 205.0], "C": [62.0, 206.0], "D": [245.0, 206.0], "E": [381.0, 207.0], "F": [381.0, 1.0], "G": [122.0, 167.0]} | [
""
] | |
<image>Find $m \angle R$. | 32 | [
"Find(MeasureOf(Angle(R)))"
] | [
"Equals(MeasureOf(Angle(R, S, Q)), 5x+4)",
"Equals(MeasureOf(Angle(S, Q, T)), 32)",
"Equals(MeasureOf(Angle(R, T, Q)), 6x-2)",
"PointLiesOnLine(A, Line(S, Q))",
"PointLiesOnLine(A, Line(R, T))",
"PointLiesOnCircle(S, Circle(C, radius_7_0))",
"PointLiesOnCircle(R, Circle(C, radius_7_0))",
"PointLiesOnC... | [
"AT",
"QA",
"QT",
"RA",
"RT",
"SA",
"SQ",
"SR"
] | {"A": [159.0, 134.0], "C": [160.0, 171.0], "Q": [12.0, 225.0], "R": [44.0, 62.0], "S": [274.0, 62.0], "T": [306.0, 226.0]} | [
"C"
] | |
<image>Find x. | 4 \sqrt { 2 } | [
"Find(x)"
] | [
"Equals(LengthOf(Line(A, Y)), 8)",
"Equals(LengthOf(Line(A, B)), x)",
"Perpendicular(Line(A, B), Line(B, Y))",
"Equals(LengthOf(Line(A, B)), LengthOf(Line(B, Y)))"
] | [
"AB",
"AY",
"BY"
] | {"A": [1.0, 0.0], "B": [2.6583676196504484, 238.00411435805466], "Y": [239.0, 240.0]} | [] | |
<image>Find x | 30 | [
"Find(x)"
] | [
"Equals(MeasureOf(Angle(H, A, E)), 3x)",
"Equals(MeasureOf(Angle(F, A, D)), 2x)",
"PointLiesOnLine(A, Line(D, E))",
"Perpendicular(Line(A, H), Line(A, D))"
] | [
"AD",
"AE",
"AF",
"AH",
"DE"
] | {"A": [118.0, 79.0], "D": [117.0, 170.0], "E": [117.0, 1.0], "F": [202.0, 126.0], "H": [0.0, 78.0]} | [
""
] | |
<image>Find the area of the figure. Round to the nearest tenth if necessary. | 340 | [
"Find(AreaOf(Shape($)))"
] | [
"Equals(LengthOf(Line(A, C)), 20)",
"Equals(LengthOf(Line(D, F)), 10)",
"Equals(LengthOf(Line(D, E)), 15)",
"Equals(LengthOf(Line(B, E)), 12)",
"PointLiesOnLine(F, Line(A, C))",
"Perpendicular(Line(D, F), Line(A, F))",
""
] | [
"AB",
"AC",
"AD",
"BC",
"BE",
"DE"
] | {"A": [1.0, 1.0], "B": [126.0, 160.0], "C": [127.0, 1.0], "D": [50.0, 62.0], "E": [51.0, 161.0]} | [
""
] | |
<image>Find the perimeter of the triangle. | 71.7 | [
"Find(PerimeterOf(Triangle($)))"
] | [
"Equals(LengthOf(Line(A, C)), 13)",
"Equals(LengthOf(Line(B, D)), 29)",
"Equals(LengthOf(Line(C, D)), 6)",
"PointLiesOnLine(D, Line(C, B))",
"Perpendicular(Line(A, D), Line(C, D))",
""
] | [
"AB",
"BD",
"AD",
"AC",
"CD",
"BC"
] | {"A": [20.0, 0.0], "B": [388.0, 137.0], "D": [19.0, 137.0]} | [
""
] | |
<image>Find the value of $x$ in the diagram. | 13 | [
"Find(x)"
] | [
"Equals(MeasureOf(Angle(H, C, B)), 6x)",
"Equals(MeasureOf(Angle(F, A, C)), 139)",
"Equals(MeasureOf(Angle(G, B, D)), 9x)",
"Equals(MeasureOf(Angle(E, D, A)), 2x)",
"PointLiesOnLine(C, Line(A, H))",
"PointLiesOnLine(B, Line(C, G))",
"PointLiesOnLine(A, Line(D, F))",
"PointLiesOnLine(A, Line(F, G))",
... | [
"AC",
"AD",
"AF",
"AG",
"AH",
"CB",
"CG",
"CH",
"DB",
"DE",
"DF",
"FG",
"GB",
"BE"
