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 $x$ in the figure. | 90 | [
"Find(x)"
] | [
"Equals(MeasureOf(Angle(E, B, A)), 4z+2)",
"Equals(MeasureOf(Angle(J, B, L)), x)",
"Equals(MeasureOf(Angle(J, B, A)), 3y-11)",
"Equals(MeasureOf(Angle(E, A, B)), y+19)",
"PointLiesOnLine(B, Line(L, E))",
"PointLiesOnLine(A, Line(E, I))",
"PointLiesOnLine(B, Line(F, H))",
"Perpendicular(Line(E, B), Lin... | [
"LB",
"BE",
"EG",
"LE",
"LG",
"BG",
"FB",
"FA",
"FH",
"BH",
"BA",
"AH",
"MB",
"MJ",
"BJ",
"EA",
"EI",
"AI"
] | {"A": [206.0, 133.0], "B": [77.0, 49.0], "C": [77.0, 195.0], "D": [205.0, 194.0], "E": [206.0, 49.0], "F": [0.0, 0.0], "G": [284.0, 50.0], "H": [284.0, 183.0], "I": [204.0, 242.0], "J": [76.0, 240.0], "K": [123.0, 56.0], "L": [0.0, 49.0], "M": [77.0, 2.0], "N": [103.0, 57.0]} | [
""
] | |
<image>A square with side length of $9$ inches is inscribed in $\odot J$. Find the exact circumference of $\odot J$. | 9 \sqrt 2 \pi | [
"Square($)",
"Equals(SideOf(Square($)),9)",
"Circle(J)",
"Find(CircumferenceOf(Circle(J)))"
] | [
"Equals(LengthOf(Line(N, S)), 9)",
"",
"PointLiesOnCircle(P, Circle(J, radius_5_0))",
"PointLiesOnCircle(S, Circle(J, radius_5_0))",
"PointLiesOnCircle(B, Circle(J, radius_5_0))",
"PointLiesOnCircle(N, Circle(J, radius_5_0))",
""
] | [
"BN",
"BJ",
"NS",
"JS",
"PB",
"SP"
] | {"B": [331.0, 214.0], "J": [174.0, 168.0], "N": [124.0, 327.0], "P": [217.0, 7.0], "S": [11.0, 122.0]} | [
"J"
] | |
<image>Find $x$. | 3 | [
"Find(x)"
] | [
"Equals(LengthOf(Line(A, C)), 5x-1)",
"Equals(LengthOf(Line(A, B)), \\frac{10}{3}x+4)",
"Equals(LengthOf(Line(B, C)), 7x-7)",
"Equals(MeasureOf(Angle(B, A, C)), MeasureOf(Angle(A, B, C)))",
"Equals(MeasureOf(Angle(A, C, B)), MeasureOf(Angle(A, B, C)))"
] | [
"AB",
"AC",
"BC"
] | {"A": [178.0, 2.0], "B": [0.0, 289.0], "C": [354.0, 289.0]} | [
""
] | |
<image>If QRST is a parallelogram, find $x$ | 5.4 | [
"Parallelogram(Q,R,S,T)",
"Find(x)"
] | [
"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>For trapezoid QRST, M and P are midpoints of the legs. If TS = 2x, PM = 20, and QR = 6x, find x. | 5 | [
"Trapezoid(Q,R,S,T)",
"IsMidpointOf(Point(M),LegOf(Trapezoid(Q,R,S,T)))",
"IsMidpointOf(Point(P),LegOf(Trapezoid(Q,R,S,T)))",
"Equals(LengthOf(Line(T,S)),2x)",
"Equals(LengthOf(Line(P,M)),20)",
"Equals(LengthOf(Line(Q,R)),6x)",
"Find(x)"
] | [
"PointLiesOnLine(P, Line(Q, T))",
"PointLiesOnLine(M, Line(R, S))"
] | [
"MS",
"PM",
"QP",
"QR",
"QT",
"RM",
"RS",
"TP",
"TS"
] | {"M": [172.0, 68.0], "P": [25.0, 70.0], "Q": [0.0, 0.0], "R": [197.0, 1.0], "S": [145.0, 140.0], "T": [50.0, 139.0]} | [
""
] | |
<image>Find the perimeter of the parallelogram. Round to the nearest tenth if necessary. | 80 | [
"Find(PerimeterOf(Parallelogram($)))"
] | [
"Equals(LengthOf(Line(E, D)), 10)",
"Equals(LengthOf(Line(C, D)), 30)",
"Equals(MeasureOf(Angle(C, E, D)), 60)"
] | [
"BC",
"BE",
"CD",
"ED"
] | {"B": [310.0, 91.0], "C": [360.0, 4.0], "D": [54.0, 1.0], "E": [0.0, 91.0]} | [
""
] | |
<image>Find x. | 14 | [
"Find(x)"
] | [
"Equals(LengthOf(Line(B, C)), x)",
"Equals(LengthOf(Line(A, C)), 7\\sqrt{2})",
"Equals(MeasureOf(Angle(C, B, A)), 45)",
"Perpendicular(Line(A, C), Line(A, B))"
] | [
"AB",
"AC",
"BC"
] | {"A": [155.0, 155.0], "B": [0.0, 156.0], "C": [158.0, 1.0]} | [
""
] | |
<image>Find $m \widehat {PO}$. | 112 | [
"Find(MeasureOf(Arc(P,O)))"
] | [
"Equals(MeasureOf(Angle(P, N, O)), 56)",
"Equals(MeasureOf(Angle(N, B, M)), 70)",
"PointLiesOnCircle(O, Circle(B, radius_5_0))",
"PointLiesOnCircle(N, Circle(B, radius_5_0))",
"PointLiesOnCircle(P, Circle(B, radius_5_0))",
"PointLiesOnCircle(M, Circle(B, radius_5_0))"
] | [
"NP",
"ON",
"PM"
] | {"B": [151.0, 150.0], "M": [4.0, 118.0], "N": [88.0, 287.0], "O": [288.0, 214.0], "P": [165.0, 1.0]} | [
"B"
] | |
<image>Find the perimeter of the triangle. Round to the nearest tenth if necessary. | 36 | [
"Find(PerimeterOf(Triangle($)))"
] | [
"Equals(LengthOf(Line(A, B)), 15)",
"Equals(LengthOf(Line(A, C)), 9)",
"Equals(LengthOf(Line(B, C)), 12)",
"Perpendicular(Line(A, C), Line(B, C))"
] | [
"AB",
"AC",
"BC"
] | {"A": [160.0, 1.0], "B": [1.0, 118.0], "C": [159.0, 118.0]} | [
""
] | |
<image>Find $m\angle D$ | 117 | [
"Find(MeasureOf(Angle(D)))"
] | [
"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>Points $P$, $Q$, and $R$ are the midpoints of $\overline{J K}$, $\overline{K L}$, and $\overline{J L}$, respectively. Find $x$ . | 5.5 | [
"IsMidpointOf(Point(P),Line(J,K))",
"IsMidpointOf(Point(Q),Line(K,L))",
"IsMidpointOf(Point(R),Line(J,L))",
"Find(x)"
] | [
"Equals(LengthOf(Line(D, R)), 28)",
"Equals(LengthOf(Line(K, D)), 6x+23)",
"PointLiesOnLine(Q, Line(L, K))",
"PointLiesOnLine(R, Line(L, J))",
"PointLiesOnLine(D, Line(L, P))",
"PointLiesOnLine(D, Line(K, R))",
"PointLiesOnLine(D, Line(J, Q))"
] | [
"DP",
"DQ",
"JD",
"JP",
"JQ",
"KD",
"KJ",
"KP",
"KQ",
"KR",
"LD",
"LJ",
"LK",
"LP",
"LQ",
"LR",
"RD",
"RJ"
] | {"D": [180.0, 136.0], "J": [380.0, 222.0], "K": [0.0, 194.0], "L": [165.0, 1.0], "P": [189.0, 202.0], "Q": [83.0, 96.0], "R": [269.0, 107.0]} | [
""
] | |
<image>Find $h$ in each triangle. | \frac { 15 \sqrt 2 } { 2 } | [
"Triangle($)",
"Find(h)"
] | [
"Equals(LengthOf(Line(B, C)), LengthOf(h))",
"Equals(MeasureOf(Angle(C, A, B)), 45)",
"Equals(LengthOf(Line(A, B)), 15)",
"Perpendicular(Line(A, C), Line(B, C))"
] | [
"AB",
"AC",
"BC"
] | {"A": [1.0, 123.0], "B": [249.0, 124.0], "C": [128.0, 3.0]} | [
""
] | |
<image>Find $XY$. | 7 | [
"Find(LengthOf(Line(X,Y)))"
] | [
"Equals(LengthOf(Line(W, X)), 7)",
"Equals(MeasureOf(Angle(X, Z, D)), 41)",
"Equals(MeasureOf(Angle(B, Z, C)), 41)",
"PointLiesOnLine(Y, Line(Z, B))",
"PointLiesOnLine(X, Line(Z, C))",
"PointLiesOnLine(W, Line(Z, D))",
"Perpendicular(Line(W, D), Line(W, X))"
] | [
"ZB",
"ZC",
"ZD",
"ZW",
"ZX",
"ZY",
"BY",
"WD",
"WX",
"XC",
"XY"
] | {"Z": [276.0, 290.0], "B": [14.0, 288.0], "C": [0.0, 51.0], "D": [235.0, 8.0], "W": [257.0, 155.0], "X": [140.0, 172.0], "Y": [141.0, 290.0]} | [
""
] | |
<image>Find the area of the shaded region. Assume that all polygons that appear to be regular are regular. Round to the nearest tenth. | 4.1 | [
"Regular(Polygon($))",
"Find(AreaOf(Shaded(Shape($))))"
] | [
"Equals(LengthOf(Line(B, D)), 1.5)",
"PointLiesOnLine(E, Line(B, D))",
"PointLiesOnCircle(A, Circle(D, radius_3_0))",
"PointLiesOnCircle(B, Circle(D, radius_3_0))",
"PointLiesOnCircle(C, Circle(D, radius_3_0))"
] | [
"AB",
"AC",
"BC",
"BD",
"BE",
"DE"
] | {"A": [43.0, 13.0], "B": [161.0, 82.0], "C": [43.0, 153.0], "D": [78.0, 82.0], "E": [125.0, 82.0]} | [
"D"
] | |
<image>Find the scale factor from $W$ to $W'$. | 0.75 | [
"Find(ScaleFactorOf(Shape(W),Shape(W')))"
] | [
"Equals(LengthOf(Line(W, W')), x)",
"Equals(LengthOf(Line(W', F)), 10.5)",
"Equals(LengthOf(Line(W, F)), 14)",
"PointLiesOnLine(C, Line(A, F))",
"PointLiesOnLine(X, Line(F, W))"
] | [
"W'C",
"W'F",
"W'R",
"CB",
"FC",
"FW",
"FW'",
"RB",
"RW",
"XB",
