id int32 1 6.98k | annotation stringclasses 132
values | source stringlengths 7 17 | problem_level int32 0 28 | problem_text_cn stringlengths 20 201 | problem_text_en stringlengths 58 424 | problem_img stringlengths 5 8 | construction_cdl listlengths 1 28 | text_cdl listlengths 0 16 | image_cdl listlengths 0 16 | goal_cdl stringlengths 8 131 | problem_answer stringclasses 906
values | theorem_seqs listlengths 0 28 | theorem_seqs_dag_json stringlengths 13 3.3k | image imagewidth (px) 48 1.6k |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
801 | NaZhu_2023-04-09 | Geometry3k-827 | 1 | 如图所示，∠EYQ=3*y+1°，∠MAQ=3*x+11°，∠YQF=4*x-5°，EF平行于MA，EF∥YQ，YM平行于QA，YQ平行于MA。求x的值。 | As shown in the diagram, ∠EYQ=3*y+1°, ∠MAQ=3*x+11°, ∠YQF=4*x-5°, EF is parallel to MA, EF is parallel to YQ, YM∥QA, YQ∥MA. Find the value of x. | 801.png | [
"Shape(EY,YQ,QF,FE)",
"Shape(YM,MA,AQ,QY)",
"Collinear(EYM)",
"Collinear(FQA)"
] | [
"Equal(MeasureOfAngle(EYQ),3*y+1)",
"Equal(MeasureOfAngle(MAQ),3*x+11)",
"Equal(MeasureOfAngle(YQF),4*x-5)",
"ParallelBetweenLine(EF,MA)",
"ParallelBetweenLine(EF,YQ)",
"ParallelBetweenLine(YM,QA)",
"ParallelBetweenLine(YQ,MA)"
] | [
"Equal(MeasureOfAngle(EYQ),3*y+1)",
"Equal(MeasureOfAngle(MAQ),3*x+11)",
"Equal(MeasureOfAngle(YQF),4*x-5)",
"ParallelBetweenLine(EF,MA)",
"ParallelBetweenLine(EF,YQ)",
"ParallelBetweenLine(YM,QA)",
"ParallelBetweenLine(YQ,MA)"
] | Value(x) | 16 | [
"parallel_property_corresponding_angle(2,AM,QY,F)"
] | {"START": ["parallel_property_corresponding_angle(2,AM,QY,F)"]} | |
802 | XiaokaiZhang_2023-03-19 | Geometry3k-828 | 1 | 如图所示，QP=17，RP=8，RQ=15，QR垂直于PR。求sin(PQ)的值。 | As shown in the diagram, QP=17, RP=8, RQ=15, QR is perpendicular to PR. Find the value of sin(PQ). | 802.png | [
"Shape(QR,RP,PQ)"
] | [
"Equal(LengthOfLine(QP),17)",
"Equal(LengthOfLine(RP),8)",
"Equal(LengthOfLine(RQ),15)",
"PerpendicularBetweenLine(QR,PR)"
] | [
"Equal(LengthOfLine(QP),17)",
"Equal(LengthOfLine(RP),8)",
"Equal(LengthOfLine(RQ),15)",
"PerpendicularBetweenLine(QR,PR)"
] | Value(Sin(MeasureOfAngle(PQR))) | 8/17 | [
"sine_theorem(1,PQR)"
] | {"START": ["sine_theorem(1,PQR)"]} | |
803 | XiaokaiZhang_2023-03-19 | Geometry3k-829 | 1 | 如图所示，∠AFB=70°，∠ECF=35°，AD⊥ED，EC垂直于BC。求∠FCB的大小。 | As shown in the diagram, ∠AFB=70°, ∠ECF=35°, AD⊥ED, EC is perpendicular to BC. Find the measure of ∠FCB. | 803.png | [
"Shape(AD,DE,EF,FA)",
"Shape(FE,EC,CF)",
"Shape(BF,FC,CB)",
"Shape(AF,FB)",
"Collinear(DEC)",
"Collinear(AFC)",
"Collinear(EFB)"
] | [
"Equal(MeasureOfAngle(AFB),70)",
"Equal(MeasureOfAngle(ECF),35)",
"PerpendicularBetweenLine(AD,ED)",
"PerpendicularBetweenLine(EC,BC)"
] | [
"Equal(MeasureOfAngle(AFB),70)",
"Equal(MeasureOfAngle(ECF),35)",
"PerpendicularBetweenLine(AD,ED)",
"PerpendicularBetweenLine(EC,BC)"
] | Value(MeasureOfAngle(FCB)) | 55 | [
"angle_addition(1,ECF,FCB)"
] | {"START": ["angle_addition(1,ECF,FCB)"]} | |
804 | XiaokaiZhang_2023-03-19 | Geometry3k-830 | 8 | 如图所示，AC=y，AD=9，BC=z，CD=x，DB=16，AD⊥CD，BC⊥AC。求y的值。 | As shown in the diagram, AC=y, AD=9, BC=z, CD=x, DB=16, AD is perpendicular to CD, BC is perpendicular to AC. Find the value of y. | 804.png | [
"Shape(CA,AD,DC)",
"Shape(CD,DB,BC)",
"Collinear(ADB)"
] | [
"Equal(LengthOfLine(AC),y)",
"Equal(LengthOfLine(AD),9)",
"Equal(LengthOfLine(BC),z)",
"Equal(LengthOfLine(CD),x)",
"Equal(LengthOfLine(DB),16)",
"PerpendicularBetweenLine(AD,CD)",
"PerpendicularBetweenLine(BC,AC)"
] | [
"Equal(LengthOfLine(AC),y)",
"Equal(LengthOfLine(AD),9)",
"Equal(LengthOfLine(BC),z)",
"Equal(LengthOfLine(CD),x)",
"Equal(LengthOfLine(DB),16)",
"PerpendicularBetweenLine(AD,CD)",
"PerpendicularBetweenLine(BC,AC)"
] | Value(y) | 15 | [
"adjacent_complementary_angle(1,ADC,CDB)",
"right_triangle_judgment_angle(1,ADC)",
"right_triangle_judgment_angle(1,CDB)",
"right_triangle_judgment_angle(1,BCA)",
"line_addition(1,AD,DB)",
"right_triangle_property_pythagorean(1,ADC)",
"right_triangle_property_pythagorean(1,CDB)",
"right_triangle_prope... | {"START": ["adjacent_complementary_angle(1,ADC,CDB)", "right_triangle_judgment_angle(1,ADC)", "right_triangle_judgment_angle(1,BCA)", "line_addition(1,AD,DB)"], "adjacent_complementary_angle(1,ADC,CDB)": ["right_triangle_judgment_angle(1,CDB)"], "right_triangle_judgment_angle(1,ADC)": ["right_triangle_property_pythagor... | |
805 | XiaokaiZhang_2023-03-19 | Geometry3k-831 | 1 | 如图所示，∠CAB=50°，∠DAC=x°。求x的值。 | As shown in the diagram, ∠CAB=50°, ∠DAC=x°. Find the value of x. | 805.png | [
"Shape(CA,AB,BC)",
"Shape(DA,AC)",
"Collinear(DAB)"
] | [
"Equal(MeasureOfAngle(CAB),50)",
"Equal(MeasureOfAngle(DAC),x)"
] | [
"Equal(MeasureOfAngle(CAB),50)",
"Equal(MeasureOfAngle(DAC),x)"
] | Value(x) | 130 | [
"adjacent_complementary_angle(1,DAC,CAB)"
] | {"START": ["adjacent_complementary_angle(1,DAC,CAB)"]} | |
806 | NaZhu_2023-04-09 | Geometry3k-832 | 2 | 如图所示，AC=15，∠AFB=y**2+26°，∠BAF=2*x+20°，∠FAD=5*x-4°，ADCB是菱形。求直线AF的长度。 | As shown in the diagram, AC=15, ∠AFB=y**2+26°, ∠BAF=2*x+20°, ∠FAD=5*x-4°, ADCB is a rhombus. Find the length of line AF. | 806.png | [
"Shape(AD,DF,FA)",
"Shape(FD,DC,CF)",
"Shape(BA,AF,FB)",
"Shape(BF,FC,CB)",
"Collinear(AFC)",
"Collinear(BFD)"
] | [
"Equal(LengthOfLine(AC),15)",
"Equal(MeasureOfAngle(AFB),y**2+26)",
"Equal(MeasureOfAngle(BAF),2*x+20)",
"Equal(MeasureOfAngle(FAD),5*x-4)",
"Rhombus(ADCB)"
] | [] | Value(LengthOfLine(AF)) | 15/2 | [
"line_addition(1,AF,FC)",
"parallelogram_property_diagonal_bisection(1,ADCB,F)"
] | {"START": ["line_addition(1,AF,FC)", "parallelogram_property_diagonal_bisection(1,ADCB,F)"]} | |
807 | XiaokaiZhang_2023-03-19 | Geometry3k-833 | 2 | 如图所示，AB=x，AC=14，BC=48，BC垂直于AC。求x的值。 | As shown in the diagram, AB=x, AC=14, BC=48, BC is perpendicular to AC. Find the value of x. | 807.png | [
"Shape(BC,CA,AB)"
] | [
"Equal(LengthOfLine(AB),x)",
"Equal(LengthOfLine(AC),14)",
"Equal(LengthOfLine(BC),48)",
"PerpendicularBetweenLine(BC,AC)"
] | [
"Equal(LengthOfLine(AB),x)",
"Equal(LengthOfLine(AC),14)",
"Equal(LengthOfLine(BC),48)",
"PerpendicularBetweenLine(BC,AC)"
] | Value(x) | 50 | [
"right_triangle_judgment_angle(1,BCA)",
"right_triangle_property_pythagorean(1,BCA)"
] | {"START": ["right_triangle_judgment_angle(1,BCA)"], "right_triangle_judgment_angle(1,BCA)": ["right_triangle_property_pythagorean(1,BCA)"]} | |
808 | NaZhu_2023-04-09 | Geometry3k-834 | 2 | 如图所示，∠MRP=5*x+7°，∠NPR=7*x-21°，LR∥NP。求∠MRP的大小。 | As shown in the diagram, ∠MRP=5*x+7°, ∠NPR=7*x-21°, LR∥NP. Find the measure of ∠MRP. | 808.png | [
"Shape(CR,RM)",
"Shape(MR,RP)",
"Shape(RP,PJ)",
"Shape(JP,PQ)",
"Shape(LR,RC)",
"Shape(PR,RL)",
"Shape(NP,PR)",
"Shape(QP,PN)",
"Collinear(CRPQ)",
"Collinear(MRL)",
"Collinear(JPN)"
] | [
"Equal(MeasureOfAngle(MRP),5*x+7)",
"Equal(MeasureOfAngle(NPR),7*x-21)",
"ParallelBetweenLine(LR,NP)"
] | [
"Equal(MeasureOfAngle(MRP),5*x+7)",
"Equal(MeasureOfAngle(NPR),7*x-21)",
"ParallelBetweenLine(LR,NP)"
] | Value(MeasureOfAngle(MRP)) | 77 | [
"vertical_angle(1,MRP,LRC)",
"parallel_property_corresponding_angle(2,PN,RL,C)"
] | {"START": ["vertical_angle(1,MRP,LRC)", "parallel_property_corresponding_angle(2,PN,RL,C)"]} | |