] | {"A": [72.73203036964512, 144.56016662664177], "B": [393.2590981343596, 151.22295657993809], "C": [271.43427346710564, 24.455457538671794], "D": [296.1305329719964, 180.85655257787147], "E": [178.0, 216.0], "F": [0.0, 132.0], "G": [435.0, 196.0], "H": [310.0, 2.0]} | [] | |
<image>$\overline{AD}$ and $\overline{CG}$ are diameters of $\odot B$. Find $m \widehat {CGD}$. | 305 | [
"IsDiameterOf(Line(A,D),Circle(B))",
"IsDiameterOf(Line(C,G),Circle(B))",
"Find(MeasureOf(Arc(C,G,D)))"
] | [
"Equals(MeasureOf(Angle(C, B, D)), 55)",
"Equals(MeasureOf(Angle(G, B, F)), 35)",
"PointLiesOnLine(B, Line(A, D))",
"PointLiesOnLine(B, Line(C, G))",
"PointLiesOnCircle(F, Circle(B, radius_6_0))",
"PointLiesOnCircle(D, Circle(B, radius_6_0))",
"PointLiesOnCircle(A, Circle(B, radius_6_0))",
"PointLiesO... | [
"AB",
"AD",
"BD",
"CB",
"CG",
"FB",
"GB"
] | {"A": [34.0, 274.0], "B": [181.0, 172.0], "C": [173.0, 1.0], "D": [316.0, 74.0], "F": [272.0, 318.0], "G": [173.0, 347.0]} | [
"B"
] | |
<image>Lines $l$, $m$, and $n$ are perpendicular bisectors of $\triangle PQR$ and meet at $T$. If $TQ = 2x$, $PT = 3y - 1$, and $TR = 8$, find $y$. | 3 | [
"IsPerpendicularBisectorOf(Line(l),Triangle(P,Q,R))",
"IsPerpendicularBisectorOf(Line(m),Triangle(P,Q,R))",
"IsPerpendicularBisectorOf(Line(n),Triangle(P,Q,R))",
"IntersectAt(Line(l),Line(m),Line(n),Point(T))",
"Equals(LengthOf(Line(T,Q)),2x)",
"Equals(LengthOf(Line(P,T)),3y-1)",
"Equals(LengthOf(Line(T... | [
"Equals(LengthOf(Line(Q, T)), 2x)",
"Equals(LengthOf(Line(T, R)), 8)",
"Equals(LengthOf(Line(P, T)), 3y-1)",
"Equals(LengthOf(Line(P, G)), z+4)",
"Equals(LengthOf(Line(G, R)), 7)",
"PointLiesOnLine(F, Line(Q, R))",
"PointLiesOnLine(G, Line(P, R))",
"PointLiesOnLine(G, Line(P, D))",
"PointLiesOnLine(... | [
"DR",
"EZ",
"FR",
"GD",
"GR",
"HQ",
"HZ",
"PD",
"PG",
"PH",
"PR",
"QF",
"QP",
"QR",
"QT",
"TP",
"TR"
] | {"A": [66.0, 146.0], "C": [199.0, 125.0], "D": [237.0, 195.0], "E": [199.0, 36.0], "F": [387.0, 96.0], "G": [199.0, 194.0], "H": [193.0, 96.0], "I": [419.0, 91.0], "M": [361.0, 10.0], "P": [2.0, 195.0], "Q": [373.0, 2.0], "R": [402.0, 195.0], "T": [167.0, 61.0]} | [
""
] | |
<image>Find x | 44.9 | [
"Find(x)"
] | [
"Equals(LengthOf(Line(A, B)), 17)",
"Equals(LengthOf(Line(A, C)), 12)",
"Equals(MeasureOf(Angle(A, B, C)), x)",
"Perpendicular(Line(A, C), Line(B, C))"
] | [
"AB",
"AC",
"BC"
] | {"A": [145.0, 1.0], "B": [0.0, 154.0], "C": [142.68601008037052, 152.0134859011034]} | [] | |
<image>Find the measure of $\angle 7$ if $\overline{A B} \perp \overline{B C}$. | 140 | [
"Perpendicular(Line(A,B),Line(B,C))",
"Find(MeasureOf(Angle(7)))"
] | [
"Equals(MeasureOf(Angle(E, B, C)), MeasureOf(angle 6))",
"Equals(MeasureOf(Angle(A, G, F)), 104)",
"Equals(MeasureOf(Angle(E, F, C)), 40)",