"XW"
] | {"W'": [326.0, 66.0], "B": [151.0, 57.0], "C": [295.0, 104.0], "F": [197.0, 216.0], "R": [135.0, 1.0], "W": [2.0, 149.0], "X": [52.0, 168.0]} | [
""
] | |
<image>Find $U B$ if $\triangle R S T \sim \triangle U V W, \overline{T A}$ and $\overline{WB}$ are medians, $T A=8$, $R A=3, W B=3 x-6,$ and $U B=x+2$ | 36 | [
"Similar(Triangle(R,S,T),Triangle(U,V,W))",
"IsMedianOf(Line(T,A),Triangle(U,V,W))",
"IsMedianOf(Line(W,B),Triangle(U,V,W))",
"Equals(LengthOf(Line(T,A)),8)",
"Equals(LengthOf(Line(R,A)),3)",
"Equals(LengthOf(Line(W,B)),3x-6)",
"Equals(LengthOf(Line(U,B)),x+2)",
"Find(LengthOf(Line(U,B)))"
] | [
"PointLiesOnLine(B, Line(U, V))",
"PointLiesOnLine(A, Line(R, S))"
] | [
"UV",
"BU",
"BV",
"BW",
"WU",
"WV"
] | {"U": [0.0, 95.0], "B": [120.0, 105.0], "V": [235.0, 115.0], "W": [69.0, 1.0]} | [
""
] | |
<image>$UJ=9, VJ=3$, and $ZT=18$. Find $JT$. | 6 | [
"Equals(LengthOf(Line(U,J)),9)",
"Equals(LengthOf(Line(V,J)),3)",
"Equals(LengthOf(Line(Z,T)),18)",
"Find(LengthOf(Line(J,T)))"
] | [
"Equals(LengthOf(Line(U, J)), 9)",
"Equals(LengthOf(Line(J, V)), 3)",
"PointLiesOnLine(V, Line(Z, U))",
"PointLiesOnLine(Y, Line(Z, S))",
"PointLiesOnLine(J, Line(Z, T))",
"PointLiesOnLine(J, Line(U, Y))",
"PointLiesOnLine(T, Line(U, S))",
"PointLiesOnLine(J, Line(S, V))",
"Equals(LengthOf(Line(S, T... | [
"JS",
"JT",
"JV",
"JY",
"ST",
"SV",
"UJ",
"US",
"UT",
"UV",
"UY",
"YS",
"ZJ",
"ZS",
"ZT",
"ZU",
"ZV",
"ZY"
] | {"J": [197.0, 164.0], "S": [27.0, 2.0], "T": [287.0, 60.0], "U": [546.0, 119.0], "V": [282.0, 245.0], "Y": [21.0, 186.0], "Z": [14.0, 372.0]} | [
""
] | |
<image>If $MNPQ \sim XYZW,$ find the perimeter of $XYZW$. | 17 | [
"Similar(Quadrilateral(M,N,P,Q),Quadrilateral(X,Y,Z,W))",
"Find(PerimeterOf(Quadrilateral(X,Y,Z,W)))"
] | [
"Equals(LengthOf(Line(N, P)), 10)",
"Equals(LengthOf(Line(W, X)), 4)",
"Equals(LengthOf(Line(Q, P)), 7)",
"Equals(LengthOf(Line(Q, M)), 8)",
"Equals(LengthOf(Line(N, M)), 9)"
] | [
"NM",
"PN",
"PQ",
"QM",
"WX",
"XY",
"YZ",
"ZW"
] | {"M": [1.0, 232.0], "N": [188.0, 1.0], "P": [512.0, 102.0], "Q": [300.0, 198.0], "W": [-200.0, 5.0], "X": [-100.0, 1.0], "Y": [-150.0, 150.0], "Z": [-300.0, 75.0]} | [
""
] | |
<image>Find the area of the figure. | 64 | [
"Find(AreaOf(Shape($)))"
] | [
"Equals(LengthOf(Line(E, H)), 8)",
"Equals(LengthOf(Line(D, E)), 8)",
"Perpendicular(Line(D, K), Line(H, K))",
"Perpendicular(Line(K, H), Line(E, H))",
"Perpendicular(Line(D, E), Line(H, E))",
"Perpendicular(Line(K, D), Line(E, D))"
] | [
"DK",
"ED",
"HE",
"KH"
] | {"D": [0.0, 175.0], "E": [174.0, 173.0], "H": [173.0, 1.0], "K": [2.0, 0.0]} | [
""
] | |
<image>Find x | 5 \sqrt { 2 } | [
"Find(y)"
] | [
"Equals(LengthOf(Line(A, C)), 10)",
"Equals(LengthOf(Line(B, C)), x)",
"Equals(MeasureOf(Angle(B, A, C)), 45)",
"Equals(LengthOf(Line(A, B)), y)",
"Perpendicular(Line(B, C), Line(A, B))"
] | [
"AB",
"AC",
"BC"
] | {"A": [159.67078189300412, 160.33539094650206], "B": [1.3332815775172762, 160.99583365680718], "C": [0.0, 0.0]} | [] | |
<image>Find $m \angle 5$. | 85 | [
"Find(MeasureOf(Angle(5)))"
] | [
"Equals(MeasureOf(Angle(C, D, E)), MeasureOf(angle 6))",
"Equals(MeasureOf(Angle(A, C, D)), 36)",
"Equals(MeasureOf(Angle(B, C, E)), 42)",
"Equals(MeasureOf(Angle(C, E, B)), MeasureOf(angle 5))",
"Equals(MeasureOf(Angle(B, E, A)), MeasureOf(angle 4))",
"Equals(MeasureOf(Angle(E, B, C)), MeasureOf(angle 7)... | [
"AB",
"AC",
"AE",
"BC",
"BD",
"BE",
"CD",
"CE",
"ED"
] | {"A": [0.0, 321.0], "B": [381.0, 265.0], "C": [440.0, 1.0], "D": [154.0, 2.0], "E": [264.0, 129.0]} | [
""
] | |
<image>Quadrilateral $WXZY$ is a rhombus. If $m \angle XZY=56$, find $m \angle YWZ$. | 28 | [
"Rhombus(W,X,Z,Y)",
"Equals(MeasureOf(Angle(X,Z,Y)),56)",
"Find(MeasureOf(Angle(Y,W,Z)))"
] | [
"Equals(MeasureOf(Angle(Y, X, W)), MeasureOf(angle 1))",
"Equals(MeasureOf(Angle(Z, X, T)), MeasureOf(angle 2))",
"Equals(MeasureOf(Angle(W, T, X)), MeasureOf(angle 3))",
"PointLiesOnLine(T, Line(Z, W))",
"PointLiesOnLine(T, Line(Y, X))"
] | [
"WT",
"WX",
"XT",
"YT",
"YW",
"YX",
"ZT",
"ZW",
"ZX",
"ZY"
] | {"T": [182.0, 103.0], "W": [2.0, 0.0], "X": [267.0, 2.0], "Y": [96.0, 204.0], "Z": [361.0, 204.0]} | [
""
] | |
<image>Find the area of the figure in feet. Round to the nearest tenth, if necessary. | 14.4 | [
"Find(AreaOf(Shape($)))"
] | [
"Equals(MeasureOf(Angle(A, B, E)), 58)",
"Equals(LengthOf(Line(B, C)), 4)",
"PointLiesOnLine(E, Line(A, C))",
"PointLiesOnLine(E, Line(B, T))",
"Equals(LengthOf(Line(B, E)), LengthOf(Line(E, T)))",
"Equals(LengthOf(Line(C, E)), LengthOf(Line(E, A)))"
] | [
"AB",
"AC",
"AT",
"BC",
"BE",
"BT",
"CE",
"EA",
"ET",
"TC"
] | {"A": [256.0, 79.0], "B": [129.0, 1.0], "C": [0.0, 80.0], "E": [128.0, 40.0], "T": [125.0, 157.0]} | [
""
] | |
<image>Find $x$. | 21 | [
"Find(x)"
] | [
"Equals(LengthOf(Line(B, A)), 7)",
"Equals(LengthOf(Line(E, D)), 7)",
"Equals(LengthOf(MeasureOf( Arc(A,O, B)) ), 105)",
"Equals(LengthOf(MeasureOf( Arc(E,O, D)) ), 5x)",
"PointLiesOnCircle(E, Circle(O, radius_4_0))",
"PointLiesOnCircle(B, Circle(O, radius_4_0))",
"PointLiesOnCircle(D, Circle(O, radius_... | [
"BA",
"ED"
] | {"A": [29.0, 61.0], "B": [271.0, 74.0], "D": [264.0, 227.0], "E": [19.0, 213.0], "O": [147.0, 145.0]} | [
"O"
] | |
<image>In $\odot P, P D=10, P Q=10,$ and $Q E=20$, find PE | 10 \sqrt { 5 } | [
"Circle(P)",
"Equals(LengthOf(Line(P,D)),10)",
"Equals(LengthOf(Line(P,Q)),10)",
"Equals(LengthOf(Line(Q,E)),20)",
"Find(LengthOf(Line(P,E)))"
] | [
"PointLiesOnLine(Q, Line(F, P))",
"PointLiesOnLine(D, Line(P, G))",
"PointLiesOnLine(D, Line(A, B))",
"PointLiesOnLine(Q, Line(E, C))",
"PointLiesOnCircle(F, Circle(P, radius_2_0))",
"PointLiesOnCircle(C, Circle(P, radius_2_0))",
"PointLiesOnCircle(E, Circle(P, radius_2_0))",
"PointLiesOnCircle(A, Cir... | [
"AB",
"DA",
"DB",
"DG",
"EC",
"EQ",
"FP",
"FQ",
"PD",
"PG",
"PQ",
"QC"
] | {"A": [100.0, 2.0], "B": [3.0, 78.0], "C": [3.0, 101.0], "D": [58.0, 44.0], "E": [103.0, 177.0], "F": [36.0, 159.0], "G": [35.0, 20.0], "P": [92.0, 88.0], "Q": [52.0, 138.0]} | [
"P"
] | |
<image>If $WXYZ$ is a kite, find $ZY$. | 8 \sqrt { 10 } | [
"Kite(W,X,Y,Z)",
"Find(LengthOf(Line(Z,Y)))"
] | [
"Equals(LengthOf(Line(X, P)), 8)",
"Equals(LengthOf(Line(W, P)), 6)",
"Equals(LengthOf(Line(Y, P)), 24)",
"Equals(LengthOf(Line(Z, P)), 8)",
"PointLiesOnLine(P, Line(Z, X))",
"PointLiesOnLine(P, Line(Y, W))"
] | [
"WP",
"XP",
"XW",
"YP",
"YW",
"YX",
"ZP",
"ZW",
"ZX",
"ZY"
] | {"P": [60.0, 76.0], "W": [1.0, 76.0], "X": [60.0, 1.0], "Y": [275.0, 77.0], "Z": [60.0, 157.0]} | [
""
] | |
<image>Find the area of the regular polygon. Round to the nearest tenth. | 15.6 | [
"Find(AreaOf(Regular(Polygon($))))"
] | [
"Equals(LengthOf(Line(A, B)), 6)"
] | [
"AB",
"AC",
"BC"
] | {"A": [2.0, 181.0], "B": [180.0, 92.0], "C": [0.0, 1.0]} | [
""
] | |
<image>If $m \angle AFB = 8x - 6$ and $m \angle BFC = 14x + 8$, find the value of $x$ so that $\angle AFC$ is a right angle. | 4 | [