809 | NaZhu_2023-04-09 | Geometry3k-835 | 2 | 如图所示，AB=15，AE=18.5，BD=q，DF=16，RE=r，RF=2，圆O的切线为BA。求q的值。 | As shown in the diagram, AB=15, AE=18.5, BD=q, DF=16, RE=r, RF=2, BA is the tangent to ⊙C. Find the value of q. | 809.png | [
"Shape(CEF,ER,RF)",
"Shape(CFD,DF)",
"Shape(CAE,EA)",
"Shape(CEF,FD,CDA,AE)",
"Shape(CDA,DB,DA)",
"Collinear(RFDB)",
"Collinear(REA)",
"Cocircular(C,EFDA)"
] | [
"Equal(LengthOfLine(AB),15)",
"Equal(LengthOfLine(AE),18.5)",
"Equal(LengthOfLine(BD),q)",
"Equal(LengthOfLine(DF),16)",
"Equal(LengthOfLine(RE),r)",
"Equal(LengthOfLine(RF),2)",
"IsTangentOfCircle(BA,C)"
] | [
"Equal(LengthOfLine(AB),15)",
"Equal(LengthOfLine(AE),18.5)",
"Equal(LengthOfLine(BD),q)",
"Equal(LengthOfLine(DF),16)",
"Equal(LengthOfLine(RE),r)",
"Equal(LengthOfLine(RF),2)"
] | Value(q) | 9 | [
"circle_property_circular_power_tangent_and_segment_line(1,BA,BDF,C)",
"line_addition(1,BD,DF)"
] | {"START": ["circle_property_circular_power_tangent_and_segment_line(1,BA,BDF,C)", "line_addition(1,BD,DF)"]} | |
810 | XiaokaiZhang_2023-03-19 | Geometry3k-836 | 3 | 如图所示，XB=12，XC=13，YE=x，YF=13/2，∠YFE=∠XCB，CB⊥XB，FE垂直于YE。求x的值。 | As shown in the diagram, XB=12, XC=13, YE=x, YF=13/2, ∠YFE=∠XCB, CB is perpendicular to XB, FE⊥YE. Find the value of x. | 810.png | [
"Shape(AY,YE,EA)",
"Shape(EY,YF,FE)",
"Collinear(AEF)",
"Shape(DX,XB,BD)",
"Shape(BX,XC,CB)",
"Collinear(DBC)"
] | [
"Equal(LengthOfLine(XB),12)",
"Equal(LengthOfLine(XC),13)",
"Equal(LengthOfLine(YE),x)",
"Equal(LengthOfLine(YF),13/2)",
"Equal(MeasureOfAngle(YFE),MeasureOfAngle(XCB))",
"PerpendicularBetweenLine(CB,XB)",
"PerpendicularBetweenLine(FE,YE)"
] | [
"Equal(LengthOfLine(XB),12)",
"Equal(LengthOfLine(XC),13)",
"Equal(LengthOfLine(YE),x)",
"Equal(LengthOfLine(YF),13/2)",
"Equal(MeasureOfAngle(YFE),MeasureOfAngle(XCB))",
"PerpendicularBetweenLine(CB,XB)",
"PerpendicularBetweenLine(FE,YE)"
] | Value(x) | 6 | [
"similar_triangle_judgment_aa(1,YFE,XCB)",
"similar_triangle_property_line_ratio(1,EYF,BXC)",
"similar_triangle_property_line_ratio(1,FEY,CBX)"
] | {"START": ["similar_triangle_judgment_aa(1,YFE,XCB)"], "similar_triangle_judgment_aa(1,YFE,XCB)": ["similar_triangle_property_line_ratio(1,FEY,CBX)", "similar_triangle_property_line_ratio(1,EYF,BXC)"]} | |
811 | NaZhu_2023-04-09 | Geometry3k-837 | 8 | 如图所示，∠ALJ=133°，∠EKI=126°，BL平行于KJ。求∠KEJ的大小。 | As shown in the diagram, ∠ALJ=133°, ∠EKI=126°, BL is parallel to KJ. Find the measure of ∠KEJ. | 811.png | [
"Shape(JK,KE,EJ)",
"Shape(DL,LA)",
"Shape(BL,LD)",
"Shape(BL,LJ)",
"Shape(JL,LA)",
"Shape(LJ,JF)",
"Shape(KJ,JL)",
"Shape(EJ,JK)",
"Shape(JE,EG)",
"Shape(GE,EC)",
"Shape(CE,EK)",
"Shape(HK,KJ)",
"Shape(IK,KH)",
"Shape(EK,KI)",
"Collinear(DLJEC)",
"Collinear(ALB)",
"Collinear(FJKI)",
... | [
"Equal(MeasureOfAngle(ALJ),133)",
"Equal(MeasureOfAngle(EKI),126)",
"ParallelBetweenLine(BL,KJ)"
] | [
"Equal(MeasureOfAngle(ALJ),133)",
"Equal(MeasureOfAngle(EKI),126)"
] | Value(MeasureOfAngle(KEJ)) | 79 | [
"parallel_property_collinear_extend(1,JK,LB,F)",
"parallel_property_collinear_extend(2,BL,JF,A)",
"parallel_property_corresponding_angle(2,LA,JF,E)",
"flat_angle(1,FJK)",
"angle_addition(1,FJE,EJK)",
"flat_angle(1,JKI)",
"angle_addition(1,JKE,EKI)",
"triangle_property_angle_sum(1,JKE)"
] | {"START": ["parallel_property_collinear_extend(1,JK,LB,F)", "flat_angle(1,FJK)", "angle_addition(1,FJE,EJK)", "flat_angle(1,JKI)", "angle_addition(1,JKE,EKI)", "triangle_property_angle_sum(1,JKE)"], "parallel_property_collinear_extend(1,JK,LB,F)": ["parallel_property_collinear_extend(2,BL,JF,A)"], "parallel_property_co... | |
812 | NaZhu_2023-04-09 | Geometry3k-838 | 1 | 如图所示，AD=9，BD=x，DC=6，DE=3。求x的值。 | As shown in the diagram, AD=9, BD=x, DC=6, DE=3. Find the value of x. | 812.png | [
"Shape(FBE,ED,DB)",
"Shape(FEA,AD,DE)",
"Shape(FAC,CD,DA)",
"Shape(FCB,BD,DC)",
"Collinear(BDA)",
"Collinear(EDC)",
"Cocircular(F,CBEA)"
] | [
"Equal(LengthOfLine(AD),9)",
"Equal(LengthOfLine(BD),x)",
"Equal(LengthOfLine(DC),6)",
"Equal(LengthOfLine(DE),3)"
] | [
"Equal(LengthOfLine(AD),9)",
"Equal(LengthOfLine(BD),x)",
"Equal(LengthOfLine(DC),6)",
"Equal(LengthOfLine(DE),3)"
] | Value(x) | 2 | [
"circle_property_circular_power_chord_and_chord(1,CDE,BDA,F)"
] | {"START": ["circle_property_circular_power_chord_and_chord(1,CDE,BDA,F)"]} | |
813 | XiaokaiZhang_2023-03-19 | Geometry3k-839 | 0 | 如图所示，KJ=7*x，KJ=LJ，KL=11*x-8，KL=KJ，LJ=x+12。求直线JK的长度。 | As shown in the diagram, KJ=7*x, KJ=LJ, KL=11*x-8, KL=KJ, LJ=x+12. Find the length of line JK. | 813.png | [
"Shape(KJ,JL,LK)"
] | [
"Equal(LengthOfLine(KJ),7*x)",
"Equal(LengthOfLine(KJ),LengthOfLine(LJ))",
"Equal(LengthOfLine(KL),11*x-8)",
"Equal(LengthOfLine(KL),LengthOfLine(KJ))",
"Equal(LengthOfLine(LJ),x+12)"
] | [
"Equal(LengthOfLine(KJ),7*x)",
"Equal(LengthOfLine(KJ),LengthOfLine(LJ))",
"Equal(LengthOfLine(KL),11*x-8)",
"Equal(LengthOfLine(KL),LengthOfLine(KJ))",
"Equal(LengthOfLine(LJ),x+12)"
] | Value(LengthOfLine(JK)) | 14 | [] | {"START": []} | |
814 | NaZhu_2023-04-09 | Geometry3k-840 | 2 | 如图所示，弧ONM的长度与⌒OPQ的长度相等，NM=26，PQ=3*x+5。求x的值。 | As shown in the diagram, the length of ⌒ONM is equal to the length of ⌒OPQ, NM=26, PQ=3*x+5. Find the value of x. | 814.png | [
"Shape(ONM,MN)",
"Shape(OPQ,QP)",
"Shape(OQN,NM,OMP,PQ)",
"Cocircular(O,NMPQ)"
] | [
"Equal(LengthOfArc(ONM),LengthOfArc(OPQ))",
"Equal(LengthOfLine(NM),26)",
"Equal(LengthOfLine(PQ),3*x+5)"
] | [
"Equal(LengthOfArc(ONM),LengthOfArc(OPQ))",
"Equal(LengthOfLine(NM),26)",
"Equal(LengthOfLine(PQ),3*x+5)"
] | Value(x) | 7 | [
"congruent_arc_judgment_length_equal(1,ONM,OPQ)",
"congruent_arc_property_chord_equal(1,ONM,OPQ)"
] | {"START": ["congruent_arc_judgment_length_equal(1,ONM,OPQ)"], "congruent_arc_judgment_length_equal(1,ONM,OPQ)": ["congruent_arc_property_chord_equal(1,ONM,OPQ)"]} | |
815 | XiaokaiZhang_2023-03-19 | Geometry3k-841 | 1 | 如图所示，∠ALC=35°，∠GCL=28°，∠LDE=25°，∠LEG=51°，CA⊥LA，LC⊥FC，LG垂直于CG。求∠LCA的大小。 | As shown in the diagram, ∠ALC=35°, ∠GCL=28°, ∠LDE=25°, ∠LEG=51°, CA is perpendicular to LA, LC⊥FC, LG⊥CG. Find the measure of ∠LCA. | 815.png | [
"Shape(LD,DE,EL)",
"Shape(LE,EG,GL)",
"Shape(LG,GC,CL)",
"Shape(LC,CA,AL)",
"Shape(AC,CF,FA)",
"Collinear(DEGC)",
"Collinear(LAF)"
] | [
"Equal(MeasureOfAngle(ALC),35)",
"Equal(MeasureOfAngle(GCL),28)",
"Equal(MeasureOfAngle(LDE),25)",
"Equal(MeasureOfAngle(LEG),51)",
"PerpendicularBetweenLine(CA,LA)",
"PerpendicularBetweenLine(LC,FC)",
"PerpendicularBetweenLine(LG,CG)"
] | [
"Equal(MeasureOfAngle(ALC),35)",
"Equal(MeasureOfAngle(GCL),28)",
"Equal(MeasureOfAngle(LDE),25)",
"Equal(MeasureOfAngle(LEG),51)",
"PerpendicularBetweenLine(CA,LA)",
"PerpendicularBetweenLine(LC,FC)",
"PerpendicularBetweenLine(LG,CG)"
] | Value(MeasureOfAngle(LCA)) | 55 | [
"triangle_property_angle_sum(1,ALC)"
] | {"START": ["triangle_property_angle_sum(1,ALC)"]} | |
816 | NaZhu_2023-04-09 | Geometry3k-843 | 7 | 如图所示，JO=12，KM=7，NL=x+3，OL=4*x-9，△KJL的内心为R，圆O的切线为JM，JO是⊙O的切线，KM是⊙O的切线，KN是⊙O的切线，⊙O的切线为LN，LO是⊙O的切线。求三角形JLK的周长。 | As shown in the diagram, JO=12, KM=7, NL=x+3, OL=4*x-9, R is the center of the inscribed circle of △KJL, the tangent to ⊙R is JM, JO is the tangent to circle R, the tangent to circle R is KM, KN is the tangent to ⊙R, the tangent to ⊙R is LN, the tangent to circle R is LO. Find the perimeter of triangle JLK. | 816.png | [
"Shape(RNM,RMO,RON)",