"Equals(MeasureOf(Angle(B, F, D)), MeasureOf(angle 4))",
"Equals(MeasureOf(Angle(B, A, D)), MeasureOf(angle 2))",
"Equals(MeasureOf(Angle(A, D, G)), 36)",
"Equals(M... | [
"AD",
"AG",
"BA",
"BC",
"BE",
"BF",
"BG",
"CD",
"CF",
"CG",
"EF",
"FD",
"FG",
"GD"
] | {"A": [114.0, 2.0], "B": [154.0, 161.0], "C": [465.0, 86.0], "D": [1.0, 84.0], "E": [346.0, 1.0], "F": [247.0, 84.0], "G": [134.0, 83.0]} | [
""
] | |
<image>Find the area of the figure. Round to the nearest tenth if necessary. | 53.4 | [
"Find(AreaOf(Shape($)))"
] | [
"Equals(LengthOf(Line(A, C)), 3.6)",
"Equals(LengthOf(Line(C, D)), 9.2)",
"Equals(LengthOf(Line(A, B)), LengthOf(Line(B, I)))",
"Perpendicular(Line(B, E), Line(C, E))",
"PointLiesOnLine(E, Line(C, D))",
"Equals(LengthOf(Line(B, E)), 8)"
] | [
"AB",
"AC",
"BI",
"CD",
"ID"
] | {"A": [1.0, 92.0], "B": [96.0, 0.0], "C": [2.0, 161.0], "D": [187.0, 164.0], "I": [188.0, 92.0]} | [
""
] | |
<image>In the figure, $m∠11 = 62$ and $m∠14 = 38$. Find the measure of $\angle 6$. | 38 | [
"Equals(MeasureOf(Angle(11)),62)",
"Equals(MeasureOf(Angle(14)),38)",
"Find(MeasureOf(Angle(6)))"
] | [
"Equals(MeasureOf(Angle(J, C, I)), MeasureOf(angle 2))",
"Equals(MeasureOf(Angle(N, C, J)), MeasureOf(angle 1))",
"Equals(MeasureOf(Angle(I, C, O)), MeasureOf(angle 3))",
"Equals(MeasureOf(Angle(K, C, N)), MeasureOf(angle 4))",
"Equals(MeasureOf(Angle(O, C, B)), MeasureOf(angle 6))",
"Equals(MeasureOf(Ang... | [
"BC",
"BF",
"BH",
"BJ",
"BK",
"BM",
"CF",
"CG",
"CI",
"CJ",
"CN",
"CO",
"FJ",
"GI",
"HK",
"HM",
"KC",
"KG",
"KI",
"KM",
"NO"
] | {"B": [288.0, 254.0], "C": [204.0, 128.0], "F": [369.0, 376.0], "G": [119.0, 380.0], "H": [434.0, 253.0], "I": [244.0, 4.0], "J": [151.0, 47.0], "K": [160.0, 255.0], "M": [0.0, 253.0], "N": [5.0, 128.0], "O": [429.0, 128.0]} | [
""
] | |
<image>Find the area of the parallelogram. Round to the nearest tenth if necessary. | 315 | [
"Find(AreaOf(Parallelogram($)))"
] | [
"Equals(LengthOf(Line(A, D)), 17)",
"Equals(LengthOf(Line(D, E)), 8)",
"Equals(LengthOf(Line(A, C)), 21)",
"Perpendicular(Line(A, E), Line(A, C))",
"PointLiesOnLine(E, Line(B, D))",
"Perpendicular(Line(A, E), Line(E, D))"
] | [
"AC",
"AD",
"AE",
"BE",
"CB",
"DB",
"DE"
] | {"A": [206.0, 173.0], "B": [74.0, 1.0], "C": [1.0, 173.0], "D": [278.0, 2.0]} | [
""
] | |
<image>Find x. Round to the nearest tenth if necessary. Assume that segments that appear
to be tangent are tangent. | 6 | [
"Tangent(Line($),Circle($))",
"Find(x)"
] | [
"Equals(LengthOf(Line(A, F)), x)",
"Equals(LengthOf(Line(D, E)), 9)",
"Equals(LengthOf(Line(A, E)), 3)",
"PointLiesOnLine(E, Line(A, B))",
"PointLiesOnLine(B, Line(A, D))",
"PointLiesOnLine(E, Line(A, D))",
"PointLiesOnLine(B, Line(D, E))",
"PointLiesOnCircle(D, Circle(B, radius_1_0))",