"Equals(MeasureOf(Angle(A,F,B)),8x-6)",
"Equals(MeasureOf(Angle(B,F,C)),14x+8)",
"RightAngle(Angle(A,F,C))",
"Find(x)"
] | [
"PointLiesOnLine(F, Line(B, E))",
"PointLiesOnLine(F, Line(A, D))",
""
] | [
"AD",
"AF",
"BE",
"BF",
"CF",
"FD",
"FE"
] | {"A": [51.0, 269.0], "B": [49.0, 202.0], "C": [144.0, 45.0], "D": [369.0, 128.0], "E": [359.0, 199.0], "F": [217.0, 198.0]} | [
""
] | |
<image>Find $x$. | 170 | [
"Find(x)"
] | [
"Equals(MeasureOf(Angle(B, C, A)), 130)",
"Equals(MeasureOf(Angle(B, C, D)), 60)",
"Equals(MeasureOf(Angle(A, C, D)), x)",
"",
"PointLiesOnCircle(A, Circle(C, radius_2_0))",
"PointLiesOnCircle(B, Circle(C, radius_2_0))",
"PointLiesOnCircle(D, Circle(C, radius_2_0))"
] | [
"AC",
"CB",
"CD",
""
] | {"A": [290.0, 172.0], "B": [73.0, 21.0], "C": [145.0, 146.0], "D": [3.0, 144.0]} | [
"C"
] | |
<image>Find $x$. | 11.2 | [
"Find(x)"
] | [
"Equals(LengthOf(Line(B, C)), 14)",
"Equals(LengthOf(Line(F, C)), 11)",
"Equals(LengthOf(Line(B, D)), x)",
"Equals(LengthOf(Line(F, B)), 20)",
"PointLiesOnLine(D, Line(B, F))",
"Equals(MeasureOf(Angle(F, C, D)), MeasureOf(Angle(D, C, B)))"
] | [
"BC",
"BD",
"CD",
"FB",
"FC",
"FD"
] | {"B": [344.0, 98.0], "C": [87.0, 2.0], "D": [174.0, 160.0], "F": [1.0, 219.0]} | [
""
] | |
<image>Find the area of the figure. Round to the nearest hundredth, if necessary. | 4 | [
"Find(AreaOf(Shape($)))"
] | [
"Equals(LengthOf(Line(C, N)), 2\\sqrt{2})",
"Equals(LengthOf(Line(C, B)), LengthOf(Line(B, N)))",
"Equals(LengthOf(Line(B, N)), LengthOf(Line(E, N)))",
"Equals(LengthOf(Line(E, N)), LengthOf(Line(C, E)))"
] | [
"BC",
"CE",
"CN",
"EN",
"NB"
] | {"B": [11.0, 196.0], "C": [9.0, 8.0], "E": [197.0, 10.0], "N": [198.0, 196.0]} | [
""
] | |
<image>Find the area of the figure. Round to the nearest tenth if necessary. | 50 | [
"Find(AreaOf(Shape($)))"
] | [
"Equals(LengthOf(Line(B, D)), 2)",
"Equals(LengthOf(Line(F, G)), 4)",
"Equals(LengthOf(Line(F, G)), 4)",
"Equals(LengthOf(Line(N, F)), 5mny44)",
"Equals(LengthOf(Line(E, A)), 5)",
"Equals(LengthOf(Line(D, C)), 48)",
"Equals(LengthOf(Line(N, C)), 4)",
"Equals(LengthOf(Line(N, C)), 4)",
"Equals(Length... | [
"BD",
"BF",
"BG",
"BN",
"DC",
"DF",
"DG",
"EA",
"FG",
"NC",
"NF",
"NG"
] | {"A": [225.0, 128.0], "B": [207.0, 12.0], "C": [-0.0, 203.0], "D": [386.0, 107.0], "E": [245.0, 120.0], "F": [194.0, 12.0], "G": [183.0, 6.0], "N": [20.0, 0.0]} | [
""
] | |
<image>Find x | 24 | [
"Find(x)"
] | [
"Equals(LengthOf(Line(M, X)), 5)",
"Equals(LengthOf(Line(M, J)), 13)",
"PointLiesOnLine(X, Line(R, S))",
"PointLiesOnCircle(J, Circle(M, radius_0_0))",
"PointLiesOnCircle(S, Circle(M, radius_0_0))",
"PointLiesOnCircle(R, Circle(M, radius_0_0))",
"Perpendicular(Line(X, R), Line(M, X))"
] | [
"JR",
"MJ",
"MR",
"MX",
"RS",
"XS",
"SR"
] | {"S": [173.0, 55.0], "J": [6.0, 87.0], "M": [95.0, 90.0], "R": [10.0, 54.0], "X": [92.0, 54.0]} | [
"M"
] | |
<image>Find x. | \frac { 13 \sqrt { 2 } } { 2 } | [
"Find(x)"
] | [
"Equals(LengthOf(Line(A, E)), 13)",
"Equals(MeasureOf(Angle(E, A, F)), y)",
"Perpendicular(Line(A, C), Line(C, E))",
"Equals(LengthOf(Line(A, C)), LengthOf(Line(C, E)))",
"Equals(LengthOf(Line(C, E)), LengthOf(Line(E, F)))",
"Equals(LengthOf(Line(E, F)), LengthOf(Line(F, A)))",
"Equals(LengthOf(Line(F, ... | [
"AC",
"CE",
"EF",
"AF",
"AE"
] | {"A": [15.0, 183.0], "C": [16.0, 13.0], "E": [184.0, 14.0], "F": [185.0, 181.0]} | [
""
] | |
<image>Find the measure of $\angle 1$. | 56 | [
"Find(MeasureOf(Angle(1)))"
] | [
"Equals(MeasureOf(Angle(B, A, C)), 124)",
"Equals(MeasureOf(Angle(C, A, E)), MeasureOf(Angle(1)))",
"PointLiesOnLine(A, Line(B, E))",
"PointLiesOnLine(A, Line(C, D))"
] | [
"AB",
"AC",
"AD",
"AE",
"AF",
"BE",
"BF",
"CD",
"EF"
] | {"A": [129.0, 65.0], "B": [1.0, 40.0], "C": [193.0, 0.0], "D": [73.0, 121.0], "E": [287.0, 97.0], "F": [23.0, 37.0]} | [
""
] | |
<image>Find $y$. | 3 | [
"Find(y)"
] | [
"Equals(LengthOf(Line(T, M)), 10-2x)",
"Equals(LengthOf(Line(M, J)), 12-3x)",
"Equals(LengthOf(Line(I, L)), y+4/5)",
"Equals(LengthOf(Line(L, C)), 2y-11/5)",
"PointLiesOnLine(L, Line(J, C))",
"PointLiesOnLine(T, Line(J, A))",
"PointLiesOnLine(J, Line(N, T))",
"PointLiesOnLine(J, Line(N, A))",
"Point... | [
"CE",
"CI",
"CL",
"CO",
"CR",
"CU",
"IM",
"IP",
"IQ",
"JA",
"JC",
"JL",
"JN",
"JR",
"JT",
"JU",
"LM",
"LR",
"LS",
"LU",
"LV",
"MP",
"MQ",
"MS",
"MV",
"NA",
"NT",
"OE",
"OI",
"PQ",
"RU",
"SV",
"TA",
"TI",
"TM",
"TP",
"TQ"
] | {"A": [435.0, 79.0], "C": [122.0, 278.0], "I": [342.0, 278.0], "J": [117.0, 81.0], "L": [121.0, 179.0], "M": [291.0, 178.0], "N": [1.0, 80.0], "O": [435.0, 277.0], "P": [386.0, 353.0], "Q": [196.0, 3.0], "R": [118.0, 25.0], "S": [435.0, 178.0], "T": [238.0, 80.0], "U": [118.0, 335.0], "V": [1.0, 179.0]} | [
""
] | |
<image>Use parallelogram ABCD to find $m\angle BCD$ | 52 | [
"Parallelogram(A,B,C,D)",
"Find(MeasureOf(Angle(B,C,D)))"
] | [
"Equals(MeasureOf(Angle(F, A, D)), 20)",
"Equals(LengthOf(Line(D, A)), 9)",
"Equals(MeasureOf(Angle(F, A, B)), 32)",
"Equals(LengthOf(Line(B, A)), 6)",
"Equals(LengthOf(Line(D, C)), 6.86)",
"Equals(MeasureOf(Angle(F, B, C)), 40.1)",
"PointLiesOnLine(F, Line(D, B))",
"PointLiesOnLine(F, Line(C, A))"
] | [
"AF",
"BA",
"BC",
"BF",
"CA",
"CF",
"DA",
"DB",
"DC",
"DF"
] | {"A": [127.0, 1.0], "B": [111.0, 83.0], "C": [0.0, 147.0], "D": [16.0, 63.0], "F": [65.0, 72.0]} | [
""
] | |
<image>In $\odot A, m \angle E A D=42$. Find $m \widehat{C D}$ | 138 | [
"Circle(A)",
"Equals(MeasureOf(Angle(E,A,D)),42)",
"Find(MeasureOf(Arc(C,D)))"
] | [
"Equals(MeasureOf(Angle(E, A, D)), 42)",
"PointLiesOnLine(A, Line(E, C))",
"PointLiesOnLine(A, Line(D, B))",
"PointLiesOnCircle(E, Circle(A, radius_2_0))",
"PointLiesOnCircle(C, Circle(A, radius_2_0))",
"PointLiesOnCircle(D, Circle(A, radius_2_0))",
"PointLiesOnCircle(B, Circle(A, radius_2_0))"
] | [
"EC",
"EA",
"ED",
"CA",
"AD",
"AB",
"DB"
] | {"E": [170.10799469013062, 92.01022434936148], "C": [3.1088217723741565, 76.19965176543187], "A": [86.23865979381443, 85.22525773195876], "D": [144.29229174629, 146.30728106661257], "B": [28.656490264952964, 24.741438483668503]} | [
"A"
] | |
<image>For the pair of similar figures, use the given areas to find the scale factor from the blue to the green figure. | \frac { \sqrt { 3 } } { 2 } | [
"Similar(Shape($1),Shape($2))",
"Find(ScaleFactorOf(Blue(Shape($)),Green(Shape($))))"
] | [
"Equals(LengthOf(Line(A, C)), 5)",
"Equals(LengthOf(Line(D, F)), x)",
"Equals(AreaOf(Triangle(A, B, C)), 40)",
"Equals(AreaOf(Triangle(D, E, F)), 30)"
] | [
"AB",
"AC",
"BC"
] | {"A": [1.0, 113.0], "B": [195.0, 113.0], "C": [6.0, 1.0]} | [
""
] | |
<image>Find x. | 4 \sqrt { 13 } | [
"Find(x)"
] | [
"Equals(LengthOf(Line(A, C)), 12)",
"Equals(LengthOf(Line(A, B)), x)",
"Equals(LengthOf(Line(B, C)), 8)",
"Perpendicular(Line(A, C), Line(B, C))"
] | [
"AB",
"AC",
"BC"
] | {"A": [80.62754695800864, 3.104802182454222], "B": [0.0, 221.0], "C": [129.356226963609, 185.0859679177698]} | [] | |