"Shape(RNM,NK,KM)",
"Shape(RMO,MJ,JO)",
"Shape(RON,OL,LN)",
"Collinear(JMK)",
"Collinear(JOL)",
"Collinear(KNL)",
"Cocircular(R,MON)"
] | [
"Equal(LengthOfLine(JO),12)",
"Equal(LengthOfLine(KM),7)",
"Equal(LengthOfLine(NL),x+3)",
"Equal(LengthOfLine(OL),4*x-9)",
"IsIncenterOfTriangle(R,KJL)",
"IsTangentOfCircle(JM,R)",
"IsTangentOfCircle(JO,R)",
"IsTangentOfCircle(KM,R)",
"IsTangentOfCircle(KN,R)",
"IsTangentOfCircle(LN,R)",
"IsTang... | [
"Equal(LengthOfLine(JO),12)",
"Equal(LengthOfLine(KM),7)",
"Equal(LengthOfLine(NL),x+3)",
"Equal(LengthOfLine(OL),4*x-9)",
"IsTangentOfCircle(JM,R)",
"IsTangentOfCircle(JO,R)",
"IsTangentOfCircle(KM,R)",
"IsTangentOfCircle(KN,R)",
"IsTangentOfCircle(LN,R)",
"IsTangentOfCircle(LO,R)"
] | Value(PerimeterOfTriangle(JLK)) | 52 | [
"tangent_of_circle_property_length_equal(1,JM,JO,R)",
"tangent_of_circle_property_length_equal(1,KM,KN,R)",
"tangent_of_circle_property_length_equal(1,LO,LN,R)",
"triangle_perimeter_formula(1,JLK)",
"line_addition(1,JM,MK)",
"line_addition(1,JO,OL)",
"line_addition(1,LN,NK)"
] | {"START": ["tangent_of_circle_property_length_equal(1,JM,JO,R)", "tangent_of_circle_property_length_equal(1,KM,KN,R)", "tangent_of_circle_property_length_equal(1,LO,LN,R)", "triangle_perimeter_formula(1,JLK)", "line_addition(1,JM,MK)", "line_addition(1,JO,OL)", "line_addition(1,LN,NK)"]} | |
817 | XiaokaiZhang_2023-03-19 | Geometry3k-844 | 6 | 如图所示，BC=BA，BC=y，BD=12，CA=x，∠BCD=60°，BD⊥AD。求x的值。 | As shown in the diagram, BC=BA, BC=y, BD=12, CA=x, ∠BCD=60°, BD⊥AD. Find the value of x. | 817.png | [
"Shape(BC,CD,DB)",
"Shape(BD,DA,AB)",
"Collinear(CDA)"
] | [
"Equal(LengthOfLine(BC),LengthOfLine(BA))",
"Equal(LengthOfLine(BC),y)",
"Equal(LengthOfLine(BD),12)",
"Equal(LengthOfLine(CA),x)",
"Equal(MeasureOfAngle(BCD),60)",
"PerpendicularBetweenLine(BD,AD)"
] | [
"Equal(LengthOfLine(BC),LengthOfLine(BA))",
"Equal(LengthOfLine(BC),y)",
"Equal(LengthOfLine(BD),12)",
"Equal(LengthOfLine(CA),x)",
"Equal(MeasureOfAngle(BCD),60)",
"PerpendicularBetweenLine(BD,AD)"
] | Value(x) | 8*sqrt(3) | [
"adjacent_complementary_angle(1,CDB,BDA)",
"triangle_property_angle_sum(1,CDB)",
"sine_theorem(1,CDB)",
"sine_theorem(1,BCD)",
"perpendicular_bisector_judgment_distance_equal(1,BD,CA)",
"line_addition(1,CD,DA)"
] | {"START": ["adjacent_complementary_angle(1,CDB,BDA)", "triangle_property_angle_sum(1,CDB)", "sine_theorem(1,CDB)", "sine_theorem(1,BCD)", "line_addition(1,CD,DA)"], "adjacent_complementary_angle(1,CDB,BDA)": ["perpendicular_bisector_judgment_distance_equal(1,BD,CA)"]} | |
818 | NaZhu_2023-04-09 | Geometry3k-845 | 1 | 如图所示，∠HPM=4*y°，∠MPR=68°，∠PRC=x°，∠SCR=5*z+2°，CR平行于MP，MC∥PR。求x的值。 | As shown in the diagram, ∠HPM=4*y°, ∠MPR=68°, ∠PRC=x°, ∠SCR=5*z+2°, CR∥MP, MC∥PR. Find the value of x. | 818.png | [
"Shape(CM,MP,PR,RC)",
"Shape(SC,CR)",
"Shape(NC,CS)",
"Shape(MC,CN)",
"Shape(CR,RG)",
"Shape(GR,RI)",
"Shape(IR,RP)",
"Shape(EM,MD)",
"Shape(DM,MC)",
"Shape(PM,ME)",
"Shape(HP,PM)",
"Shape(LP,PH)",
"Shape(RP,PL)",
"Collinear(NCRI)",
"Collinear(DMPL)",
"Collinear(EMCS)",
"Collinear(HP... | [
"Equal(MeasureOfAngle(HPM),4*y)",
"Equal(MeasureOfAngle(MPR),68)",
"Equal(MeasureOfAngle(PRC),x)",
"Equal(MeasureOfAngle(SCR),5*z+2)",
"ParallelBetweenLine(CR,MP)",
"ParallelBetweenLine(MC,PR)"
] | [
"Equal(MeasureOfAngle(HPM),4*y)",
"Equal(MeasureOfAngle(MPR),68)",
"Equal(MeasureOfAngle(PRC),x)",
"Equal(MeasureOfAngle(SCR),5*z+2)",
"ParallelBetweenLine(CR,MP)",
"ParallelBetweenLine(MC,PR)"
] | Value(x) | 112 | [
"parallel_property_ipsilateral_internal_angle(1,PM,RC)"
] | {"START": ["parallel_property_ipsilateral_internal_angle(1,PM,RC)"]} | |
819 | NaZhu_2023-04-09 | Geometry3k-846 | 4 | 如图所示，PQ=10，∠QPM=81°，P是⊙P的圆心。求扇形PQM的面积。 | As shown in the diagram, PQ=10, ∠QPM=81°, the center of circle P is P. Find the area of the sector PQM. | 819.png | [
"Shape(PQM,MP,PQ)",
"Shape(PMQ,QP,PM)",
"Cocircular(P,QM)"
] | [
"Equal(LengthOfLine(PQ),10)",
"Equal(MeasureOfAngle(QPM),81)",
"IsCentreOfCircle(P,P)"
] | [
"Equal(LengthOfLine(PQ),10)",
"Equal(MeasureOfAngle(QPM),81)",
"IsCentreOfCircle(P,P)"
] | Value(AreaOfSector(PQM)) | 155*pi/2 | [
"radius_of_circle_property_length_equal(1,PQ,P)",
"round_angle(1,MPQ,QPM)",
"arc_property_center_angle(1,PQM,P)",
"sector_area_formula(1,PQM)"
] | {"START": ["radius_of_circle_property_length_equal(1,PQ,P)", "round_angle(1,MPQ,QPM)", "arc_property_center_angle(1,PQM,P)", "sector_area_formula(1,PQM)"]} | |
820 | NaZhu_2023-04-09 | Geometry3k-847 | 9 | 如图所示，AN=19，BF=10，CE=2，FN=20，AE垂直于FE，EF垂直于NF，AN和CB是梯形ACBN的平行边。求ACBN的面积。 | As shown in the diagram, AN=19, BF=10, CE=2, FN=20, AE is perpendicular to FE, EF⊥NF, AC and BN are the non-parallel sides (legs) of trapezoid ACBN. Find the area of quadrilateral ACBN. | 820.png | [
"Shape(AC,CE,EA)",
"Shape(AE,EF,FN,NA)",
"Shape(NF,FB,BN)",
"Collinear(CEFB)"
] | [
"Equal(LengthOfLine(AN),19)",
"Equal(LengthOfLine(BF),10)",
"Equal(LengthOfLine(CE),2)",
"Equal(LengthOfLine(FN),20)",
"PerpendicularBetweenLine(AE,FE)",
"PerpendicularBetweenLine(EF,NF)",
"Trapezoid(ACBN)"
] | [
"Equal(LengthOfLine(AN),19)",
"Equal(LengthOfLine(BF),10)",
"Equal(LengthOfLine(CE),2)",
"Equal(LengthOfLine(FN),20)",
"PerpendicularBetweenLine(AE,FE)",
"PerpendicularBetweenLine(EF,NF)"
] | Value(AreaOfQuadrilateral(ACBN)) | 500 | [
"parallel_judgment_ipsilateral_internal_angle(1,EA,FN)",
"parallel_property_collinear_extend(3,BC,NA,E)",
"parallel_property_collinear_extend(1,EC,NA,F)",
"parallelogram_judgment_parallel_and_parallel(1,AEFN)",
"parallelogram_property_opposite_line_equal(1,EFNA)",
"line_addition(1,CE,EF)",
"line_additio... | {"START": ["parallel_judgment_ipsilateral_internal_angle(1,EA,FN)", "parallel_property_collinear_extend(3,BC,NA,E)", "line_addition(1,CE,EF)", "line_addition(1,CF,FB)", "altitude_of_quadrilateral_judgment_right_vertex(2,NF,ACBN)", "trapezoid_area_formula(1,ACBN)"], "parallel_judgment_ipsilateral_internal_angle(1,EA,FN)... | |
821 | NaZhu_2023-04-09 | Geometry3k-849 | 1 | 如图所示，四边形MPON是菱形。求∠MRN的大小。 | As shown in the diagram, MPON is a rhombus. Find the measure of ∠MRN. | 821.png | [
"Shape(MP,PR,RM)",
"Shape(RP,PO,OR)",
"Shape(RO,ON,NR)",
"Shape(RN,NM,MR)",
"Collinear(MRO)",
"Collinear(PRN)"
] | [
"Rhombus(MPON)"
] | [] | Value(MeasureOfAngle(MRN)) | 90 | [
"kite_property_diagonal_perpendicular_bisection(1,MPON,R)"
] | {"START": ["kite_property_diagonal_perpendicular_bisection(1,MPON,R)"]} | |
822 | XiaokaiZhang_2023-03-19 | Geometry3k-850 | 7 | 如图所示，QT=18，RT=34，QS垂直于RS，RT垂直于ST。求△SRQ的高。 | As shown in the diagram, QT=18, RT=34, QS⊥RS, RT⊥ST. Find the height of triangle SRQ. | 822.png | [
"Shape(SR,RT,TS)",
"Shape(ST,TQ,QS)",
"Collinear(RTQ)"
] | [
"Equal(LengthOfLine(QT),18)",
"Equal(LengthOfLine(RT),34)",
"PerpendicularBetweenLine(QS,RS)",
"PerpendicularBetweenLine(RT,ST)"
] | [
"Equal(LengthOfLine(QT),18)",
"Equal(LengthOfLine(RT),34)",
"PerpendicularBetweenLine(QS,RS)",
"PerpendicularBetweenLine(RT,ST)"
] | Value(HeightOfTriangle(SRQ)) | 6*sqrt(17) | [
"mirror_similar_triangle_judgment_aa(1,SRT,QSR)",
"altitude_of_triangle_judgment(1,ST,SRQ)",
"line_addition(1,QT,TR)",
"right_triangle_judgment_angle(1,RTS)",
"right_triangle_property_pythagorean(1,RTS)",