"PointLiesOn... | [
"AB",
"AD",
"AE",
"AF",
"BD",
"BE",
"DE"
] | {"A": [254.0, 86.0], "B": [88.0, 85.0], "D": [1.0, 84.0], "E": [173.0, 85.0], "F": [124.0, 9.0]} | [
"B"
] | |
<image>Find y. | 3 | [
"Find(y)"
] | [
"Equals(LengthOf(Line(C, D)), y)",
"Equals(LengthOf(Line(A, B)), x)",
"Equals(LengthOf(Line(B, D)), 9)",
"Equals(LengthOf(Line(A, D)), 3\\sqrt{3})",
"Equals(MeasureOf(Angle(A, B, C)), 30)",
"Equals(MeasureOf(Angle(B, C, A)), 60)",
"PointLiesOnLine(D, Line(B, C))",
"Perpendicular(Line(C, D), Line(A, D)... | [
"AB",
"AC",
"AD",
"BC",
"BD",
"CD"
] | {"A": [0.0, 148.0], "B": [259.0, 149.0], "C": [1.0, 2.0], "D": [65.0, 38.0]} | [
""
] | |
<image>Find the perimeter of the parallelogram. Round to the nearest tenth if necessary. | 46 | [
"Find(PerimeterOf(Parallelogram($)))"
] | [
"Equals(LengthOf(Line(A, D)), 13)",
"Equals(LengthOf(Line(A, C)), 10)",
"Equals(MeasureOf(Angle(C, B, D)), 45)"
] | [
"AC",
"AD",
"CB",
"DB"
] | {"A": [2.0, 0.0], "B": [336.0, 118.0], "C": [117.0, 117.0], "D": [220.0, 3.0]} | [
""
] | |
<image>Find x. | 15 | [
"Find(x)"
] | [
"Equals(LengthOf(Line(A, D)), x)",
"Equals(LengthOf(Line(D, E)), 9)",
"Equals(LengthOf(Line(A, E)), 12)",
"PointLiesOnLine(E, Line(A, B))",
"PointLiesOnLine(E, Line(C, D))",
"Perpendicular(Line(D, E), Line(A, E))"
] | [
"AB",
"AC",
"AD",
"AE",
"BC",
"BD",
"BE",
"CD",
"CE",
"DE"
] | {"A": [185.84032327304624, 1.3245519277160724], "B": [0.5001101873523908, 186.50532032114205], "C": [185.66314003583818, 187.3208654034164], "D": [1.4919786264340615, 0.5026727434648279], "E": [93.24932975871313, 94.2520107238606]} | [] | |
<image>Find $x$. | 50 | [
"Find(x)"
] | [
"Equals(MeasureOf(Angle(D, F, C)), 145)",
"Equals(MeasureOf(Angle(B, F, D)), 165)",
"Equals(MeasureOf(Angle(C, F, B)), x)",
"",
"PointLiesOnCircle(B, Circle(F, radius_5_0))",
"PointLiesOnCircle(C, Circle(F, radius_5_0))",
"PointLiesOnCircle(D, Circle(F, radius_5_0))"
] | [
"BF",
"DF",
"FC"
] | {"B": [197.0, 240.0], "C": [259.0, 148.0], "D": [36.0, 42.0], "F": [134.0, 132.0]} | [
"F"
] | |
<image>If $R T=16, Q P=24,$ and $S T=9,$ find $P S$ | 13.5 | [
"Equals(LengthOf(Line(R,T)),16)",
"Equals(LengthOf(Line(Q,P)),24)",
"Equals(LengthOf(Line(S,T)),9)",
"Find(LengthOf(Line(P,S)))"
] | [
"PointLiesOnLine(T, Line(P, S))",
"PointLiesOnLine(R, Line(Q, S))",
"Parallel(Line(P, Q), Line(T, R))"
] | [
"PQ",
"QS",
"PS",
"QR",
"RS",
"PT",
"TS"
] | {"S": [209.0, 149.0], "P": [0.0, 148.0], "Q": [49.0, 8.0], "R": [96.0, 46.0], "T": [59.0, 148.0]} | [
""
] | |
<image>Find $CP$. | 8 | [
"Find(LengthOf(Line(C,P)))"
] | [
"Equals(LengthOf(Line(P, A)), 8)",
"PointLiesOnLine(A, Line(B, Y))",
"PointLiesOnLine(C, Line(B, D))",
"PointLiesOnLine(P, Line(B, E))",
"Perpendicular(Line(B, C), Line(P, C))",