<image>Find $x$. (Hint: Draw an auxiliary line.) | 130 | [
"Find(x)"
] | [
"Equals(MeasureOf(Angle(B, C, J)), 105)",
"Equals(MeasureOf(Angle(C, I, K)), x)",
"Equals(MeasureOf(Angle(K, N, F)), 125)",
"PointLiesOnLine(E, Line(F, H))",
"PointLiesOnLine(C, Line(B, D))",
"PointLiesOnLine(C, Line(I, J))",
"PointLiesOnLine(E, Line(I, K))",
"Parallel(Line(D, B), Line(F, H))"
] | [
"BC",
"BD",
"CD",
"CI",
"CJ",
"EF",
"EH",
"EI",
"EK",
"FH",
"IJ",
"IK"
] | {"B": [1.0, 92.0], "C": [226.0, 91.0], "D": [492.0, 92.0], "E": [242.0, 197.0], "F": [2.0, 199.0], "H": [492.0, 199.0], "I": [207.0, 145.0], "J": [245.0, 1.0], "K": [292.0, 271.0]} | [
""
] | |
<image>Find z | 2 \sqrt { 6 } | [
"Find(z)"
] | [
"Equals(LengthOf(Line(B, C)), z)",
"Equals(LengthOf(Line(B, D)), 8)",
"Equals(LengthOf(Line(A, B)), y)",
"Equals(MeasureOf(Angle(B, A, C)), 45)",
"Equals(MeasureOf(Angle(B, D, A)), 60)",
"Equals(LengthOf(Line(A, D)), x)",
"Perpendicular(Line(B, C), Line(A, C))",
"Perpendicular(Line(A, B), Line(A, D))"... | [
"AB",
"AC",
"AD",
"BC",
"BD"
] | {"A": [128.83179792830194, 219.33187314906218], "B": [128.67133250256126, 1.5536783226083486], "C": [236.50898890208444, 110.49547483516933], "D": [0.19786910197868224, 220.33333333333331]} | [] | |
<image>In $\odot F$, find $m \widehat {AED}$. | 153 | [
"Circle(F)",
"Find(MeasureOf(Arc(A,E,D)))"
] | [
"Equals(MeasureOf(Angle(E, F, A)), 63)",
"PointLiesOnLine(F, Line(A, C))",
"PointLiesOnCircle(D, Circle(F, radius_7_0))",
"PointLiesOnCircle(E, Circle(F, radius_7_0))",
"PointLiesOnCircle(B, Circle(F, radius_7_0))",
"PointLiesOnCircle(A, Circle(F, radius_7_0))",
"PointLiesOnCircle(C, Circle(F, radius_7_... | [
"AC",
"AF",
"CF",
"DC",
"DF",
"EB",
"FB",
"FE"
] | {"A": [87.0, 328.0], "B": [83.0, 29.0], "C": [265.0, 26.0], "D": [323.0, 80.0], "E": [273.0, 324.0], "F": [175.0, 179.0]} | [
"F"
] | |
<image>Find $y$ in the figure. | 26.3 | [
"Find(y)"
] | [
"Equals(MeasureOf(Angle(K, H, A)), 6y+10)",
"Equals(MeasureOf(Angle(H, I, M)), 8x-12)",
"Equals(MeasureOf(Angle(G, I, D)), 4x)",
"PointLiesOnLine(H, Line(A, G))",
"PointLiesOnLine(I, Line(A, G))",
"PointLiesOnLine(H, Line(A, I))",
"PointLiesOnLine(I, Line(G, H))",
"PointLiesOnLine(I, Line(D, M))",
"... | [
"AG",
"AH",
"AI",
"DM",
"GH",
"GI",
"HI",
"HK",
"HO",
"ID",
"IM",
"OC"
] | {"A": [131.0, 3.0], "D": [6.0, 147.0], "G": [204.0, 225.0], "H": [154.0, 74.0], "I": [178.0, 145.0], "K": [2.0, 73.0], "M": [319.0, 145.0], "O": [320.0, 72.0]} | [
""
] | |
<image>For trapezoid $ABCD$, $S$ and $T$ are midpoints of the legs. If $AB=3x$, $ST=15$, and $CD=9x$, find $x$. | 2.5 | [
"Trapezoid(A,B,C,D)",
"IsMidpointOf(Point(S),LegOf(Trapezoid(A,B,C,D)))",
"IsMidpointOf(Point(T),LegOf(Trapezoid(A,B,C,D)))",
"Equals(LengthOf(Line(A,B)),3x)",
"Equals(LengthOf(Line(S,T)),15)",
"Equals(LengthOf(Line(C,D)),9x)",
"Find(x)"
] | [
"PointLiesOnLine(S, Line(A, C))",
"PointLiesOnLine(T, Line(B, D))"
] | [
"AB",
"AC",
"AS",
"BD",
"BT",
"CD",
"CS",
"DT",
"ST"
] | {"A": [75.0, 1.0], "B": [238.0, 2.0], "C": [0.0, 228.0], "D": [316.0, 229.0], "S": [37.0, 118.0], "T": [278.0, 120.0]} | [
""
] | |
<image>If $BT = 5$ and $TC = 8$, find $CD$. | \sqrt { 89 } | [
"Equals(LengthOf(Line(B,T)),5)",
"Equals(LengthOf(Line(T,C)),8)",
"Find(LengthOf(Line(C,D)))"
] | [
"PointLiesOnLine(T, Line(C, A))",
"PointLiesOnLine(T, Line(D, B))",
"Equals(LengthOf(Line(A, B)), LengthOf(Line(A, D)))",
"Equals(LengthOf(Line(C, B)), LengthOf(Line(C, D)))"
] | [
"AB",
"AD",
"AT",
"BT",
"CA",
"CB",
"CD",
"CT",
"DB",
"DT"
] | {"A": [0.0, 165.0], "B": [169.0, 12.0], "C": [287.0, 11.0], "D": [221.0, 111.0], "T": [194.0, 61.0]} | [
""
] | |
<image>If $QR=2$, $XW=12$, $QW=15$, and $ST=5$, find $RS$. | 8 | [
"Equals(LengthOf(Line(Q,R)),2)",
"Equals(LengthOf(Line(X,W)),12)",
"Equals(LengthOf(Line(Q,W)),15)",
"Equals(LengthOf(Line(S,T)),5)",
"Find(LengthOf(Line(R,S)))"
] | [
"PointLiesOnLine(R, Line(Q, S))",
"PointLiesOnLine(R, Line(T, Q))",
"PointLiesOnLine(S, Line(R, T))",
"PointLiesOnLine(S, Line(Q, T))",
"PointLiesOnLine(X, Line(Q, W))",
"PointLiesOnLine(X, Line(Q, V))",
"PointLiesOnLine(W, Line(X, V))",
"PointLiesOnLine(W, Line(Q, V))",
"Parallel(Line(R, X), Line(S... | [
"QR",
"QS",
"QT",
"QV",
"QW",
"QX",
"RS",
"RT",
"RX",
"ST",
"SW",
"WV",
"XV",
"XW"
] | {"Q": [1.0, 222.0], "R": [96.0, 163.0], "S": [284.0, 61.0], "T": [392.0, 0.0], "V": [495.0, 222.0], "W": [342.0, 223.0], "X": [114.0, 222.0]} | [
""
] | |
<image>In $\odot W$, $\widehat {RS} \cong \widehat {TV}$. Find $RS$. | 11 | [
"Circle(W)",
"Congruent(Arc(R,S),Arc(T,V))",
"Find(LengthOf(Line(R,S)))"
] | [
"Equals(LengthOf(Line(S, R)), 5x-9)",
"Equals(LengthOf(Line(T, V)), 2x+3)",
"",
"PointLiesOnCircle(S, Circle(A, radius_1_0))",
"PointLiesOnCircle(T, Circle(A, radius_1_0))",
"PointLiesOnCircle(R, Circle(A, radius_1_0))",
"PointLiesOnCircle(V, Circle(A, radius_1_0))"
] | [
"SR",
"TV"
] | {"A": [137.0, 138.0], "R": [1.0, 126.0], "S": [155.0, 2.0], "T": [245.0, 52.0], "V": [233.0, 215.0]} | [
"A"
] | |
<image>In the figure, $m∠8 = 96$ and $m∠12 = 42$. Find the measure of $\angle 9$. | 84 | [
"Equals(MeasureOf(Angle(8)),96)",
"Equals(MeasureOf(Angle(12)),42)",
"Find(MeasureOf(Angle(9)))"
] | [
"Equals(MeasureOf(Angle(K, D, L)), MeasureOf(angle 1))",
"Equals(MeasureOf(Angle(L, D, H)), MeasureOf(angle 2))",
"Equals(MeasureOf(Angle(D, H, I)), MeasureOf(angle 3))",
"Equals(MeasureOf(Angle(I, H, C)), MeasureOf(angle 4))",
"Equals(MeasureOf(Angle(A, D, K)), MeasureOf(angle 8))",
"Equals(MeasureOf(Ang... | [
"AD",
"AE",
"AG",
"AJ",
"AL",
"AM",
"CH",
"CK",
"DC",
"DH",
"DK",
"DL",
"DM",
"EG",
"EJ",
"FI",
"GF",
"GH",
"GI",
"GJ",
"HF",
"HI",
"HK",
"LM"
] | {"A": [125.0, 179.0], "C": [388.0, 88.0], "D": [110.0, 88.0], "E": [0.0, 179.0], "F": [202.0, 267.0], "G": [239.0, 178.0], "H": [276.0, 89.0], "I": [313.0, 1.0], "J": [388.0, 178.0], "K": [0.0, 88.0], "L": [95.0, 4.0], "M": [140.0, 264.0]} | [
""
] | |
<image>$\overline{JH}, \overline{JP},$ and $\overline{PH}$ are midsegments of $\triangle KLM$. Find $x$. | 50 | [
"IsMidsegmentOf(Line(J,H),Triangle(K,L,M))",
"IsMidsegmentOf(Line(J,P),Triangle(K,L,M))",
"IsMidsegmentOf(Line(P,H),Triangle(K,L,M))",
"Find(x)"
] | [
"Equals(LengthOf(Line(H, P)), 25)",
"PointLiesOnLine(P, Line(L, K))",
"PointLiesOnLine(H, Line(L, M))",
"PointLiesOnLine(J, Line(K, M))"
] | [
"HJ",
"HL",
"HM",
"HP",
"JK",
"JP",
"LK",
"ML",
"PK",
"PL",
"JH",
"JM"
] | {"H": [392.0, 123.0], "J": [191.0, 125.0], "K": [85.0, 7.0], "L": [470.0, 1.0], "M": [291.0, 264.0], "P": [271.0, 4.0]} | [
""
] | |
<image>Find y | \frac { 5 } { 3 } | [
"Find(y)"
] | [
"Equals(LengthOf(Line(B, C)), z)",
"Equals(LengthOf(Line(B, A)), 15)",
"Equals(LengthOf(Line(A, C)), 5)",
"PointLiesOnLine(Y, Line(A, B))",
"Perpendicular(Line(B, Y), Line(C, Y))",
"Perpendicular(Line(B, C), Line(C, A))"
] | [
"AB",
"AC",
"AY",
"BC",
"BY",
"CY"
] | {"A": [78.0, 247.0], "B": [78.0, 0.0], "C": [1.0, 218.0], "Y": [79.0, 220.0]} | [