"mirror_similar_triangle_property_line_ratio(1,SRT,QSR)",
"mirror_similar_triangle_property_line_ra... | {"START": ["mirror_similar_triangle_judgment_aa(1,SRT,QSR)", "altitude_of_triangle_judgment(1,ST,SRQ)", "line_addition(1,QT,TR)", "right_triangle_judgment_angle(1,RTS)"], "mirror_similar_triangle_judgment_aa(1,SRT,QSR)": ["mirror_similar_triangle_property_line_ratio(1,SRT,QSR)", "mirror_similar_triangle_property_line_r... | |
823 | NaZhu_2023-04-09 | Geometry3k-851 | 2 | 如图所示，VW=3，XV=2*a-2，ZV=Div(5*a+1,4)，∠ZYV=53°，XWZY是菱形。求∠VZY的大小。 | As shown in the diagram, VW=3, XV=2*a-2, ZV=Div(5*a+1,4), ∠ZYV=53°, XWZY is a rhombus. Find the measure of ∠VZY. | 823.png | [
"Shape(XV,VY,YX)",
"Shape(XW,WV,VX)",
"Shape(YV,VZ,ZY)",
"Shape(VW,WZ,ZV)",
"Collinear(XVZ)",
"Collinear(YVW)"
] | [
"Equal(LengthOfLine(VW),3)",
"Equal(LengthOfLine(XV),2*a-2)",
"Equal(LengthOfLine(ZV),Div(5*a+1,4))",
"Equal(MeasureOfAngle(ZYV),53)",
"Rhombus(XWZY)"
] | [
"Equal(LengthOfLine(VW),3)",
"Equal(LengthOfLine(XV),2*a-2)",
"Equal(LengthOfLine(ZV),Div(5*a+1,4))",
"Equal(MeasureOfAngle(ZYV),53)"
] | Value(MeasureOfAngle(VZY)) | 37 | [
"kite_property_diagonal_perpendicular_bisection(1,ZYXW,V)",
"triangle_property_angle_sum(1,YVZ)"
] | {"START": ["kite_property_diagonal_perpendicular_bisection(1,ZYXW,V)", "triangle_property_angle_sum(1,YVZ)"]} | |
824 | NaZhu_2023-04-09 | Geometry3k-853 | 5 | 如图所示，GJ=40，圆L的半径为32，圆L的圆心为L，JK垂直于LK。求直线LK的长度。 | As shown in the diagram, GJ=40, the radius of circle L is 32, L is the center of circle L, JK⊥LK. Find the length of line LK. | 824.png | [
"Shape(LHG,GK,KH)",
"Shape(LJH,HK,KJ)",
"Shape(KL,LJ,JK)",
"Shape(LGJ,JL,LK,KG)",
"Collinear(GKJ)",
"Collinear(LKH)",
"Cocircular(L,JHG)"
] | [
"Equal(LengthOfLine(GJ),40)",
"Equal(RadiusOfCircle(L),32)",
"IsCentreOfCircle(L,L)",
"PerpendicularBetweenLine(JK,LK)"
] | [
"IsCentreOfCircle(L,L)",
"PerpendicularBetweenLine(JK,LK)"
] | Value(LengthOfLine(LK)) | 4*sqrt(39) | [
"line_addition(1,GK,KJ)",
"circle_property_chord_perpendicular_bisect_chord(1,L,LK,JG)",
"radius_of_circle_property_length_equal(1,LJ,L)",
"right_triangle_judgment_angle(1,JKL)",
"right_triangle_property_pythagorean(1,JKL)"
] | {"START": ["line_addition(1,GK,KJ)", "circle_property_chord_perpendicular_bisect_chord(1,L,LK,JG)", "radius_of_circle_property_length_equal(1,LJ,L)", "right_triangle_judgment_angle(1,JKL)"], "right_triangle_judgment_angle(1,JKL)": ["right_triangle_property_pythagorean(1,JKL)"]} | |
825 | XiaokaiZhang_2023-03-19 | Geometry3k-854 | 0 | 如图所示，AC=2/3*x-4，AD=1/3*x+2，AD=AC，DE=5*y，EB=7/3*y+8，ED=EB。求x的值。 | As shown in the diagram, AC=2/3*x-4, AD=1/3*x+2, AD=AC, DE=5*y, EB=7/3*y+8, ED=EB. Find the value of x. | 825.png | [
"Shape(DA,AE,ED)",
"Shape(AC,CB,BE,EA)",
"Collinear(DAC)",
"Collinear(DEB)"
] | [
"Equal(LengthOfLine(AC),2/3*x-4)",
"Equal(LengthOfLine(AD),1/3*x+2)",
"Equal(LengthOfLine(AD),LengthOfLine(AC))",
"Equal(LengthOfLine(DE),5*y)",
"Equal(LengthOfLine(EB),7/3*y+8)",
"Equal(LengthOfLine(ED),LengthOfLine(EB))"
] | [
"Equal(LengthOfLine(AC),2/3*x-4)",
"Equal(LengthOfLine(AD),1/3*x+2)",
"Equal(LengthOfLine(AD),LengthOfLine(AC))",
"Equal(LengthOfLine(DE),5*y)",
"Equal(LengthOfLine(EB),7/3*y+8)",
"Equal(LengthOfLine(ED),LengthOfLine(EB))"
] | Value(x) | 18 | [] | {"START": []} | |
826 | NaZhu_2023-04-09 | Geometry3k-855 | 1 | 如图所示，∠AEJ=3*y+44°，∠CAE=94°，∠EJC=2*x°，∠JCA=96°，AC平行于EJ。求y的值。 | As shown in the diagram, ∠AEJ=3*y+44°, ∠CAE=94°, ∠EJC=2*x°, ∠JCA=96°, AC∥EJ. Find the value of y. | 826.png | [
"Shape(CA,AE,EJ,JC)"
] | [
"Equal(MeasureOfAngle(AEJ),3*y+44)",
"Equal(MeasureOfAngle(CAE),94)",
"Equal(MeasureOfAngle(EJC),2*x)",
"Equal(MeasureOfAngle(JCA),96)",
"ParallelBetweenLine(AC,EJ)"
] | [
"Equal(MeasureOfAngle(AEJ),3*y+44)",
"Equal(MeasureOfAngle(CAE),94)",
"Equal(MeasureOfAngle(EJC),2*x)",
"Equal(MeasureOfAngle(JCA),96)",
"ParallelBetweenLine(AC,EJ)"
] | Value(y) | 14 | [
"parallel_property_ipsilateral_internal_angle(1,AC,EJ)"
] | {"START": ["parallel_property_ipsilateral_internal_angle(1,AC,EJ)"]} | |
827 | XiaokaiZhang_2023-03-19 | Geometry3k-856 | 6 | 如图所示，BD=2*x，CD=3*x-7，ED=14，BD垂直于ED，CE垂直于BE。求x的值。 | As shown in the diagram, BD=2*x, CD=3*x-7, ED=14, BD⊥ED, CE is perpendicular to BE. Find the value of x. | 827.png | [
"Shape(CE,ED,DC)",
"Shape(DE,EB,BD)",
"Collinear(CDB)"
] | [
"Equal(LengthOfLine(BD),2*x)",
"Equal(LengthOfLine(CD),3*x-7)",
"Equal(LengthOfLine(ED),14)",
"PerpendicularBetweenLine(BD,ED)",
"PerpendicularBetweenLine(CE,BE)"
] | [
"Equal(LengthOfLine(BD),2*x)",
"Equal(LengthOfLine(CD),3*x-7)",
"Equal(LengthOfLine(ED),14)",
"PerpendicularBetweenLine(BD,ED)",
"PerpendicularBetweenLine(CE,BE)"
] | Value(x) | 7 | [
"mirror_similar_triangle_judgment_aa(1,EBD,CEB)",
"line_addition(1,CD,DB)",
"right_triangle_judgment_angle(1,BDE)",
"right_triangle_property_pythagorean(1,BDE)",
"mirror_similar_triangle_property_line_ratio(1,EBD,CEB)",
"mirror_similar_triangle_property_line_ratio(1,DEB,EBC)"
] | {"START": ["mirror_similar_triangle_judgment_aa(1,EBD,CEB)", "line_addition(1,CD,DB)", "right_triangle_judgment_angle(1,BDE)"], "mirror_similar_triangle_judgment_aa(1,EBD,CEB)": ["mirror_similar_triangle_property_line_ratio(1,EBD,CEB)", "mirror_similar_triangle_property_line_ratio(1,DEB,EBC)"], "right_triangle_judgment... | |
828 | XiaokaiZhang_2023-03-19 | Geometry3k-857 | 8 | 如图所示，AC=5，BA=15，BC=z，YA=y，BC⊥AC，CY垂直于BY。求y的值。 | As shown in the diagram, AC=5, BA=15, BC=z, YA=y, BC is perpendicular to AC, CY is perpendicular to BY. Find the value of y. | 828.png | [
"Shape(CA,AY,YC)",
"Shape(CY,YB,BC)",
"Collinear(AYB)"
] | [
"Equal(LengthOfLine(AC),5)",
"Equal(LengthOfLine(BA),15)",
"Equal(LengthOfLine(BC),z)",
"Equal(LengthOfLine(YA),y)",
"PerpendicularBetweenLine(BC,AC)",
"PerpendicularBetweenLine(CY,BY)"
] | [
"Equal(LengthOfLine(AC),5)",
"Equal(LengthOfLine(BA),15)",
"Equal(LengthOfLine(BC),z)",
"Equal(LengthOfLine(YA),y)",
"PerpendicularBetweenLine(BC,AC)",
"PerpendicularBetweenLine(CY,BY)"
] | Value(y) | 5/3 | [
"adjacent_complementary_angle(1,AYC,CYB)",
"right_triangle_judgment_angle(1,AYC)",
"right_triangle_judgment_angle(1,CYB)",
"right_triangle_judgment_angle(1,BCA)",
"line_addition(1,AY,YB)",
"right_triangle_property_pythagorean(1,AYC)",
"right_triangle_property_pythagorean(1,CYB)",
"right_triangle_prope... | {"START": ["adjacent_complementary_angle(1,AYC,CYB)", "right_triangle_judgment_angle(1,CYB)", "right_triangle_judgment_angle(1,BCA)", "line_addition(1,AY,YB)"], "adjacent_complementary_angle(1,AYC,CYB)": ["right_triangle_judgment_angle(1,AYC)"], "right_triangle_judgment_angle(1,AYC)": ["right_triangle_property_pythagor... | |
829 | NaZhu_2023-04-09 | Geometry3k-858 | 4 | 如图所示，AE=4，DE=7，YE=11，YC和YA是风筝形DCYA的一组临边。求四边形ADCY的面积。 | As shown in the diagram, AE=4, DE=7, YE=11, quadrilateral DCYA is a kite. Find the area of ADCY. | 829.png | [
"Shape(AD,DE,EA)",
"Shape(DC,CE,ED)",
"Shape(AE,EY,YA)",
"Shape(EC,CY,YE)",
"Collinear(DEY)",
"Collinear(AEC)"
] | [
"Equal(LengthOfLine(AE),4)",
"Equal(LengthOfLine(DE),7)",
"Equal(LengthOfLine(YE),11)",
"Kite(DCYA)"
] | [
"Equal(LengthOfLine(AE),4)",
"Equal(LengthOfLine(DE),7)",
"Equal(LengthOfLine(YE),11)"
] | Value(AreaOfQuadrilateral(ADCY)) | 72 | [