"Perpendicular(Line(B, A), Line(P, A))",
"Equals(MeasureOf(Angle(A, B, P)), MeasureOf(Angle(P, B, C)))"
] | [
"AD",
"AY",
"BA",
"BC",
"BD",
"BE",
"BP",
"BY",
"CD",
"PA",
"PC",
"PE"
] | {"A": [39.0, 192.0], "B": [0.0, 1.0], "C": [169.0, 89.0], "D": [229.0, 121.0], "E": [164.0, 223.0], "P": [125.0, 169.0], "Y": [54.0, 259.0]} | [
""
] | |
<image>Circle $O$ has a radius of $10, A B=10$ and $m \overrightarrow{A B}=60 .$ Find $m\widehat{A Y}$ | 30 | [
"Equals(RadiusOf(Circle(O)),10)",
"Equals(LengthOf(Line(A,B)),10)",
"Equals(LengthOf(Line(A,B)),60)",
"Find(MeasureOf(Arc(A,Y)))"
] | [
"PointLiesOnLine(X, Line(A, B))",
"PointLiesOnLine(X, Line(Y, O))",
"PointLiesOnCircle(A, Circle(O, radius_2_0))",
"PointLiesOnCircle(Y, Circle(O, radius_2_0))",
"PointLiesOnCircle(B, Circle(O, radius_2_0))",
"Perpendicular(Line(B, X), Line(X, O))"
] | [
"AB",
"AX",
"BX",
"XY",
"XO",
"YO"
] | {"A": [5.5651567090242295, 60.264242572680914], "Y": [56.0, 15.0], "O": [90.25, 91.75], "B": [119.60284616281973, 6.656781718332603], "X": [63.98751244231291, 32.80074201429735]} | [
"O"
] | |
<image>Let $\overline{J K}$ be the median of $A B T Q .$ Find $J K$ | 78 | [
"IsMedianOf(Line(J,K),Quadrilateral(A,B,T,Q))",
"Find(LengthOf(Line(J,K)))"
] | [
"Equals(LengthOf(Line(Q, T)), 86)",
"Equals(LengthOf(Line(R, S)), 54)",
"PointLiesOnLine(A, Line(R, Q))",
"PointLiesOnLine(B, Line(S, T))",
"Equals(LengthOf(Line(Q, A)), LengthOf(Line(R, A)))",
"Equals(LengthOf(Line(B, T)), LengthOf(Line(S, B)))"
] | [
"AB",
"BT",
"QA",
"QT",
"RA",
"RQ",
"RS",
"SB",
"ST"
] | {"A": [17.0, 34.0], "B": [245.0, 35.0], "Q": [0.0, 65.0], "R": [36.0, 2.0], "S": [211.0, 2.0], "T": [274.0, 64.0]} | [
""
] | |
<image>Find the perimeter of each polygon for the given information.
$B Y=C Z=A X=2.5$
diameter of $\odot G=5$ | 15 \sqrt { 3 } | [
"Equals(LengthOf(Line(B,Y)),2.5)",
"Equals(LengthOf(Line(C,Z)),2.5)",
"Equals(LengthOf(Line(A,X)),2.5)",
"Equals(DiameterOf(Circle(G)),5)",
"Find(PerimeterOf(Polygon($)))"
] | [
"PointLiesOnLine(I, Line(C, G))",
"PointLiesOnLine(Z, Line(C, G))",
"PointLiesOnLine(D, Line(C, A))",
"PointLiesOnLine(G, Line(C, F))",
"PointLiesOnLine(I, Line(C, F))",
"PointLiesOnLine(Z, Line(C, F))",
"PointLiesOnLine(Z, Line(C, I))",
"PointLiesOnLine(E, Line(C, B))",
"PointLiesOnLine(Y, Line(G, ... | [
"AB",
"AD",
"AE",
"AF",
"AX",
"BE",
"BK",
"BY",
"CA",
"CB",
"CD",
"CE",
"CF",
"CG",
"CH",
"CI",
"CZ",
"DB",
"DK",
"DX",
"DY",
"EX",
"FB",
"FI",
"FX",
"FZ",
"GA",
"GB",
"GD",
"GE",
"GF",
"GH",
"GI",
"GK",
"GX",
"GY",
"GZ",
"HA",
"HB",
"HD"... | {"A": [123.0, 2.0], "B": [246.0, 214.0], "C": [0.0, 214.0], "D": [61.0, 105.0], "E": [122.0, 211.0], "F": [184.0, 104.0], "G": [124.0, 142.0], "H": [216.0, 195.0], "I": [91.0, 161.0], "K": [97.0, 128.0], "X": [124.0, 70.0], "Y": [185.0, 178.0], "Z": [61.0, 177.0]} | [