""
] | |
<image>Find x to the nearest tenth. Assume that segments that appear to be tangent are tangent. | 14 | [
"Find(x)"
] | [
"Equals(LengthOf(Line(B, K)), x+2)",
"Equals(LengthOf(Line(B, J)), x)",
"Equals(LengthOf(Line(L, B)), x+10)",
"Equals(LengthOf(Line(B, M)), x+7)",
"PointLiesOnLine(B, Line(L, J))",
"PointLiesOnLine(B, Line(M, K))",
"PointLiesOnCircle(L, Circle(O, radius_5_0))",
"PointLiesOnCircle(M, Circle(O, radius_5... | [
"BJ",
"BK",
"BL",
"BM",
"LJ",
"MK"
] | {"A": [78.0, 77.0], "O": [96.0, 94.0], "J": [38.0, 19.0], "K": [114.0, 2.0], "L": [147.0, 176.0], "M": [34.0, 168.0]} | [
"O"
] | |
<image>If $c=8,$ find $b$ | 4 \sqrt { 3 } | [
"Equals(c,8)",
"Find(b)"
] | [
"Equals(LengthOf(Line(A, B)), c)",
"Equals(MeasureOf(Angle(A, B, C)), 60)",
"Equals(LengthOf(Line(C, B)), a)",
"Equals(MeasureOf(Angle(C, A, B)), 30)",
"Equals(LengthOf(Line(C, A)), b)",
"Perpendicular(Line(C, B), Line(C, A))"
] | [
"AB",
"CA",
"CB"
] | {"A": [262.0, 96.0], "B": [0.0, 0.0], "C": [2.0, 96.0]} | [
""
] | |
<image>Find $m\angle T$ | 58 | [
"Find(MeasureOf(Angle(T)))"
] | [
"Equals(MeasureOf(Angle(S, T, Q)), x)",
"Equals(MeasureOf(Angle(T, Q, R)), 2x+5)",
"Equals(MeasureOf(Angle(R, S, T)), 2x+7)",
"Equals(MeasureOf(Angle(Q, R, S)), x)"
] | [
"RQ",
"SR",
"ST",
"TQ"
] | {"Q": [176.0, 3.0], "R": [0.0, 144.0], "S": [214.0, 241.0], "T": [382.0, 147.0]} | [
""
] | |
<image>Find the area of the shaded region. Round to the nearest tenth. | 68.7 | [
"Find(AreaOf(Shaded(Shape($))))"
] | [
"Equals(LengthOf(Line(E, D)), 12)",
"Equals(LengthOf(Line(A, E)), 9)",
"PointLiesOnLine(F, Line(A, E))",
"PointLiesOnCircle(A, Circle(B, radius_1_0))",
"PointLiesOnCircle(C, Circle(B, radius_1_0))",
"PointLiesOnCircle(D, Circle(B, radius_1_0))",
"PointLiesOnCircle(E, Circle(B, radius_1_0))"
] | [
"AC",
"AE",
"AF",
"CD",
"ED",
"EF"
] | {"A": [160.0, 180.0], "B": [98.0, 102.0], "C": [158.0, 22.0], "D": [40.0, 25.0], "E": [39.0, 181.0], "F": [54.0, 179.0]} | [
"B"
] | |
<image>Segments $\overline{AD}$ and $\overline{CE}$ are medians of $\triangle ACB, \overline{AD} \perp \overline{CE}, AB=10$, and $CE=9$. Find $CA$ | \sqrt { 52 } | [
"IsMedianOf(Line(A,D),Triangle(A,C,B))",
"IsMedianOf(Line(C,E),Triangle(A,C,B))",
"Perpendicular(Line(A,D),Line(C,E))",
"Equals(LengthOf(Line(A,B)),10)",
"Equals(LengthOf(Line(C,E)),9)",
"Find(LengthOf(Line(C,A)))"
] | [
"PointLiesOnLine(D, Line(C, B))",
"PointLiesOnLine(F, Line(C, E))",
"PointLiesOnLine(E, Line(A, B))",
"PointLiesOnLine(F, Line(A, D))"
] | [
"AB",
"AD",
"AE",
"CA",
"CB",
"CD",
"CE",
"DB",
"EB",
"FA",
"FC",
"FD",
"FE"
] | {"A": [169.0, 1.0], "B": [570.0, 106.0], "C": [2.0, 303.0], "D": [284.0, 205.0], "E": [371.0, 54.0], "F": [244.0, 140.0]} | [
""
] | |
<image>In $\triangle RST$, $Z$ is the centroid and $RZ=18$. Find $ZV$. | 9 | [
"Triangle(R,S,T)",
"IsCentroidOf(Point(Z),Triangle(R,S,T))",
"Equals(LengthOf(Line(R,Z)),18)",
"Find(LengthOf(Line(Z,V)))"
] | [
"Equals(LengthOf(Line(R, W)), 14)",
"Equals(LengthOf(Line(R, Z)), 18)",
"Equals(LengthOf(Line(U, Z)), 5)",
"PointLiesOnLine(Z, Line(U, S))",
"PointLiesOnLine(W, Line(S, R))",
"PointLiesOnLine(V, Line(S, T))",
"PointLiesOnLine(U, Line(R, T))",
"PointLiesOnLine(Z, Line(R, V))",
"PointLiesOnLine(Z, Lin... | [
"SR",
"ST",
"SV",
"SW",
"SZ",
"RT",
"RV",
"RW",
"RZ",
"TV",
"TW",
"TZ",
"US",
"UR",
"UT",
"UZ",
"ZV",
"ZW"
] | {"S": [1.0, 301.0], "R": [389.0, 1.0], "T": [328.0, 412.0], "U": [355.0, 231.0], "V": [179.0, 362.0], "W": [190.0, 154.0], "Z": [243.0, 253.0]} | [
""
] | |
<image>Find x. | 45 | [
"Find(x)"
] | [
"Equals(LengthOf(Line(A, C)), 12)",
"Equals(MeasureOf(Angle(B, A, C)), x)",
"Equals(LengthOf(Line(A, B)), y)",
"Perpendicular(Line(B, C), Line(B, D))",
"Equals(LengthOf(Line(A, D)), LengthOf(Line(B, D)))"
] | [
"AB",
"AC",
"AD",
"BC",
"BD"
] | {"A": [1.0, 0.0], "B": [145.0, 148.0], "C": [145.0, 1.0], "D": [3.0, 148.0]} | [
""
] | |
<image>In $\odot B$, the diameter is 20 units long, and $m \angle A C E=45$ Find $x$ | \sqrt { 2 } | [
"Circle(B)",
"Equals(DiameterOf(Circle($)),2)",
"Equals(MeasureOf(Angle(A,C,E)),45)",
"Find(x)"
] | [
"Equals(LengthOf(Line(B, D)), 5x)",
"PointLiesOnLine(D, Line(E, C))",
"PointLiesOnCircle(C, Circle(B, radius_1_0))",
"PointLiesOnCircle(A, Circle(B, radius_1_0))",
"PointLiesOnCircle(E, Circle(B, radius_1_0))",
"Perpendicular(Line(D, E), Line(B, D))"
] | [
"AB",
"BC",
"AC",
"BD",
"DE",
"DC",
"EC"
] | {"A": [23.0, 29.0], "B": [87.0, 88.0], "C": [154.0, 147.0], "D": [161.0, 85.0], "E": [154.0, 30.0]} | [
"B"
] | |
<image>Find $x$. | 145 | [
"Find(x)"
] | [
"Equals(MeasureOf(Angle(A, G, C)), 85)",
"Equals(MeasureOf(Angle(A, G, D)), 40)",
"Equals(MeasureOf(Angle(E, G, C)), x)",
"PointLiesOnCircle(A, G)",
"PointLiesOnCircle(C, G)",
"PointLiesOnCircle(D, G)",
"PointLiesOnCircle(E, G)",
"Perpendicular(Line(G, D), Line(G, E))"
] | [
"AG",
"GA",
"GD",
"GE",
"GC"
] | {"A": [27.0, 53.0], "C": [214.0, 34.0], "D": [2.0, 150.0], "E": [147.0, 258.0], "G": [130.0, 126.0]} | [
"G"
] | |
<image>Find $m \angle T$. | 135 | [
"Find(MeasureOf(Angle(T)))"
] | [
"Equals(MeasureOf(Angle(W, R, T)), 4x)",
"Equals(MeasureOf(Angle(R, W, Z)), 45)",
"Equals(MeasureOf(Angle(W, Z, T)), 2x+30)",
"PointLiesOnCircle(R, Circle(O, radius_4_0))",
"PointLiesOnCircle(T, Circle(O, radius_4_0))",
"PointLiesOnCircle(W, Circle(O, radius_4_0))",
"PointLiesOnCircle(Z, Circle(O, radiu... | [
"RT",
"TZ",
"WR",
"WZ"
] | {"O": [157.0, 159.0], "R": [197.0, 6.0], "T": [302.0, 96.0], "W": [1.0, 168.0], "Z": [309.0, 197.0]} | [
"O"
] | |
<image>Find $x$. | 7 | [
"Find(x)"
] | [
"Equals(MeasureOf(Angle(C, A, B)), 80)",
"Equals(LengthOf(Line(A, B)), 4y-2)",
"Equals(LengthOf(Line(A, C)), 2y+2)",
"Equals(MeasureOf(Angle(B, C, A)), 6x+8)",
"Equals(LengthOf(Line(A, B)), LengthOf(Line(A, C)))"
] | [
"AB",
"AC",
"BC"
] | {"A": [221.49172576832152, 0.19858156028368512], "B": [0.5848597220661702, 250.6694385136907], "C": [422.0, 252.0]} | [] | |
<image>In the figure, $m ∠ 12 = 64$. Find the measure of $ \angle 5$. | 64 | [
"Equals(MeasureOf(Angle(12)),64)",
"Find(MeasureOf(Angle(13)))"
] | [
"Equals(MeasureOf(Angle(I, D, W)), MeasureOf(angle 1))",
"Equals(MeasureOf(Angle(W, D, L)), MeasureOf(angle 2))",
"Equals(MeasureOf(Angle(D, U, A)), MeasureOf(angle 3))",
"Equals(MeasureOf(Angle(A, U, L)), MeasureOf(angle 4))",
"Equals(MeasureOf(Angle(H, D, I)), MeasureOf(angle 5))",
"Equals(MeasureOf(Ang... | [
"AE",
"AG",
"AK",
"AQ",
"AU",
"BH",
"BN",
"DI",
"DJ",
"DL",
"DW",
"EB",
"EG",
"EH",
"EK",
"EN",
"EU",
"GK",
"GQ",
"GU",
"HD",
"HJ",
"HN",
"HW",
"IL",
"KU",
"QD",
"QI",
"QL",
"QU",
"UD",
"UI",
"UL",
"WJ"
] | {"A": [378.0, 3.0], "B": [7.0, 171.0], "D": [176.0, 72.0], "E": [277.0, 171.0], "G": [353.0, 43.0], "H": [118.0, 170.0], "I": [27.0, 71.0], "J": [72.0, 249.0], "K": [231.0, 249.0], "L": [518.0, 73.0], "N": [492.0, 172.0], "Q": [353.0, 73.0], "S": [503.0, 72.0], "T": [460.0, 73.0], "U": [334.0, 73.0], "W": [218.0, 2.0]} | [