"kite_property_diagonal_perpendicular_bisection(1,DCYA,E)",
"line_addition(1,DE,EY)",
"line_addition(1,AE,EC)",
"kite_area_formula_diagonal(1,DCYA)"
] | {"START": ["kite_property_diagonal_perpendicular_bisection(1,DCYA,E)", "line_addition(1,DE,EY)", "line_addition(1,AE,EC)", "kite_area_formula_diagonal(1,DCYA)"]} | |
830 | NaZhu_2023-04-09 | Geometry3k-859 | 1 | 如图所示，AB=3*y，AD=2*x，BC=5*x-18，DC=96-y，ADCB是▱。求x的值。 | As shown in the diagram, AB=3*y, AD=2*x, BC=5*x-18, DC=96-y, DA and CB are opposite sides of the parallelogram ADCB. Find the value of x. | 830.png | [
"Shape(AD,DC,CB,BA)"
] | [
"Equal(LengthOfLine(AB),3*y)",
"Equal(LengthOfLine(AD),2*x)",
"Equal(LengthOfLine(BC),5*x-18)",
"Equal(LengthOfLine(DC),96-y)",
"Parallelogram(ADCB)"
] | [
"Equal(LengthOfLine(AB),3*y)",
"Equal(LengthOfLine(AD),2*x)",
"Equal(LengthOfLine(BC),5*x-18)",
"Equal(LengthOfLine(DC),96-y)"
] | Value(x) | 6 | [
"parallelogram_property_opposite_line_equal(1,ADCB)"
] | {"START": ["parallelogram_property_opposite_line_equal(1,ADCB)"]} | |
831 | XiaokaiZhang_2023-03-19 | Geometry3k-860 | 6 | 如图所示，PS=3，RY=5，WX=10，WY=8，XY=6，RP∥XW，BQ垂直于SQ，SY⊥BY，YS⊥PS。求直线SY的长度。 | As shown in the diagram, PS=3, RY=5, WX=10, WY=8, XY=6, RP∥XW, BQ⊥SQ, SY is perpendicular to BY, YS⊥PS. Find the length of line SY. | 831.png | [
"Shape(YR,RB,BY)",
"Shape(YB,BQ,QS,SY)",
"Shape(PY,YS,SP)",
"Shape(BX,XQ,QB)",
"Shape(SQ,QW,WS)",
"Collinear(RYP)",
"Collinear(YSW)",
"Collinear(RBQ)",
"Collinear(XQW)",
"Collinear(YBX)",
"Collinear(PSQ)"
] | [
"Equal(LengthOfLine(PS),3)",
"Equal(LengthOfLine(RY),5)",
"Equal(LengthOfLine(WX),10)",
"Equal(LengthOfLine(WY),8)",
"Equal(LengthOfLine(XY),6)",
"ParallelBetweenLine(RP,XW)",
"PerpendicularBetweenLine(BQ,SQ)",
"PerpendicularBetweenLine(SY,BY)",
"PerpendicularBetweenLine(YS,PS)"
] | [
"PerpendicularBetweenLine(BQ,SQ)",
"PerpendicularBetweenLine(SY,BY)",
"PerpendicularBetweenLine(YS,PS)"
] | Value(LengthOfLine(SY)) | 4 | [
"cosine_theorem(1,XWY)",
"parallel_property_collinear_extend(3,RP,XW,Y)",
"parallel_property_ipsilateral_internal_angle(1,YP,XW)",
"angle_addition(1,PYS,SYB)",
"triangle_property_angle_sum(1,PYS)",
"sine_theorem(1,SPY)"
] | {"START": ["cosine_theorem(1,XWY)", "parallel_property_collinear_extend(3,RP,XW,Y)", "angle_addition(1,PYS,SYB)", "triangle_property_angle_sum(1,PYS)", "sine_theorem(1,SPY)"], "parallel_property_collinear_extend(3,RP,XW,Y)": ["parallel_property_ipsilateral_internal_angle(1,YP,XW)"]} | |
832 | NaZhu_2023-04-09 | Geometry3k-861 | 4 | 如图所示，弧FCB的角度为75，⌒FDE的角度为45。求∠EAD的大小。 | As shown in the diagram, the measure of arc FCB is 75, the measure of arc FDE is 45. Find the measure of ∠EAD. | 832.png | [
"Shape(FDE,EA,AD)",
"Shape(FEC,CE)",
"Shape(FCB,BA,AC)",
"Shape(FBD,DA,AB)",
"Shape(AE,EC,CA)",
"Collinear(EAB)",
"Collinear(DAC)",
"Cocircular(F,DECB)"
] | [
"Equal(MeasureOfArc(FCB),75)",
"Equal(MeasureOfArc(FDE),45)"
] | [
"Equal(MeasureOfArc(FCB),75)",
"Equal(MeasureOfArc(FDE),45)"
] | Value(MeasureOfAngle(EAD)) | 60 | [
"arc_property_circumference_angle_external(1,FDE,C)",
"arc_property_circumference_angle_external(1,FCB,E)",
"triangle_property_angle_sum(1,ECA)",
"adjacent_complementary_angle(1,CAE,EAD)"
] | {"START": ["arc_property_circumference_angle_external(1,FDE,C)", "arc_property_circumference_angle_external(1,FCB,E)", "triangle_property_angle_sum(1,ECA)", "adjacent_complementary_angle(1,CAE,EAD)"]} | |
833 | XiaokaiZhang_2023-03-19 | Geometry3k-862 | 2 | 如图所示，JQ=2*y-1，PQ=8.1，∠JMH=14*y-8°，∠MHJ=13*y°，∠PJQ=53°，三角形JMH镜像全等于三角形JQP。求y的值。 | As shown in the diagram, JQ=2*y-1, PQ=8.1, ∠JMH=14*y-8°, ∠MHJ=13*y°, ∠PJQ=53°, △JMH is mirror congruent to △JQP. Find the value of y. | 833.png | [
"Shape(MH,HJ,JM)",
"Shape(JQ,QP,PJ)",
"Collinear(HJP)",
"Collinear(MJQ)"
] | [
"Equal(LengthOfLine(JQ),2*y-1)",
"Equal(LengthOfLine(PQ),8.1)",
"Equal(MeasureOfAngle(JMH),14*y-8)",
"Equal(MeasureOfAngle(MHJ),13*y)",
"Equal(MeasureOfAngle(PJQ),53)",
"MirrorCongruentBetweenTriangle(JMH,JQP)"
] | [
"Equal(LengthOfLine(JQ),2*y-1)",
"Equal(LengthOfLine(PQ),8.1)",
"Equal(MeasureOfAngle(JMH),14*y-8)",
"Equal(MeasureOfAngle(MHJ),13*y)",
"Equal(MeasureOfAngle(PJQ),53)"
] | Value(y) | 5 | [
"mirror_congruent_triangle_property_angle_equal(1,JMH,JQP)",
"triangle_property_angle_sum(1,JMH)"
] | {"START": ["mirror_congruent_triangle_property_angle_equal(1,JMH,JQP)", "triangle_property_angle_sum(1,JMH)"]} | |
834 | XiaokaiZhang_2023-03-19 | Geometry3k-863 | 3 | 如图所示，TS=5，∠ATS=135°，RS垂直于TS。求直线RT的长度。 | As shown in the diagram, TS=5, ∠ATS=135°, RS is perpendicular to TS. Find the length of line RT. | 834.png | [
"Shape(RS,ST,TR)",
"Shape(AT,TS)",
"Collinear(RTA)"
] | [
"Equal(LengthOfLine(TS),5)",
"Equal(MeasureOfAngle(ATS),135)",
"PerpendicularBetweenLine(RS,TS)"
] | [
"Equal(LengthOfLine(TS),5)",
"Equal(MeasureOfAngle(ATS),135)",
"PerpendicularBetweenLine(RS,TS)"
] | Value(LengthOfLine(RT)) | 5*sqrt(2) | [
"adjacent_complementary_angle(1,ATS,STR)",
"triangle_property_angle_sum(1,RST)",
"sine_theorem(1,TRS)"
] | {"START": ["adjacent_complementary_angle(1,ATS,STR)", "triangle_property_angle_sum(1,RST)", "sine_theorem(1,TRS)"]} | |
835 | NaZhu_2023-04-09 | Geometry3k-864 | 1 | 如图所示，AB=21，SC=17，BSAC是风筝形。求四边形ACBS的面积。 | As shown in the diagram, AB=21, SC=17, quadrilateral BSAC is a kite. Find the area of quadrilateral ACBS. | 835.png | [
"Shape(AC,CO,OA)",
"Shape(OC,CB,BO)",
"Shape(AO,OS,SA)",
"Shape(SO,OB,BS)",
"Collinear(AOB)",
"Collinear(COS)"
] | [
"Equal(LengthOfLine(AB),21)",
"Equal(LengthOfLine(SC),17)",
"Kite(BSAC)"
] | [
"Equal(LengthOfLine(AB),21)",
"Equal(LengthOfLine(SC),17)"
] | Value(AreaOfQuadrilateral(ACBS)) | 357/2 | [
"kite_area_formula_diagonal(1,ACBS)"
] | {"START": ["kite_area_formula_diagonal(1,ACBS)"]} | |
836 | NaZhu_2023-04-09 | Geometry3k-865 | 3 | 如图所示，∠UEN=64°，DU平行于HE，HD平行于EU。求∠HDI的大小。 | As shown in the diagram, ∠UEN=64°, DU is parallel to HE, HD is parallel to EU. Find the measure of ∠HDI. | 836.png | [
"Shape(DH,HE,EU,UD)",
"Shape(ID,DW)",
"Shape(WD,DU)",
"Shape(DU,UA)",
"Shape(AU,UL)",
"Shape(HD,DI)",
"Shape(UD,DH)",
"Shape(EU,UD)",
"Shape(LU,UE)",
"Shape(BH,HD)",
"Shape(DH,HE)",
"Shape(HE,EU)",
"Shape(UE,EN)",
"Shape(JH,HB)",
"Shape(EH,HJ)",
"Shape(KE,EH)",
"Shape(NE,EK)",
"Col... | [
"Equal(MeasureOfAngle(UEN),64)",
"ParallelBetweenLine(DU,HE)",
"ParallelBetweenLine(HD,EU)"
] | [
"ParallelBetweenLine(DU,HE)",
"ParallelBetweenLine(HD,EU)"
] | Value(MeasureOfAngle(HDI)) | 64 | [
"parallel_property_corresponding_angle(2,HD,EU,N)",
"parallel_property_corresponding_angle(1,DU,HE,W)",
"vertical_angle(1,WDU,HDI)"
] | {"START": ["parallel_property_corresponding_angle(2,HD,EU,N)", "parallel_property_corresponding_angle(1,DU,HE,W)", "vertical_angle(1,WDU,HDI)"]} | |
837 | XiaokaiZhang_2023-03-19 | Geometry3k-866 | 2 | 如图所示，AC=x，BC=17，∠ABC=39°，∠BCA=96°。求x的值。 | As shown in the diagram, AC=x, BC=17, ∠ABC=39°, ∠BCA=96°. Find the value of x. | 837.png | [
"Shape(AB,BC,CA)"
] | [
"Equal(LengthOfLine(AC),x)",
"Equal(LengthOfLine(BC),17)",
"Equal(MeasureOfAngle(ABC),39)",
"Equal(MeasureOfAngle(BCA),96)"
] | [
"Equal(LengthOfLine(AC),x)",
"Equal(LengthOfLine(BC),17)",
"Equal(MeasureOfAngle(ABC),39)",
"Equal(MeasureOfAngle(BCA),96)"