"G"
] | |
<image>In the figure, $a ∥ b, c ∥ d$, and $m∠4 = 57$. Find the measure of $\angle 1$. | 123 | [
"Parallel(Line(a),Line(b))",
"Parallel(Line(c),Line(d))",
"Equals(MeasureOf(Angle(4)),57)",
"Find(MeasureOf(Angle(1)))"
] | [
"Equals(MeasureOf(Angle(I, F, A)), MeasureOf(angle 1))",
"Equals(MeasureOf(Angle(A, F, E)), MeasureOf(angle 2))",
"Equals(MeasureOf(Angle(I, E, B)), MeasureOf(angle 3))",
"Equals(MeasureOf(Angle(B, E, C)), MeasureOf(angle 4))",
"Equals(MeasureOf(Angle(D, F, I)), MeasureOf(angle 8))",
"Equals(MeasureOf(Ang... | [
"BE",
"BH",
"BK",
"EF",
"EH",
"EI",
"EC",
"EK",
"HD",
"HG",
"HJ",
"HK",
"DF",
"DG",
"DJ",
"DA",
"DL",
"FI",
"FC",
"FA",
"FL",
"GJ",
"AL",
"IC"
] | {"B": [404.0, 1.0], "D": [98.43010752688173, 141.0], "E": [353.10239713367645, 45.65544402422728], "F": [207.542468791765, 45.269343418545816], "G": [0.0, 141.0], "H": [244.43010752688173, 141.0], "I": [106.0, 45.0], "J": [377.0, 141.0], "C": [483.0, 46.0], "K": [192.0, 187.0], "A": [258.0, 1.0], "L": [46.0, 187.0]} | [] | |
<image>Find y | 30 | [
"Find(y)"
] | [
"Equals(MeasureOf(Angle(A, B, C)), y)",
"Equals(LengthOf(Line(A, B)), 20)",
"Equals(LengthOf(Line(B, C)), x)",
"Equals(LengthOf(Line(A, C)), 10)",
"Perpendicular(Line(B, C), Line(A, C))"
] | [
"AB",
"AC",
"BC"
] | {"A": [114.0, 194.0], "B": [0.0, 0.0], "C": [1.9899592944369076, 192.6840343735866]} | [] | |
<image>Find $m \angle T$. | 55 | [
"Find(MeasureOf(Angle(T)))"
] | [
"Equals(MeasureOf(Angle(S, U, V)), 2x+15)",
"Equals(MeasureOf(Angle(S, T, B)), 3x-5)",
"PointLiesOnLine(B, Line(V, T))",
"PointLiesOnLine(B, Line(U, S))",
"PointLiesOnCircle(V, Circle(A, radius_4_0))",
"PointLiesOnCircle(T, Circle(A, radius_4_0))",
"PointLiesOnCircle(S, Circle(A, radius_4_0))",
"Point... | [
"BS",
"TB",
"TS",
"UB",
"US",
"VB",
"VT",
"VU"
] | {"A": [168.0, 160.0], "B": [147.0, 93.0], "S": [69.0, 281.0], "T": [82.0, 28.0], "U": [184.0, 3.0], "V": [303.0, 241.0]} | [
"A"
] | |
<image>Find $m\angle B$ | 120 | [
"Find(MeasureOf(Angle(B)))"
] | [
"Equals(MeasureOf(Angle(B, C, D)), x+25)",
"Equals(MeasureOf(Angle(C, D, E)), 2x-22)",
"Equals(MeasureOf(Angle(C, B, A)), 1.5x+3)",
"Equals(MeasureOf(Angle(B, A, E)), x+27)",
"Equals(MeasureOf(Angle(A, E, D)), x)"
] | [
"AB",
"BC",
"CD",
"EA",
"ED"
] | {"A": [17.0, 126.0], "B": [2.0, 70.0], "C": [47.0, 8.0], "D": [114.0, 46.0], "E": [133.0, 128.0]} | [
""
] | |
<image>Find $m \angle 3$. | 51 | [
"Find(MeasureOf(Angle(3)))"
] | [
"Equals(MeasureOf(Angle(A, B, F)), 48)",
"Equals(MeasureOf(Angle(F, A, B)), MeasureOf(angle 1))",
"Equals(MeasureOf(Angle(C, F, D)), MeasureOf(angle 2))",
"Equals(MeasureOf(Angle(F, D, C)), MeasureOf(angle 3))",