""
] | |
<image>Triangle $CDE$ has an area of 336 square inches. Find $CE$. | 28 | [
"Equals(AreaOf(Triangle(C,D,E)),336)",
"Find(LengthOf(Line(C,E)))"
] | [
"Equals(LengthOf(Line(D, F)), 24)",
"PointLiesOnLine(F, Line(C, E))",
"Perpendicular(Line(D, F), Line(C, F))"
] | [
"CD",
"CE",
"DE",
"CF",
"EF",
"DF"
] | {"C": [1.0, 94.0], "D": [27.0, 1.0], "E": [206.0, 96.0]} | [
""
] | |
<image>Find $x$ in the figure. | 16 | [
"Find(x)"
] | [
"Equals(MeasureOf(Angle(G, O, I)), 3y+1)",
"Equals(MeasureOf(Angle(O, I, E)), 4x-5)",
"Equals(MeasureOf(Angle(I, B, H)), 3x+11)",
"PointLiesOnLine(O, Line(G, H))",
"PointLiesOnLine(I, Line(E, B))",
"Parallel(Line(G,E), Line(O,I))",
"Parallel(Line(O,I), Line(H,B))",
"Parallel(Line(G,E), Line(O,I))",
... | [
"EB",
"EI",
"GE",
"GH",
"GO",
"HB",
"IB",
"OH",
"OI"
] | {"A": [195.0, 8.0], "B": [471.0, 148.0], "C": [50.0, 91.0], "D": [285.0, 145.0], "E": [387.0, 7.0], "F": [232.0, 77.0], "G": [1.0, 6.0], "H": [87.0, 146.0], "I": [428.0, 78.0], "J": [398.0, 146.0], "K": [303.0, 6.0], "L": [229.0, 146.0], "M": [455.0, 122.0], "N": [285.0, 78.0], "O": [41.0, 75.0], "P": [71.0, 122.0], "Q... | [
""
] | |
<image>Find the area of the shaded region. Round to the nearest tenth. | 449.0 | [
"Find(AreaOf(Shaded(Shape($))))"
] | [
"Equals(LengthOf(Line(E, D)), 10.5)",
"Equals(LengthOf(PerimeterOf(Cirle(B))), 35)",
"PointLiesOnLine(C, Line(A, B))",
"PointLiesOnCircle(A, Circle(B, radius_1_0))",
"PointLiesOnCircle(C, Circle(B, radius_1_1))"
] | [
"AB",
"AC",
"BC"
] | {"A": [208.0, 73.0], "B": [110.0, 107.0], "C": [151.0, 92.0]} | [
"B"
] | |
<image>Find the area of the parallelogram. Round to the nearest tenth if necessary. | 259.9 | [
"Find(AreaOf(Parallelogram($)))"
] | [
"Equals(LengthOf(Line(E, D)), 10)",
"Equals(LengthOf(Line(C, D)), 30)",
"Equals(MeasureOf(Angle(C, E, D)), 60)"
] | [
"BC",
"BE",
"CD",
"ED"
] | {"B": [310.0, 91.0], "C": [360.0, 4.0], "D": [54.0, 1.0], "E": [0.0, 91.0]} | [
""
] | |
<image>Polygon ABCD ~ polygon AEFG, $m\angle AGF = 108$, GF = 14, AD = 12, DG = 4.5, EF = 8, and AB = 26.
Find the perimeter of trapezoid ABCD | 73.2 | [
"Similar(Quadrilateral(A,B,C,D),Quadrilateral(A,E,F,G))",
"Equals(MeasureOf(Angle(A,G,F)),108)",
"Equals(LengthOf(Line(G,F)),14)",
"Equals(LengthOf(Line(A,D)),12)",
"Equals(LengthOf(Line(D,G)),4.5)",
"Equals(LengthOf(Line(E,F)),8)",
"Equals(LengthOf(Line(A,B)),26)",
"Find(PerimeterOf(Trapezoid(A,B,C,D... | [
"Equals(LengthOf(Line(B, A)), 26)",
"Equals(LengthOf(Line(E, F)), 8)",
"Equals(MeasureOf(Angle(A, G, F)), 108)",
"Equals(LengthOf(Line(D, A)), 12)",
"Equals(LengthOf(Line(G, F)), 14)",
"Equals(LengthOf(Line(D, G)), 4.5)",
"PointLiesOnLine(E, Line(B, A))",
"PointLiesOnLine(G, Line(D, A))"
] | [
"AE",
"AG",
"BA",
"BE",
"CB",
"CD",
"DA",
"DG",
"EF",
"GF"
] | {"A": [1.0, 27.0], "B": [250.0, 1.0], "C": [249.0, 133.0], "D": [37.0, 132.0], "E": [156.0, 9.0], "F": [159.0, 87.0], "G": [22.0, 89.0]} | [
""
] | |
<image>In $\odot B$, $m \angle LBM = 3x$ and $m \angle LBQ = 4x + 61$. What is the measure of $\angle PBQ$? | 51 | [
"Circle(B)",
"Equals(MeasureOf(Angle(L,B,M)),3x)",
"Equals(MeasureOf(Angle(L,B,Q)),4x+61)",
"Find(MeasureOf(Angle(P,B,Q)))"
] | [
"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": [149.0, 154.0], "K": [150.0, 2.0], "L": [47.0, 45.0], "M": [2.0, 153.0], "P": [258.0, 261.0], "Q": [303.0, 152.0]} | [
"B"
] | |
<image>Find x. Round the angle measure to the nearest degree. | 40 | [
"Find(x)"
] | [
"Equals(MeasureOf(Angle(A, B, C)), x)",
"Equals(LengthOf(Line(B, C)), 36)",
"Equals(LengthOf(Line(A, B)), 59)",
"Equals(LengthOf(Line(A, C)), 39)"
] | [
"AB",
"AC",
"BC"
] | {"A": [175.5836255151627, 276.424940025835], "B": [135.0, 0.0], "C": [1.361989838989416, 142.94432932725513]} | [] | |
<image>Find $m \angle 1$. | 108 | [
"Find(MeasureOf(Angle(1)))"
] | [
"Equals(MeasureOf(Angle(C, D, A)), MeasureOf(angle 1))",
"Equals(MeasureOf(Angle(C, B, A)), 42)",
"Equals(MeasureOf(Angle(A, C, B)), 38)",
"Equals(MeasureOf(Angle(A, D, B)), 72)",
"Equals(MeasureOf(Angle(D, A, C)), MeasureOf(angle 2))",
"Equals(MeasureOf(Angle(B, A, D)), MeasureOf(angle 3))",
"PointLies... | [
"AB",
"AC",
"AD",
"BC",
"BD",
"CD"
] | {"A": [202.0, 184.0], "B": [-0.0, 1.0], "C": [434.0, 2.0], "D": [262.0, 2.0]} | [
""
] | |
<image>Find x | 22 | [
"Find(x)"
] | [
"Equals(MeasureOf(Arc(G, C)), 148)",
"Equals(MeasureOf(Arc(H, B)), 128)",
"PointLiesOnLine(G, Line(C, F))",
"PointLiesOnLine(H, Line(C, F))",
"PointLiesOnLine(H, Line(C, G))",
"PointLiesOnLine(B, Line(E, F))",
"PointLiesOnLine(G, Line(F, H))",
"PointLiesOnCircle(B, Circle(D, radius_3_0))",
"PointLie... | [
"BE",
"BF",
"CD",
"CF",
"CG",
"CH",
"EF",
"FG",
"FH",
"GH"
] | {"B": [158.0, 107.0], "C": [247.0, 26.0], "D": [169.0, 56.0], "E": [221.0, 120.0], "F": [0.0, 75.0], "G": [114.0, 52.0], "H": [217.0, 32.0]} | [
"D"
] | |
<image>If $m\angle ZYW = 2x - 7$ and $m \angle WYX = 2x + 5$, find $m\angle ZYW$. | 39 | [
"Equals(MeasureOf(Angle(Z,Y,W)),2x-7)",
"Equals(MeasureOf(Angle(W,Y,X)),2x+5)",
"Find(MeasureOf(Angle(Z,Y,W)))"
] | [
"PointLiesOnLine(P, Line(X, Z))",
"PointLiesOnLine(P, Line(W, Y))",
"Perpendicular(Line(W, Z), Line(X, W))"
] | [
"WP",
"WY",
"WZ",
"XP",
"XW",
"XY",
"XZ",
"YP",
"YZ",
"ZP"
] | {"P": [137.0, 69.0], "W": [1.0, 137.0], "X": [275.0, 136.0], "Y": [273.0, 2.0], "Z": [0.0, 0.0]} | [
""
] | |
<image>Find $m\angle B$ | 90 | [
"Find(MeasureOf(Angle(A)))"
] | [
"Equals(MeasureOf(Angle(A, E, D)), 2x+10)",
"Equals(MeasureOf(Angle(C, D, E)), x)",
"Equals(MeasureOf(Angle(B, C, D)), 2x-20)",
"PointLiesOnLine(F, Line(A, E))",
"Perpendicular(Line(F, A), Line(A, B))",
"Perpendicular(Line(A, B), Line(C, B))"
] | [
"AB",
"AE",
"CB",
"DC",
"ED",
"FA",
"FE"
] | {"A": [1.0, 1.0], "B": [2.0, 208.0], "C": [210.0, 206.0], "D": [334.0, 80.0], "E": [211.0, 2.0], "F": [122.0, 3.0]} | [
""
] | |
<image>Find $x$ in each figure. | 31 | [
"Find(x)"
] | [
"Equals(MeasureOf(Angle(H, N, K)), 4x)",
"Equals(MeasureOf(Angle(N, M, O)), 56)",
"Equals(MeasureOf(Angle(T, L, G)), 3y-11)",
"PointLiesOnLine(J, Line(A, C))",
"PointLiesOnLine(U, Line(A, C))",
"PointLiesOnLine(U, Line(A, L))",
"PointLiesOnLine(L, Line(A, Q))",
"PointLiesOnLine(U, Line(A, Q))",
"Poi... | [
"HG",
"HL",
"HM",
"HN",
"KF",
"KN",
"LG",
"LT",
"MG",
"ML",
"MO",
"NF",
"NG",
"NL",
"NM",
"PM",
"PO",
"QL",
"QT"
] | {"A": [166.0, 199.0], "C": [324.0, 83.0], "D": [98.0, 154.0], "E": [248.0, 45.0], "F": [4.0, 131.0], "G": [344.0, 175.0], "H": [1.0, 37.0], "I": [118.0, 138.0], "J": [198.0, 174.0], "K": [191.0, 0.0], "L": [254.0, 139.0], "M": [170.0, 105.0], "N": [87.0, 72.0], "O": [81.0, 169.0], "P": [268.0, 38.0], "Q": [344.0, 77.0]... | [
""
] | |
<image>Find $m \angle 6$. | 45 | [
"Find(MeasureOf(Angle(6)))"
] | [
"Equals(MeasureOf(Angle(N, C, A)), MeasureOf(angle 8))",
"Equals(MeasureOf(Angle(C, E, B)), 110)",
"Equals(MeasureOf(Angle(A, E, C)), MeasureOf(angle 7))",