] | Value(x) | -17*sqrt(30*sqrt(5)+150)/16-17*sqrt(2*sqrt(5)+10)/16+17/8+17*sqrt(3)/8+17*sqrt(5)/8+17*sqrt(6*sqrt(5)+30)/16+17*sqrt(15)/8+17*sqrt(10*sqrt(5)+50)/16 | [
"triangle_property_angle_sum(1,ABC)",
"sine_theorem(1,CAB)"
] | {"START": ["triangle_property_angle_sum(1,ABC)", "sine_theorem(1,CAB)"]} | |
838 | XiaokaiZhang_2023-03-19 | Geometry3k-867 | 4 | 如图所示，MP=MR，MR=RP。求∠MRP的大小。 | As shown in the diagram, MP=MR, MR=RP. Find the measure of ∠MRP. | 838.png | [
"Shape(MR,RP,PM)"
] | [
"Equal(LengthOfLine(MP),LengthOfLine(MR))",
"Equal(LengthOfLine(MR),LengthOfLine(RP))"
] | [
"Equal(LengthOfLine(MP),LengthOfLine(MR))",
"Equal(LengthOfLine(MR),LengthOfLine(RP))"
] | Value(MeasureOfAngle(MRP)) | 60 | [
"isosceles_triangle_judgment_line_equal(1,MRP)",
"isosceles_triangle_judgment_line_equal(1,RPM)",
"equilateral_triangle_judgment_isosceles_and_isosceles(1,MRP)",
"equilateral_triangle_property_angle(1,RPM)"
] | {"START": ["isosceles_triangle_judgment_line_equal(1,MRP)", "isosceles_triangle_judgment_line_equal(1,RPM)"], "equilateral_triangle_judgment_isosceles_and_isosceles(1,MRP)": ["equilateral_triangle_property_angle(1,RPM)"], "isosceles_triangle_judgment_line_equal(1,MRP)": ["equilateral_triangle_judgment_isosceles_and_iso... | |
839 | XiaokaiZhang_2023-03-19 | Geometry3k-868 | 2 | 如图所示，BC=11，BO=35，CA=23，OA=25，OA⊥CA。求△OCB的面积。 | As shown in the diagram, BC=11, BO=35, CA=23, OA=25, OA is perpendicular to CA. Find the area of triangle OCB. | 839.png | [
"Shape(OA,AC,CO)",
"Shape(OC,CB,BO)",
"Collinear(ACB)"
] | [
"Equal(LengthOfLine(BC),11)",
"Equal(LengthOfLine(BO),35)",
"Equal(LengthOfLine(CA),23)",
"Equal(LengthOfLine(OA),25)",
"PerpendicularBetweenLine(OA,CA)"
] | [
"Equal(LengthOfLine(BC),11)",
"Equal(LengthOfLine(BO),35)",
"Equal(LengthOfLine(CA),23)",
"Equal(LengthOfLine(OA),25)",
"PerpendicularBetweenLine(OA,CA)"
] | Value(AreaOfTriangle(OCB)) | 275/2 | [
"altitude_of_triangle_judgment(2,OA,OCB)",
"triangle_area_formula_common(1,OCB)"
] | {"START": ["altitude_of_triangle_judgment(2,OA,OCB)", "triangle_area_formula_common(1,OCB)"]} | |
840 | XiaokaiZhang_2023-03-19 | Geometry3k-869 | 2 | 如图所示，∠UVS=27°，∠VWS=72°，TU⊥VU，UV垂直于WV。求∠WSV的大小。 | As shown in the diagram, ∠UVS=27°, ∠VWS=72°, TU⊥VU, UV is perpendicular to WV. Find the measure of ∠WSV. | 840.png | [
"Shape(TU,US,ST)",
"Shape(UV,VS,SU)",
"Shape(SV,VW,WS)",
"Collinear(USW)",
"Collinear(TSV)"
] | [
"Equal(MeasureOfAngle(UVS),27)",
"Equal(MeasureOfAngle(VWS),72)",
"PerpendicularBetweenLine(TU,VU)",
"PerpendicularBetweenLine(UV,WV)"
] | [
"Equal(MeasureOfAngle(UVS),27)",
"Equal(MeasureOfAngle(VWS),72)"
] | Value(MeasureOfAngle(WSV)) | 45 | [
"angle_addition(1,UVS,SVW)",
"triangle_property_angle_sum(1,SVW)"
] | {"START": ["angle_addition(1,UVS,SVW)", "triangle_property_angle_sum(1,SVW)"]} | |
841 | NaZhu_2023-04-09 | Geometry3k-870 | 6 | 如图所示，∠AHN=37°，∠EKL=110°，AQ平行于HN，KL∥AQ，KL平行于HN。求∠HAK的大小。 | As shown in the diagram, ∠AHN=37°, ∠EKL=110°, AQ is parallel to HN, KL is parallel to AQ, KL is parallel to HN. Find the measure of ∠HAK. | 841.png | [
"Shape(JK,KE)",
"Shape(EK,KL)",
"Shape(AK,KJ)",
"Shape(LK,KA)",
"Shape(CH,HA)",
"Shape(AH,HN)",
"Shape(FH,HC)",
"Shape(NH,HF)",
"Shape(PA,AK)",
"Shape(KA,AQ)",
"Shape(HA,AP)",
"Shape(QA,AH)",
"Collinear(JKL)",
"Collinear(CHN)",
"Collinear(EKA)",
"Collinear(AHF)",
"Collinear(PAQ)"
] | [
"Equal(MeasureOfAngle(AHN),37)",
"Equal(MeasureOfAngle(EKL),110)",
"ParallelBetweenLine(AQ,HN)",
"ParallelBetweenLine(KL,AQ)",
"ParallelBetweenLine(KL,HN)"
] | [
"Equal(MeasureOfAngle(AHN),37)",
"Equal(MeasureOfAngle(EKL),110)",
"ParallelBetweenLine(AQ,HN)",
"ParallelBetweenLine(KL,AQ)",
"ParallelBetweenLine(KL,HN)"
] | Value(MeasureOfAngle(HAK)) | 107 | [
"parallel_property_corresponding_angle(1,KL,AQ,E)",
"flat_angle(1,PAQ)",
"angle_addition(1,PAK,KAQ)",
"parallel_property_collinear_extend(1,AQ,HN,P)",
"parallel_property_alternate_interior_angle(2,PA,HN)",
"angle_addition(1,HAP,PAK)"
] | {"START": ["parallel_property_corresponding_angle(1,KL,AQ,E)", "flat_angle(1,PAQ)", "angle_addition(1,PAK,KAQ)", "parallel_property_collinear_extend(1,AQ,HN,P)", "angle_addition(1,HAP,PAK)"], "parallel_property_collinear_extend(1,AQ,HN,P)": ["parallel_property_alternate_interior_angle(2,PA,HN)"]} | |
842 | NaZhu_2023-04-09 | Geometry3k-871 | 4 | 如图所示，CD=30，ED=10，∠DEF=60°，四边形DEBC是平行四边形，EF垂直于DF。求DEBC的面积。 | As shown in the diagram, CD=30, ED=10, ∠DEF=60°, ED and BC are opposite sides of the ▱ DEBC, EF is perpendicular to DF. Find the area of quadrilateral DEBC. | 842.png | [
"Shape(DE,EF,FD)",
"Shape(DF,FB,BC,CD)",
"Collinear(EFB)"
] | [
"Equal(LengthOfLine(CD),30)",
"Equal(LengthOfLine(ED),10)",
"Equal(MeasureOfAngle(DEF),60)",
"Parallelogram(DEBC)",
"PerpendicularBetweenLine(EF,DF)"
] | [
"Equal(LengthOfLine(CD),30)",
"Equal(LengthOfLine(ED),10)",
"Equal(MeasureOfAngle(DEF),60)",
"PerpendicularBetweenLine(EF,DF)"
] | Value(AreaOfQuadrilateral(DEBC)) | 150*sqrt(3) | [
"sine_theorem(1,DEF)",
"altitude_of_quadrilateral_judgment_left_vertex(1,DF,DEBC)",
"parallelogram_property_opposite_line_equal(1,EBCD)",
"parallelogram_area_formula_common(1,DEBC)"
] | {"START": ["sine_theorem(1,DEF)", "altitude_of_quadrilateral_judgment_left_vertex(1,DF,DEBC)", "parallelogram_property_opposite_line_equal(1,EBCD)", "parallelogram_area_formula_common(1,DEBC)"]} | |
843 | NaZhu_2023-04-09 | Geometry3k-872 | 1 | 如图所示，∠BAD=2*x+33°，∠CBA=4*x+11°，CD和BA是等腰梯形CBAD的平行边。求x的值。 | As shown in the diagram, ∠BAD=2*x+33°, ∠CBA=4*x+11°, quadrilateral CBAD is a isosceles trapezoid. Find the value of x. | 843.png | [
"Shape(BA,AP,PB)",
"Shape(AD,DP,PA)",
"Shape(BP,PC,CB)",
"Shape(PD,DC,CP)",
"Collinear(APC)",
"Collinear(DPB)"
] | [
"Equal(MeasureOfAngle(BAD),2*x+33)",
"Equal(MeasureOfAngle(CBA),4*x+11)",
"IsoscelesTrapezoid(CBAD)"
] | [] | Value(x) | 11 | [
"isosceles_trapezoid_property_angle_equal(1,CBAD)"
] | {"START": ["isosceles_trapezoid_property_angle_equal(1,CBAD)"]} | |
844 | NaZhu_2023-04-09 | Geometry3k-873 | 3 | 如图所示，AC=5，AD=9，∠DAC=60°，DACB是▱，AC⊥DC。求四边形DACB的面积。 | As shown in the diagram, AC=5, AD=9, ∠DAC=60°, AD and CB are opposite sides of the ▱ DACB, AC⊥DC. Find the area of DACB. | 844.png | [
"Shape(DA,AC,CD)",
"Shape(DC,CB,BD)"
] | [
"Equal(LengthOfLine(AC),5)",
"Equal(LengthOfLine(AD),9)",
"Equal(MeasureOfAngle(DAC),60)",
"Parallelogram(DACB)",
"PerpendicularBetweenLine(AC,DC)"
] | [
"Equal(LengthOfLine(AC),5)",
"Equal(LengthOfLine(AD),9)",
"Equal(MeasureOfAngle(DAC),60)",
"PerpendicularBetweenLine(AC,DC)"
] | Value(AreaOfQuadrilateral(DACB)) | 45*sqrt(3)/2 | [
"sine_theorem(1,DAC)",
"altitude_of_quadrilateral_judgment_diagonal(1,DACB)",
"parallelogram_area_formula_common(1,DACB)"
] | {"START": ["sine_theorem(1,DAC)", "altitude_of_quadrilateral_judgment_diagonal(1,DACB)", "parallelogram_area_formula_common(1,DACB)"]} | |
845 | NaZhu_2023-04-09 | Geometry3k-874 | 6 | 如图所示，AB=12，DA=5，P是圆P的圆心，⊙P的直径为BD，⊙P的直径为DB。求圆P的周长。 | As shown in the diagram, AB=12, DA=5, P is the center of ⊙P, the diameter of ⊙P is BD, the diameter of ⊙P is DB. Find the circumference of the circle P. | 845.png | [
"Shape(PBC,CB)",
"Shape(PDA,AD)",
"Shape(PCD,DC)",
"Shape(PAB,BA)",
"Shape(CD,DP,PB,BC)",
"Shape(AB,BP,PD,DA)",
"Collinear(DPB)",
"Cocircular(P,ABCD)"