"Equals(MeasureOf(Angle(E, D, F)), 39)",
"PointLiesOnLine(B, Line(F, C))",
"PointLiesOnLine(... | [
"BA",
"CB",
"DC",
"DE",
"DF",
"EA",
"FA",
"FB",
"FC",
"FE"
] | {"A": [82.0, 1.0], "B": [198.0, 110.0], "C": [410.0, 108.0], "D": [409.0, 375.0], "E": [81.0, 374.0], "F": [82.0, 108.0]} | [
""
] | |
<image>Find x. | 7 \sqrt { 2 } | [
"Find(x)"
] | [
"Equals(LengthOf(Line(A, B)), y)",
"Equals(MeasureOf(Angle(C, B, A)), 45)",
"Equals(LengthOf(Line(A, C)), x)",
"Equals(LengthOf(Line(B, C)), 7\\sqrt{2})",
"Perpendicular(Line(A, C), Line(B, C))"
] | [
"AB",
"AC",
"BC"
] | {"A": [0.0, 0.0], "B": [275.0, 34.0], "C": [125.0, 156.0]} | [
""
] | |
<image>Each pair of polygons is similar. Find y | 48.45 | [
"Similar(Polygon($1),Polygon($2))",
"Find(y)"
] | [
"Equals(LengthOf(Line(R, T)), y+3)",
"Equals(LengthOf(Line(T, S)), 49)",
"Equals(LengthOf(Line(R, S)), x)",
"Equals(LengthOf(Line(W, V)), 20)",
"Equals(LengthOf(Line(U, V)), 21)",
"Equals(LengthOf(Line(W, U)), 29)",
"Equals(MeasureOf(Angle(R, T, S)), MeasureOf(Angle(W, V, U)))",
"Equals(MeasureOf(Angl... | [
"RT",
"SR",
"ST",
"UV",
"WU",
"WV"
] | {"R": [0.0, 0.0], "S": [257.0, 245.0], "T": [256.0, 1.0]} | [
""
] | |
<image>Find $CD$ if $AB=6, AF=8, BC=x, CD=y$, $DE=2y-3,$ and $FE=x+\frac{10}{3}$ | 9 | [
"Equals(LengthOf(Line(A,B)),6)",
"Equals(LengthOf(Line(A,F)),8)",
"Equals(LengthOf(Line(B,C)),x)",
"Equals(LengthOf(Line(C,D)),y)",
"Equals(LengthOf(Line(D,E)),2y-3)",
"Equals(LengthOf(Line(F,E)),x+\\frac{10}{3})",
"Find(LengthOf(Line(C,D)))"
] | [
"PointLiesOnLine(D, Line(E, C))",
"PointLiesOnLine(F, Line(E, A))",
"PointLiesOnLine(B, Line(C, A))",
"Parallel(Line(C, D), Line(F, B))",
"Parallel(Line(C, B), Line(F, D))"
] | [
"AB",
"CA",
"CB",
"CD",
"EA",
"EC",
"ED",
"EF",
"FA",
"FB",
"FD"
] | {"A": [73.0, 1.0], "B": [44.0, 66.0], "C": [0.0, 166.0], "D": [93.0, 166.0], "E": [228.0, 166.0], "F": [134.0, 68.0]} | [
""
] | |
<image>Find the measure of $ \angle 1$ in the figure. | 80 | [
"Find(MeasureOf(Angle(1)))"
] | [
"Equals(MeasureOf(Angle(A, D, C)), 100)",
"Equals(MeasureOf(Angle(C, D, B)), MeasureOf(angle 1))",
"Equals(MeasureOf(Angle(A, B, E)), MeasureOf(angle 3))",
"Equals(MeasureOf(Angle(E, F, C)), 75)",
"Equals(MeasureOf(Angle(C, F, B)), MeasureOf(angle 2))",
"PointLiesOnLine(D, Line(A, B))",
"PointLiesOnLine... | [
"AB",
"AD",
"BD",
"BE",
"BF",
"CG",
"DC",
"DF",
"DG",
"EF",
"FC",
"FG"
] | {"A": [175.0, 3.0], "B": [0.0, 133.0], "C": [147.0, 184.0], "D": [90.0, 66.0], "E": [220.0, 97.0], "F": [113.0, 114.0], "G": [58.0, 0.0]} | [
""
] | |
<image>Find $x$ so that the quadrilateral is a parallelogram | 3 | [
"Parallelogram($)",
"Find(x)"
] | [
"Equals(LengthOf(Line(C, D)), x+5)",
"Equals(LengthOf(Line(B, D)), 2y+5)",
"Equals(LengthOf(Line(A, C)), y+10)",