"Equals(MeasureOf(Angle(C, A, B)), MeasureOf(angle 6))",
"Equals(MeasureOf(Angle(N, E, D)), MeasureOf(angle 5))",
"Equals(MeasureOf(Angle(E, A, H)), M... | [
"AB",
"AC",
"AD",
"AE",
"AF",
"AG",
"AH",
"AI",
"AK",
"AL",
"AM",
"BD",
"BI",
"BK",
"BL",
"CE",
"CF",
"DI",
"DK",
"DL",
"EB",
"ED",
"EI",
"EK",
"EL",
"FJ",
"GD",
"GH",
"GM",
"HJ",
"HM",
"IK",
"IL",
"JM",
"KL",
"NA",
"NC",
"NE",
"NF",
"NG"... | {"A": [133.0, 186.0], "B": [476.0, 63.0], "C": [219.0, 0.0], "D": [23.0, 230.0], "E": [325.0, 120.0], "F": [73.0, 313.0], "G": [47.0, 155.0], "H": [531.0, 326.0], "I": [519.0, 49.0], "J": [556.0, 315.0], "K": [498.0, 57.0], "L": [3.0, 233.0], "M": [551.0, 339.0], "N": [484.0, 314.0]} | [
""
] | |
<image>Find x | \frac { 17 \sqrt { 2 } } { 2 } | [
"Find(x)"
] | [
"Equals(LengthOf(Line(B, C)), x)",
"Equals(MeasureOf(Angle(A, B, D)), y)",
"Equals(LengthOf(Line(A, B)), 17)",
"Perpendicular(Line(B, D), Line(A, D))",
"Equals(LengthOf(Line(B, D)), LengthOf(Line(A, C)))",
"Equals(LengthOf(Line(A, C)), LengthOf(Line(A, D)))",
"Equals(LengthOf(Line(A, C)), LengthOf(Line(... | [
"AB",
"AC",
"AD",
"BC",
"BD"
] | {"A": [154.0, 166.0], "B": [1.0, 12.0], "C": [155.0, 13.0], "D": [2.0, 167.0]} | [
""
] | |
<image>If $FK = 3x - 1$, $KG = 4y + 3$, $JK = 6y - 2$, and $KH = 2x + 3$, find $x$ so that the quadrilateral is a parallelogram. | 4 | [
"Equals(LengthOf(Line(F,K)),3x-1)",
"Equals(LengthOf(Line(K,G)),4y+3)",
"Equals(LengthOf(Line(J,K)),6y-2)",
"Equals(LengthOf(Line(K,H)),2x+3)",
"Parallelogram($)",
"Find(x)"
] | [
"PointLiesOnLine(K, Line(G, J))",
"PointLiesOnLine(K, Line(F, H))"
] | [
"FH",
"FJ",
"FK",
"GF",
"GH",
"GJ",
"GK",
"HJ",
"HK",
"JK"
] | {"F": [90.0, 1.0], "G": [361.0, 1.0], "H": [274.0, 130.0], "J": [0.0, 130.0], "K": [182.0, 65.0]} | [
""
] | |
<image>In $\odot X, A B=30, C D=30,$ and $m \widehat{C Z}=40$
Find $m\widehat{A B}$ | 80 | [
"Circle(X)",
"Equals(LengthOf(Line(A,B)),30)",
"Equals(LengthOf(Line(C,D)),30)",
"Equals(MeasureOf(Arc(C,Z)),40)",
"Find(MeasureOf(Arc(A,B)))"
] | [
"PointLiesOnLine(N, Line(X, Z))",
"PointLiesOnLine(M, Line(X, Y))",
"PointLiesOnLine(N, Line(C, D))",
"PointLiesOnLine(M, Line(A, B))",
"PointLiesOnCircle(Z, Circle(X, radius_0_0))",
"PointLiesOnCircle(A, Circle(X, radius_0_0))",
"PointLiesOnCircle(C, Circle(X, radius_0_0))",
"PointLiesOnCircle(D, Cir... | [
"AB",
"CD",
"MA",
"MB",
"NC",
"ND",
"XM",
"XN",
"XY",
"XZ",
"YM",
"ZN"
] | {"A": [35.0, 18.0], "B": [144.0, 32.0], "C": [78.0, 175.0], "D": [176.0, 100.0], "M": [96.0, 26.0], "N": [128.0, 137.0], "X": [90.0, 91.0], "Y": [98.0, 0.0], "Z": [143.0, 156.0]} | [
"X"
] | |
<image>Find $m \angle Q$ | 70 | [
"Find(MeasureOf(Angle(Q)))"
] | [
"Equals(LengthOf(Line(R, Q)), 8)",
"Equals(MeasureOf(Angle(S, R, Q)), 110)",
"Equals(LengthOf(Line(S, T)), 8)"
] | [
"QR",
"RS",
"ST",
"TQ"
] | {"Q": [2.0, 0.0], "R": [237.0, 0.0], "S": [245.0, 83.0], "T": [11.0, 138.0]} | [
""
] | |
<image>Triangle $WXY$ has an area of 248 square inches. Find the length of the base. | 31 | [
"Equals(AreaOf(Triangle(W,X,Y)),248)",
"Find(LengthOf(BaseOf(Polygon($))))"
] | [
"Perpendicular(Line(X,A), Line(W,Y))",
"Equals(LengthOf(Line(A,X)), 16)"
] | [
"WX",
"WY",
"YX",
"XA"
] | {"A": [65.0, 131.0], "W": [1.0, 153.0], "X": [66.0, 1.0], "Y": [302.0, 154.0]} | [
""
] | |
<image>Find $x$ to the nearest tenth. Assume that segments that appear to be tangent are tangent. | 5.6 | [
"Find(x)"
] | [
"Equals(LengthOf(Line(B, E)), 3x)",
"Equals(LengthOf(Line(A, H)), x+2)",
"Equals(LengthOf(Line(D, B)), x)",
"Equals(LengthOf(Line(D, H)), 8)",
"PointLiesOnLine(H, Line(A, D))",
"PointLiesOnLine(B, Line(D, E))",
"PointLiesOnLine(I, Line(C, G))",
"PointLiesOnCircle(A, Circle(F, radius_5_0))",
"PointLi... | [
"AD",
"AH",
"BE",
"DB",
"DE",
"DH",
"HB"
] | {"A": [217.0, 66.0], "B": [52.0, 154.0], "D": [1.0, 232.0], "E": [149.0, 5.0], "F": [129.0, 100.0], "H": [74.0, 176.0]} | [
"F"
] | |
<image>Find $m \angle 4$. | 125 | [
"Find(MeasureOf(Angle(4)))"
] | [
"Equals(MeasureOf(Angle(B, C, A)), MeasureOf(angle 3))",
"Equals(MeasureOf(Angle(C, D, S)), 125)",
"Equals(MeasureOf(Angle(A, D, C)), MeasureOf(angle 1))",
"Equals(MeasureOf(Angle(C, A, D)), 62)",
"Equals(MeasureOf(Angle(B, D, A)), MeasureOf(angle 4))",
"Equals(MeasureOf(Angle(S, A, B)), MeasureOf(angle 2... | [
"AB",
"AC",
"AS",
"AD",
"BC",
"BD",
"CD",
"SD"
] | {"A": [0.7728039539852603, 285.3951788306993], "B": [496.4070112893643, 439.3562091503268], "C": [129.64755649926752, 0.7090349447763913], "S": [509.0, 247.0], "D": [345.77362740812964, 259.1858588575463]} | [] | |
<image>If WXYZ is a kite, find $YZ$ | \sqrt { 89 } | [
"Kite(W,X,Y,Z)",
"Find(LengthOf(Line(Y,Z)))"
] | [
"Equals(LengthOf(Line(A, X)), 5)",
"Equals(LengthOf(Line(Y, W)), 8)",
"PointLiesOnLine(A, Line(Y, W))",
"PointLiesOnLine(A, Line(X, Z))"
] | [
"AY",
"AX",
"AZ",
"AW",
"YX",
"YZ",
"YW",
"XZ",
"XW",
"ZW"
] | {"A": [85.33516483516483, 61.0], "Y": [86.0, 182.0], "X": [173.3542956891318, 60.70432098765432], "Z": [0.6394339304175247, 60.7001287001287], "W": [85.13726487252654, 0.9345877522131758]} | [] | |
<image>Find x. Round to the nearest tenth. | 5.2 | [
"Find(x)"
] | [
"Equals(LengthOf(Line(A, B)), x)",
"Equals(LengthOf(Line(C, A)), 11)",
"Equals(MeasureOf(Angle(B, C, A)), 28)",
"Equals(LengthOf(Line(C, B)), 9)"
] | [
"CA",
"CB",
"AB"
] | {"C": [212.90779360818593, 190.91561704177852], "A": [19.712239583333336, 1.0416666666666714], "B": [0.09960620801483344, 128.69585360203845]} | [] | |
<image>Find the measure of $\angle B$ | 46 | [
"Find(MeasureOf(Angle(D)))"
] | [
"Equals(MeasureOf(Angle(D, A, B)), 4x-26)",
"Equals(MeasureOf(Angle(A, B, C)), x)",
"Equals(MeasureOf(Angle(C, D, A)), x)",
"Equals(MeasureOf(Angle(B, C, D)), 2x+18)"
] | [
"AB",
"AD",
"BC",
"DC"
] | {"A": [268.0, 2.0], "B": [3.0, 119.0], "C": [227.0, 322.0], "D": [493.0, 121.0]} | [
""
] | |
<image>Find $m \angle A G E$ | 120 | [
"Find(MeasureOf(Angle(A,G,E)))"
] | [
"Equals(MeasureOf(Angle(A, G, C)), 60)",
"PointLiesOnLine(G, Line(E, C))",
"PointLiesOnLine(G, Line(A, B))",
"PointLiesOnCircle(E, Circle(G, radius_0_0))",
"PointLiesOnCircle(C, Circle(G, radius_0_0))",
"PointLiesOnCircle(A, Circle(G, radius_0_0))",
"PointLiesOnCircle(D, Circle(G, radius_0_0))",
"Poin... | [
"AB",
"EC",
"GA",
"GB",
"GC",
"GD",
"GE"
] | {"A": [1.0, 82.0], "B": [169.0, 86.0], "C": [44.0, 12.0], "D": [85.0, 159.0], "E": [126.0, 158.0], "G": [86.0, 85.0]} | [
"G"
] | |
<image>Find the area of the parallelogram. Round to the nearest tenth if necessary. | 665.1 | [
"Find(AreaOf(Parallelogram($)))"
] | [
"Equals(LengthOf(Line(A, C)), 32)",
"Equals(MeasureOf(Angle(E, C, S)), 30)",
"Equals(LengthOf(Line(E, S)), 12)",
"PointLiesOnLine(E, Line(B, S))",
"Perpendicular(Line(C, E), Line(E, B))"
] | [
"AB",
"AC",
"BS",
"CS",
"CE",
"BE",
"SE"
] | {"A": [1.0, 1.0], "B": [69.0, 113.0], "C": [212.0, 1.0], "S": [276.0, 112.0]} | [
""
] | |
<image>$a=14, b=48,$ and $c=50$, find $tanB$ | 3.43 | [