] | [
"Equal(LengthOfLine(AB),12)",
"Equal(LengthOfLine(DA),5)",
"IsCentreOfCircle(P,P)",
"IsDiameterOfCircle(BD,P)",
"IsDiameterOfCircle(DB,P)"
] | [
"Equal(LengthOfLine(AB),12)",
"Equal(LengthOfLine(DA),5)",
"IsCentreOfCircle(P,P)",
"IsDiameterOfCircle(BD,P)",
"IsDiameterOfCircle(DB,P)"
] | Value(PerimeterOfCircle(P)) | 13*pi | [
"diameter_of_circle_property_right_angle(1,DAB,P)",
"right_triangle_judgment_angle(1,DAB)",
"right_triangle_property_pythagorean(1,DAB)",
"diameter_of_circle_property_length_equal(1,BD,P)",
"circle_property_length_of_radius_and_diameter(1,P)",
"circle_perimeter_formula(1,P)"
] | {"START": ["diameter_of_circle_property_right_angle(1,DAB,P)", "diameter_of_circle_property_length_equal(1,BD,P)", "circle_property_length_of_radius_and_diameter(1,P)", "circle_perimeter_formula(1,P)"], "diameter_of_circle_property_right_angle(1,DAB,P)": ["right_triangle_judgment_angle(1,DAB)"], "right_triangle_judgmen... | |
846 | XiaokaiZhang_2023-03-19 | Geometry3k-875 | 8 | 如图所示，AB=25，CA=z，CB=y，DA=8，DC=x，BA⊥CA，DC垂直于BC。求y的值。 | As shown in the diagram, AB=25, CA=z, CB=y, DA=8, DC=x, BA is perpendicular to CA, DC⊥BC. Find the value of y. | 846.png | [
"Shape(DC,CA,AD)",
"Shape(AC,CB,BA)",
"Collinear(DAB)"
] | [
"Equal(LengthOfLine(AB),25)",
"Equal(LengthOfLine(CA),z)",
"Equal(LengthOfLine(CB),y)",
"Equal(LengthOfLine(DA),8)",
"Equal(LengthOfLine(DC),x)",
"PerpendicularBetweenLine(BA,CA)",
"PerpendicularBetweenLine(DC,BC)"
] | [
"Equal(LengthOfLine(AB),25)",
"Equal(LengthOfLine(CA),z)",
"Equal(LengthOfLine(CB),y)",
"Equal(LengthOfLine(DA),8)",
"Equal(LengthOfLine(DC),x)",
"PerpendicularBetweenLine(BA,CA)",
"PerpendicularBetweenLine(DC,BC)"
] | Value(y) | 5*sqrt(33) | [
"adjacent_complementary_angle(1,BAC,CAD)",
"right_triangle_judgment_angle(1,DCB)",
"right_triangle_judgment_angle(1,BAC)",
"right_triangle_judgment_angle(1,CAD)",
"line_addition(1,DA,AB)",
"right_triangle_property_pythagorean(1,DCB)",
"right_triangle_property_pythagorean(1,BAC)",
"right_triangle_prope... | {"START": ["adjacent_complementary_angle(1,BAC,CAD)", "right_triangle_judgment_angle(1,DCB)", "right_triangle_judgment_angle(1,BAC)", "line_addition(1,DA,AB)"], "adjacent_complementary_angle(1,BAC,CAD)": ["right_triangle_judgment_angle(1,CAD)"], "right_triangle_judgment_angle(1,BAC)": ["right_triangle_property_pythagor... | |
847 | XiaokaiZhang_2023-03-19 | Geometry3k-876 | 5 | 如图所示，AB=BC，AC=12，BD=h，∠CAB=60°，CD垂直于BD。求h的值。 | As shown in the diagram, AB=BC, AC=12, BD=h, ∠CAB=60°, CD is perpendicular to BD. Find the value of h. | 847.png | [
"Shape(BC,CD,DB)",
"Shape(BD,DA,AB)",
"Collinear(CDA)"
] | [
"Equal(LengthOfLine(AB),LengthOfLine(BC))",
"Equal(LengthOfLine(AC),12)",
"Equal(LengthOfLine(BD),h)",
"Equal(MeasureOfAngle(CAB),60)",
"PerpendicularBetweenLine(CD,BD)"
] | [
"Equal(LengthOfLine(AB),LengthOfLine(BC))",
"Equal(LengthOfLine(AC),12)",
"Equal(LengthOfLine(BD),h)",
"Equal(MeasureOfAngle(CAB),60)",
"PerpendicularBetweenLine(CD,BD)"
] | Value(h) | 6*sqrt(3) | [
"perpendicular_bisector_judgment_distance_equal(1,BD,CA)",
"line_addition(1,CD,DA)",
"triangle_property_angle_sum(1,BDA)",
"sine_theorem(1,BDA)",
"sine_theorem(1,ABD)"
] | {"START": ["perpendicular_bisector_judgment_distance_equal(1,BD,CA)", "line_addition(1,CD,DA)", "triangle_property_angle_sum(1,BDA)", "sine_theorem(1,BDA)", "sine_theorem(1,ABD)"]} | |
848 | NaZhu_2023-04-09 | Geometry3k-877 | 4 | 如图所示，BE=7，四边形AFDE是正方形。求AFDE的面积。 | As shown in the diagram, BE=7, AFDE is a square. Find the area of AFDE. | 848.png | [
"Shape(FB,BA,AF)",
"Shape(FD,DB,BF)",
"Shape(BD,DE,EB)",
"Shape(AB,BE,EA)",
"Collinear(FBE)",
"Collinear(DBA)"
] | [
"Equal(LengthOfLine(BE),7)",
"Square(AFDE)"
] | [
"Equal(LengthOfLine(BE),7)"
] | Value(AreaOfQuadrilateral(AFDE)) | 98 | [
"parallelogram_property_diagonal_bisection(1,FDEA,B)",
"rectangle_property_diagonal_equal(1,AFDE)",
"line_addition(1,FB,BE)",
"kite_area_formula_diagonal(1,AFDE)"
] | {"START": ["parallelogram_property_diagonal_bisection(1,FDEA,B)", "rectangle_property_diagonal_equal(1,AFDE)", "line_addition(1,FB,BE)", "kite_area_formula_diagonal(1,AFDE)"]} | |
849 | XiaokaiZhang_2023-03-19 | Geometry3k-878 | 2 | 如图所示，AB=18，AD=y，BC=x，BD=z，∠ABC=45°，∠ADB=60°，BA⊥DA，BC⊥AC。求y的值。 | As shown in the diagram, AB=18, AD=y, BC=x, BD=z, ∠ABC=45°, ∠ADB=60°, BA is perpendicular to DA, BC⊥AC. Find the value of y. | 849.png | [
"Shape(DB,BA,AD)",
"Shape(AB,BC,CA)"
] | [
"Equal(LengthOfLine(AB),18)",
"Equal(LengthOfLine(AD),y)",
"Equal(LengthOfLine(BC),x)",
"Equal(LengthOfLine(BD),z)",
"Equal(MeasureOfAngle(ABC),45)",
"Equal(MeasureOfAngle(ADB),60)",
"PerpendicularBetweenLine(BA,DA)",
"PerpendicularBetweenLine(BC,AC)"
] | [
"Equal(LengthOfLine(AB),18)",
"Equal(LengthOfLine(AD),y)",
"Equal(LengthOfLine(BC),x)",
"Equal(LengthOfLine(BD),z)",
"Equal(MeasureOfAngle(ABC),45)",
"Equal(MeasureOfAngle(ADB),60)",
"PerpendicularBetweenLine(BA,DA)",
"PerpendicularBetweenLine(BC,AC)"
] | Value(y) | 6*sqrt(3) | [
"triangle_property_angle_sum(1,DBA)",
"sine_theorem(1,ADB)"
] | {"START": ["triangle_property_angle_sum(1,DBA)", "sine_theorem(1,ADB)"]} | |
850 | NaZhu_2023-04-09 | Geometry3k-879 | 2 | 如图所示，四边形ADCB的面积为150，四边形FJHG的面积为54，CD=10，JH=x，ADCB是平行四边形，四边形FJHG是▱，ADCB相似于FJHG。求x的值。 | As shown in the diagram, the area of ADCB is 150, the area of quadrilateral FJHG is 54, CD=10, JH=x, quadrilateral ADCB is a ▱, FG and JH are opposite sides of the ▱ FJHG, ADCB is similar to FJHG. Find the value of x. | 850.png | [
"Shape(FJ,JH,HG,GF)",
"Shape(AD,DC,CB,BA)"
] | [
"Equal(AreaOfQuadrilateral(ADCB),150)",
"Equal(AreaOfQuadrilateral(FJHG),54)",
"Equal(LengthOfLine(CD),10)",
"Equal(LengthOfLine(JH),x)",
"Parallelogram(ADCB)",
"Parallelogram(FJHG)",
"SimilarBetweenQuadrilateral(ADCB,FJHG)"
] | [
"Equal(LengthOfLine(CD),10)",
"Equal(LengthOfLine(JH),x)"
] | Value(x) | 6 | [
"similar_quadrilateral_property_line_ratio(1,DCBA,JHGF)",
"similar_quadrilateral_property_area_square_ratio(1,ADCB,FJHG)"
] | {"START": ["similar_quadrilateral_property_line_ratio(1,DCBA,JHGF)", "similar_quadrilateral_property_area_square_ratio(1,ADCB,FJHG)"]} | |
851 | NaZhu_2023-04-09 | Geometry3k-880 | 6 | 如图所示，CA=10，圆C的圆心为C，四边形ABFD是正方形。求⊙C的面积减去四边形ABFD的面积。 | As shown in the diagram, CA=10, C is the center of circle C, quadrilateral ABFD is a square. Find the area of the circle C minus the area of quadrilateral ABFD. | 851.png | [
"Shape(CBF,FB)",
"Shape(CFD,DF)",
"Shape(CDA,AD)",
"Shape(CAB,BA)",
"Shape(CB,BF,FC)",
"Shape(CA,AB,BC)",
"Shape(CD,DA,AC)",
"Shape(CF,FD,DC)",
"Collinear(FCA)",
"Collinear(DCB)",
"Cocircular(C,BFDA)"
] | [
"Equal(LengthOfLine(CA),10)",
"IsCentreOfCircle(C,C)",
"Square(ABFD)"
] | [
"Equal(LengthOfLine(CA),10)",
"IsCentreOfCircle(C,C)"
] | Value(Sub(AreaOfCircle(C),AreaOfQuadrilateral(ABFD))) | -200+100*pi | [
"radius_of_circle_property_length_equal(1,CA,C)",
"radius_of_circle_property_length_equal(1,CF,C)",
"line_addition(1,FC,CA)",
"rectangle_property_diagonal_equal(1,ABFD)",
"kite_area_formula_diagonal(1,ABFD)",
"circle_area_formula(1,C)"
] | {"START": ["radius_of_circle_property_length_equal(1,CA,C)", "radius_of_circle_property_length_equal(1,CF,C)", "line_addition(1,FC,CA)", "rectangle_property_diagonal_equal(1,ABFD)", "kite_area_formula_diagonal(1,ABFD)", "circle_area_formula(1,C)"]} | |