"Equals(LengthOf(Line(A, B)), 2x+2)"
] | [
"AB",
"AC",
"BD",
"CD"
] | {"A": [0.0, 226.0], "B": [263.0, 169.0], "C": [1.0, 59.0], "D": [264.0, 1.0]} | [
""
] | |
<image>What is the area of the shaded region if $r=4$. | 64 - 16 \pi | [
"Equals(r,4)",
"Find(AreaOf(Shaded(Shape($))))"
] | [
"PointLiesOnLine(B, Line(L, P))",
"PointLiesOnLine(B, Line(Q, M))",
"PointLiesOnCircle(L, Circle(B, radius_6_0))",
"PointLiesOnCircle(M, Circle(B, radius_6_0))",
"PointLiesOnCircle(K, Circle(B, radius_6_0))",
"PointLiesOnCircle(P, Circle(B, radius_6_0))",
"PointLiesOnCircle(Q, Circle(B, radius_6_0))"
] | [
"BK",
"BM",
"BQ",
"LB",
"LP",
"MQ",
"PB"
] | {} | [
"B"
] | |
<image>Rhombus $RSTU$ has an area of 675 square meters. Find $SU$. | 45 | [
"Equals(AreaOf(Rhombus(R,S,T,U)),675)",
"Find(LengthOf(Line(S,U)))"
] | [
"Equals(LengthOf(Line(R, B)), 15)",
"Equals(LengthOf(Line(T, B)), 15)",
"PointLiesOnLine(B, Line(A, U))",
"PointLiesOnLine(B, Line(R, T))",
"Perpendicular(Line(U, B), Line(T, B))"
] | [
"AB",
"AR",
"AT",
"AU",
"RB",
"RT",
"RU",
"TB",
"UB",
"UT"
] | {"A": [245.0, 1.0], "B": [123.0, 87.0], "R": [59.0, 1.0], "T": [186.0, 172.0], "U": [1.0, 174.0]} | [
""
] | |
<image>Use parallelogram $JKLM$ to find JK | 6 | [
"Parallelogram(J,K,L,M)",
"Find(LengthOf(Line(J,K)))"
] | [
"Equals(MeasureOf(Angle(L, K, J)), 109)",
"Equals(LengthOf(Line(L, M)), 6)"
] | [
"KJ",
"LK",
"LM",
"MJ"
] | {"J": [1.0, 1.0], "K": [282.0, 28.0], "L": [355.0, 213.0], "M": [77.0, 186.0]} | [
""
] | |
<image>Find x in the figure. | 16 | [
"Find(x)"
] | [
"Equals(MeasureOf(Angle(Y, Q, F)), 4x-5)",
"Equals(MeasureOf(Angle(E, Y, Q)), 3y+1)",
"Equals(MeasureOf(Angle(M, A, Q)), 3x+11)",
"PointLiesOnLine(Y, Line(E, M))",
"PointLiesOnLine(Q, Line(F, A))",
"Parallel(Line(E, F), Line(Y, Q))",
"Parallel(Line(Y, Q), Line(M, A))",
"Parallel(Line(M, A), Line(E, F)... | [
"EF",
"YQ",
"AM",
"EY",
"YM",
"EM",
"FQ",
"QA",
"AF",
""
] | {"A": [378.0, 120.0], "E": [1.0, 8.0], "F": [311.0, 7.0], "M": [66.0, 119.0], "Q": [345.0, 66.0], "Y": [37.0, 67.0]} | [
""
] | |
<image>$\overline{QS}$ is a diameter of $\odot V$. Find $m \widehat {STP}$. | 147 | [
"IsDiameterOf(Line(Q,S),Circle(V))",
"Find(MeasureOf(Arc(S,T,P)))"
] | [
"Equals(MeasureOf(Angle(T, V, S)), 75)",
"Equals(MeasureOf(Angle(P, V, T)), 72)",
"PointLiesOnLine(V, Line(S, Q))",
"PointLiesOnCircle(S, Circle(V, radius_1_0))",
"PointLiesOnCircle(P, Circle(V, radius_1_0))",
"PointLiesOnCircle(R, Circle(V, radius_1_0))",
"PointLiesOnCircle(T, Circle(V, radius_1_0))",
... | [
"QP",
"SQ",
"SV",
"VP",
"VQ",
"VR",
"VT"
] | {"P": [25.0, 235.0], "Q": [1.0, 150.0], "R": [151.0, -0.0], "S": [304.0, 153.0], "T": [191.0, 292.0], "V": [156.0, 149.0]} | [
"V"
] |
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