"Equals(a,14)",
"Equals(b,48)",
"Equals(c,50)",
"Find(TanOf(Angle(B)))"
] | [
"Equals(LengthOf(Line(B, A)), LengthOf(c))",
"Equals(LengthOf(Line(B, C)), LengthOf(a))",
"Equals(LengthOf(Line(A, C)), LengthOf(b))",
"Perpendicular(Line(B, C), Line(A, C))"
] | [
"BA",
"BC",
"AC"
] | {"B": [0.0, 4.0], "A": [162.0, 90.0], "C": [0.9878021062201725, 87.99977271005379]} | [] | |
<image>Find x | 22.5 | [
"Find(x)"
] | [
"Equals(MeasureOf(Angle(B, C, A)), 5x)",
"Equals(MeasureOf(Angle(D, C, B)), 3x)",
"PointLiesOnLine(C, Line(A, D))"
] | [
"AC",
"AD",
"CB",
"CD"
] | {"A": [208.0, 1.0], "B": [18.0, 0.0], "C": [105.5455737704918, 63.66950819672131], "D": [0.0, 127.0]} | [] | |
<image>Find $x$. | 6 | [
"Find(x)"
] | [
"Equals(LengthOf(Line(A, E)), 2x+1)",
"Equals(LengthOf(Line(B, E)), 3x-5)",
"PointLiesOnLine(E, Line(A, B))",
"PointLiesOnLine(D, Line(B, C))",
"Perpendicular(Line(A, C), Line(A, E))",
"Perpendicular(Line(E, D), Line(B, E))",
"Equals(LengthOf(Line(C, D)), LengthOf(Line(B, D)))"
] | [
"AB",
"AC",
"AE",
"BC",
"BD",
"BE",
"CD",
"ED"
] | {"A": [2.6617683149770244, 269.67205585523317], "B": [493.614222877242, 270.6691584430449], "C": [1.6680221513406082, 1.1829904309836365], "D": [227.0214664064678, 123.7942570393086], "E": [230.9856351773672, 270.4685015625857]} | [] | |
<image>Find $x$ so that the quadrilateral is a parallelogram | 11 | [
"Parallelogram($)",
"Find(x)"
] | [
"Equals(LengthOf(Line(C, A)), 2y+3)",
"Equals(LengthOf(Line(E, A)), 2x+4)",
"Equals(LengthOf(Line(D, A)), 3y+5)",
"Equals(LengthOf(Line(B, A)), 4y-11)",
"PointLiesOnLine(A, Line(B, C))",
"PointLiesOnLine(A, Line(D, E))"
] | [
"BA",
"BC",
"BD",
"BE",
"CA",
"CD",
"CE",
"DA",
"DE",
"EA"
] | {"A": [224.0, 115.0], "B": [1.0, 229.0], "C": [449.0, 1.0], "D": [29.0, 26.0], "E": [422.0, 205.0]} | [
""
] | |
<image>Use rhombus XYZW with $m \angle W Y Z=53, V W=3$, $X V=2 a-2,$ and $Z V=\frac{5 a+1}{4}$
Find XZ | 8 | [
"Rhombus(X,Y,Z,W)",
"Equals(MeasureOf(Angle(W,Y,Z)),53)",
"Equals(LengthOf(Line(V,W)),3)",
"Equals(LengthOf(Line(X,V)),2a-2)",
"Equals(LengthOf(Line(Z,V)),\\frac{5a+1}{4})",
"Find(LengthOf(Line(X,Z)))"
] | [
"Equals(MeasureOf(Angle(W, Y, Z)), 53)",
"PointLiesOnLine(V, Line(X, Z))",
"PointLiesOnLine(V, Line(W, Y))"
] | [
"VY",
"WV",
"WY",
"XV",
"XW",
"XY",
"XZ",
"ZV",
"ZW",
"ZY"
] | {"V": [87.0, 48.0], "W": [68.0, 95.0], "X": [0.0, 0.0], "Y": [106.0, 1.0], "Z": [175.0, 96.0]} | [
""
] | |
<image>Use rhombus XYZW with $m \angle W Y Z=53, V W=3$, $X V=2 a-2,$ and $Z V=\frac{5 a+1}{4}$
Find $m \angle X Y W$ | 53 | [
"Rhombus(X,Y,Z,W)",
"Equals(MeasureOf(Angle(W,Y,Z)),53)",
"Equals(LengthOf(Line(V,W)),3)",
"Equals(LengthOf(Line(X,V)),2a-2)",
"Equals(LengthOf(Line(Z,V)),\\frac{5a+1}{4})",
"Find(MeasureOf(Angle(X,Y,W)))"
] | [
"Equals(MeasureOf(Angle(W, Y, Z)), 53)",
"PointLiesOnLine(V, Line(X, Z))",
"PointLiesOnLine(V, Line(W, Y))"
] | [
"VY",
"WV",
"WY",
"XV",
"XW",
"XY",
"XZ",
"ZV",
"ZW",
"ZY"
] | {"V": [87.0, 48.0], "W": [68.0, 95.0], "X": [0.0, 0.0], "Y": [106.0, 1.0], "Z": [175.0, 96.0]} | [
""
] | |
<image>In the figure, $QR \| TS$, $QT \| RS$, and $m \angle 1=131$. Find the measure of $\angle 7$. | 49 | [
"Parallel(Line(Q,R),Line(T,S))",
"Parallel(Line(Q,T),Line(R,S))",
"Equals(MeasureOf(Angle(1)),131)",
"Find(MeasureOf(Angle(2)))"
] | [
"Equals(MeasureOf(Angle(J, Q, R)), MeasureOf(angle 1))",
"Equals(MeasureOf(Angle(R, Q, C)), MeasureOf(angle 2))",
"Equals(MeasureOf(Angle(H, R, Q)), MeasureOf(angle 5))",
"Equals(MeasureOf(Angle(Q, T, S)), MeasureOf(angle 3))",
"Equals(MeasureOf(Angle(Q, R, S)), MeasureOf(angle 6))",
"Equals(MeasureOf(Ang... | [
"AB",
"AH",
"AL",
"AM",
"AO",
"AR",
"BH",
"BL",
"BM",
"BO",
"BR",
"CF",
"CJ",
"CK",
"CN",
"CP",
"CQ",
"EG",
"EP",
"FJ",
"FK",
"FN",
"FP",
"FQ",
"GA",
"GB",
"GH",
"GL",
"GM",
"GO",
"GP",
"GR",
"HL",
"HM",
"HO",
"IS",
"JK",
"JN",
"JP",
"JQ"... | {"A": [147.0, 69.0], "B": [148.0, 263.0], "C": [8.0, 266.0], "E": [76.0, 173.0], "F": [13.0, 208.0], "G": [146.0, 215.0], "H": [147.0, 1.0], "I": [78.0, 97.0], "J": [9.0, 1.0], "K": [10.0, 186.0], "L": [148.0, 44.0], "M": [146.0, 114.0], "N": [8.0, 224.0], "O": [145.0, 88.0], "P": [22.0, 143.0], "Q": [6.0, 55.0], "R": ... | [
""
] | |
<image>Find x | 8 \sqrt { 3 } | [
"Find(x)"
] | [
"Equals(LengthOf(Line(A, B)), y)",
"Equals(LengthOf(Line(A, C)), 8)",
"Equals(MeasureOf(Angle(A, B, C)), 30)",
"Equals(LengthOf(Line(B, C)), x)",
"Perpendicular(Line(A, C), Line(B, C))"
] | [
"AB",
"AC",
"BC"
] | {"A": [238.3323417870322, 0.5308248234127575], "B": [1.3429131550758768, 139.9943410335698], "C": [239.0, 138.0]} | [] | |
<image>Find $x$. | 25 | [
"Find(x)"
] | [
"Equals(MeasureOf(Angle(B, N, M)), 2x-5)",
"PointLiesOnLine(B, Line(N, L))",
"PointLiesOnCircle(L, Circle(B, radius_3_0))",
"PointLiesOnCircle(M, Circle(B, radius_3_0))",
"PointLiesOnCircle(N, Circle(B, radius_3_0))",
"Equals(LengthOf(Line(L, M)), LengthOf(Line(M, N)))"
] | [
"LB",
"LM",
"NB",
"NL",
"NM"
] | {"B": [159.0, 157.0], "L": [132.0, 3.0], "M": [313.0, 131.0], "N": [186.0, 312.0]} | [
"B"
] | |
<image>In the figure, $ \triangle ABC$ is isosceles, $ \triangle DCE$ is equilateral, and $ \triangle FCG$ is isosceles. Find the measures of the $\angle 3$ at vertex $C$. | 26 | [
"Isosceles(Triangle(A,B,C))",
"Equilateral(Triangle(D,C,E))",
"Isosceles(Triangle(F,C,G))",
"Equals(Angle(3),Angle(C))",
"Find(MeasureOf(Angle(3)))"
] | [
"Equals(MeasureOf(Angle(D, C, A)), MeasureOf(angle 1))",
"Equals(MeasureOf(Angle(F, C, D)), MeasureOf(angle 2))",
"Equals(MeasureOf(Angle(G, C, F)), MeasureOf(angle 3))",
"Equals(MeasureOf(Angle(E, C, G)), MeasureOf(angle 4))",
"Equals(MeasureOf(Angle(B, C, E)), MeasureOf(angle 5))",
"Equals(MeasureOf(Ang... | [
"AB",
"AC",
"AD",
"AE",
"AF",
"AG",
"BD",
"BE",
"BF",
"CB",
"CD",
"CE",
"CF",
"CG",
"DE",
"DF",
"FE",
"GB",
"GD",
"GE",
"GF"
] | {"A": [0.0, 233.0], "B": [518.0, 233.0], "C": [259.0, 1.0], "D": [103.0, 233.0], "E": [413.0, 233.0], "F": [208.0, 233.0], "G": [311.0, 233.0]} | [
""
] | |
<image>Find the area of the parallelogram. Round to the nearest tenth if necessary. | 103.9 | [
"Find(AreaOf(Parallelogram($)))"
] | [
"Equals(LengthOf(Line(A, D)), 12)",
"Equals(MeasureOf(Angle(D, C, B)), 60)",
"Equals(LengthOf(Line(D, C)), 10)"
] | [
"AB",
"AD",
"BC",
"DC"
] | {"A": [308.0, 2.0], "B": [212.0, 165.0], "C": [0.0, 168.0], "D": [98.0, 0.0]} | [
""
] | |
<image>Find y. | \frac { 3 } { 2 } | [
"Find(y)"
] | [
"Equals(LengthOf(Line(F, B)), x) ",
"Equals(LengthOf(Line(A, B)), y)",
"Equals(LengthOf(Line(A, F)), z)",
"Equals(LengthOf(Line(F, L)), 2\\sqrt{3})",
"Equals(LengthOf(Line(A, L)), \\sqrt{3})",
"PointLiesOnLine(B, Line(F, L))",
"Perpendicular(Line(F, B), Line(A, B))",
"Perpendicular(Line(A, F), Line(A,... | [
"AB",
"AF",
"AL",
"FB",
"FL",
"LB"
] | {"A": [222.0, 249.0], "B": [99.0, 138.0], "F": [223.0, 1.0], "L": [1.0, 249.0]} | [
""
] |
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