852 | XiaokaiZhang_2023-03-19 | Geometry3k-881 | 2 | 如图所示，AB=c，CA=b，CB=a，∠ABC=60°，∠CAB=30°，b=18，BC⊥AC。求a的值。 | As shown in the diagram, AB=c, CA=b, CB=a, ∠ABC=60°, ∠CAB=30°, b=18, BC⊥AC. Find the value of a. | 852.png | [
"Shape(BC,CA,AB)"
] | [
"Equal(LengthOfLine(AB),c)",
"Equal(LengthOfLine(CA),b)",
"Equal(LengthOfLine(CB),a)",
"Equal(MeasureOfAngle(ABC),60)",
"Equal(MeasureOfAngle(CAB),30)",
"Equal(b,18)",
"PerpendicularBetweenLine(BC,AC)"
] | [
"Equal(LengthOfLine(AB),c)",
"Equal(LengthOfLine(CA),b)",
"Equal(LengthOfLine(CB),a)",
"Equal(MeasureOfAngle(ABC),60)",
"Equal(MeasureOfAngle(CAB),30)",
"PerpendicularBetweenLine(BC,AC)"
] | Value(a) | 6*sqrt(3) | [
"sine_theorem(1,BCA)",
"sine_theorem(1,ABC)"
] | {"START": ["sine_theorem(1,BCA)", "sine_theorem(1,ABC)"]} | |
853 | NaZhu_2023-04-09 | Geometry3k-882 | 2 | 如图所示，AB=12，EB=9，∠EBA=55°，ADCB是菱形。求∠ECB的大小。 | As shown in the diagram, AB=12, EB=9, ∠EBA=55°, quadrilateral ADCB is a rhombus. Find the measure of ∠ECB. | 853.png | [
"Shape(AE,EB,BA)",
"Shape(AD,DE,EA)",
"Shape(ED,DC,CE)",
"Shape(BE,EC,CB)",
"Collinear(AEC)",
"Collinear(DEB)"
] | [
"Equal(LengthOfLine(AB),12)",
"Equal(LengthOfLine(EB),9)",
"Equal(MeasureOfAngle(EBA),55)",
"Rhombus(ADCB)"
] | [] | Value(MeasureOfAngle(ECB)) | 180*asin(3/4)/pi | [
"kite_property_diagonal_perpendicular_bisection(1,CBAD,E)",
"sine_theorem(1,BEC)"
] | {"START": ["kite_property_diagonal_perpendicular_bisection(1,CBAD,E)", "sine_theorem(1,BEC)"]} | |
854 | NaZhu_2023-04-09 | Geometry3k-883 | 16 | 如图所示，RP=6，SQ=4，圆R的圆心为R，S是⊙S的圆心，PQ是⊙O的切线，QP是圆O的切线，MA垂直于RA。求直线PQ的长度。 | As shown in the diagram, RP=6, SQ=4, the center of ⊙R is R, S is the center of circle S, the tangent to circle S is PQ, QP is the tangent to ⊙R, MA is perpendicular to RA. Find the length of line PQ. | 854.png | [
"Shape(RPC,CR,RA,AP)",
"Shape(RCM,MA,AR,RC)",
"Shape(RMP,PA,AM)",
"Shape(RCM,SNC,NM)",
"Shape(SNC,CS,SN)",
"Shape(SQN,NS,SQ)",
"Shape(SCQ,QS,SC)",
"Shape(RMP,MN,SQN,QP)",
"Collinear(RAP)",
"Collinear(RCS)",
"Collinear(AMNS)",
"Cocircular(R,CMP)",
"Cocircular(S,QNC)"
] | [
"Equal(LengthOfLine(RP),6)",
"Equal(LengthOfLine(SQ),4)",
"IsCentreOfCircle(R,R)",
"IsCentreOfCircle(S,S)",
"IsTangentOfCircle(PQ,S)",
"IsTangentOfCircle(QP,R)",
"PerpendicularBetweenLine(MA,RA)"
] | [
"Equal(LengthOfLine(RP),6)",
"Equal(LengthOfLine(SQ),4)",
"IsCentreOfCircle(R,R)",
"IsCentreOfCircle(S,S)",
"PerpendicularBetweenLine(MA,RA)"
] | Value(LengthOfLine(PQ)) | 4*sqrt(6) | [
"radius_of_circle_property_length_equal(1,RP,R)",
"radius_of_circle_property_length_equal(1,RC,R)",
"radius_of_circle_property_length_equal(1,SQ,S)",
"radius_of_circle_property_length_equal(1,SC,S)",
"line_addition(1,RC,CS)",
"tangent_of_circle_property_perpendicular(1,PQ,S,S)",
"tangent_of_circle_prope... | {"START": ["radius_of_circle_property_length_equal(1,RP,R)", "radius_of_circle_property_length_equal(1,RC,R)", "radius_of_circle_property_length_equal(1,SQ,S)", "radius_of_circle_property_length_equal(1,SC,S)", "line_addition(1,RC,CS)", "tangent_of_circle_property_perpendicular(1,PQ,S,S)", "tangent_of_circle_property_p... | |
855 | NaZhu_2023-04-09 | Geometry3k-884 | 4 | 如图所示，AN=12，BN=6，∠NBE=45°，四边形ANBC是平行四边形，AE⊥BE。求ANBC的面积。 | As shown in the diagram, AN=12, BN=6, ∠NBE=45°, ANBC is a ▱, AE⊥BE. Find the area of ANBC. | 855.png | [
"Shape(BC,CA,AE,EB)",
"Shape(BE,EN,NB)",
"Collinear(AEN)"
] | [
"Equal(LengthOfLine(AN),12)",
"Equal(LengthOfLine(BN),6)",
"Equal(MeasureOfAngle(NBE),45)",
"Parallelogram(ANBC)",
"PerpendicularBetweenLine(AE,BE)"
] | [
"Equal(LengthOfLine(AN),12)",
"Equal(LengthOfLine(BN),6)",
"Equal(MeasureOfAngle(NBE),45)",
"PerpendicularBetweenLine(AE,BE)"
] | Value(AreaOfQuadrilateral(ANBC)) | 36*sqrt(2) | [
"triangle_property_angle_sum(1,BEN)",
"sine_theorem(1,BEN)",
"altitude_of_quadrilateral_judgment_right_vertex(1,BE,CANB)",
"parallelogram_area_formula_common(1,CANB)"
] | {"START": ["triangle_property_angle_sum(1,BEN)", "sine_theorem(1,BEN)", "altitude_of_quadrilateral_judgment_right_vertex(1,BE,CANB)", "parallelogram_area_formula_common(1,CANB)"]} | |
856 | XiaokaiZhang_2023-03-19 | Geometry3k-885 | 3 | 如图所示，AB=h，AC=15，∠ACB=60°，BA垂直于CA。求h的值。 | As shown in the diagram, AB=h, AC=15, ∠ACB=60°, BA⊥CA. Find the value of h. | 856.png | [
"Shape(AC,CB,BA)"
] | [
"Equal(LengthOfLine(AB),h)",
"Equal(LengthOfLine(AC),15)",
"Equal(MeasureOfAngle(ACB),60)",
"PerpendicularBetweenLine(BA,CA)"
] | [
"Equal(LengthOfLine(AB),h)",
"Equal(LengthOfLine(AC),15)",
"Equal(MeasureOfAngle(ACB),60)",
"PerpendicularBetweenLine(BA,CA)"
] | Value(h) | 15*sqrt(3) | [
"triangle_property_angle_sum(1,BAC)",
"sine_theorem(1,BAC)",
"sine_theorem(1,CBA)"
] | {"START": ["triangle_property_angle_sum(1,BAC)", "sine_theorem(1,BAC)", "sine_theorem(1,CBA)"]} | |
857 | XiaokaiZhang_2023-03-19 | Geometry3k-886 | 3 | 如图所示，XY=8，XZ=8，∠YXZ=60°。求∠ZYX的大小。 | As shown in the diagram, XY=8, XZ=8, ∠YXZ=60°. Find the measure of ∠ZYX. | 857.png | [
"Shape(XZ,ZY,YX)"
] | [
"Equal(LengthOfLine(XY),8)",
"Equal(LengthOfLine(XZ),8)",
"Equal(MeasureOfAngle(YXZ),60)"
] | [
"Equal(LengthOfLine(XY),8)",
"Equal(LengthOfLine(XZ),8)",
"Equal(MeasureOfAngle(YXZ),60)"
] | Value(MeasureOfAngle(ZYX)) | 60 | [
"isosceles_triangle_judgment_line_equal(1,XZY)",
"isosceles_triangle_property_angle_equal(1,XZY)",
"triangle_property_angle_sum(1,XZY)"
] | {"START": ["isosceles_triangle_judgment_line_equal(1,XZY)", "triangle_property_angle_sum(1,XZY)"], "isosceles_triangle_judgment_line_equal(1,XZY)": ["isosceles_triangle_property_angle_equal(1,XZY)"]} | |
858 | NaZhu_2023-04-09 | Geometry3k-887 | 4 | 如图所示，∠FEI=2*x+70°，∠HGL=2*y-20°，∠RFX=3*x+40°，GF平行于HE，HG平行于EF。求y的值。 | As shown in the diagram, ∠FEI=2*x+70°, ∠HGL=2*y-20°, ∠RFX=3*x+40°, GF∥HE, HG is parallel to EF. Find the value of y. | 858.png | [
"Shape(RF,FX)",
"Shape(XF,FE)",
"Shape(FE,EI)",
"Shape(IE,EO)",
"Shape(GF.FR)",
"Shape(EF,FG)",
"Shape(HE,EF)",
"Shape(OE,EH)",
"Shape(QG,GF)",
"Shape(FG,GH)",
"Shape(GH,HE)",
"Shape(EH,HM)",
"Shape(LG,GQ)",
"Shape(HG,GL)",
"Shape(JH,HG)",
"Shape(MH,HJ)",
"Collinear(RFEO)",
"Collin... | [
"Equal(MeasureOfAngle(FEI),2*x+70)",
"Equal(MeasureOfAngle(HGL),2*y-20)",
"Equal(MeasureOfAngle(RFX),3*x+40)",
"ParallelBetweenLine(GF,HE)",
"ParallelBetweenLine(HG,EF)"
] | [
"ParallelBetweenLine(GF,HE)",
"ParallelBetweenLine(HG,EF)"
] | Value(y) | 75 | [
"vertical_angle(1,RFX,EFG)",
"vertical_angle(1,FEI,OEH)",
"parallel_property_corresponding_angle(1,EH,FG,O)",
"parallel_property_corresponding_angle(2,FE,GH,L)"
] | {"START": ["vertical_angle(1,RFX,EFG)", "vertical_angle(1,FEI,OEH)", "parallel_property_corresponding_angle(1,EH,FG,O)", "parallel_property_corresponding_angle(2,FE,GH,L)"]} | |
859 | NaZhu_2023-04-09 | Geometry3k-888 | 3 | 如图所示，RT=6，S是圆S的圆心，PT垂直于QT。求直线PR的长度。 | As shown in the diagram, RT=6, S is the center of ⊙S, PT⊥QT. Find the length of line PR. | 859.png | [
"Shape(SRQ,QT,TR)",
"Shape(SQP,PT,TQ)",
"Shape(SPR,RT,TS,ST,TP)",
"Collinear(PTR)",
"Collinear(QTS)",
"Cocircular(S,RQP)"
] | [
"Equal(LengthOfLine(RT),6)",
"IsCentreOfCircle(S,S)",
"PerpendicularBetweenLine(PT,QT)"
] | [
"Equal(LengthOfLine(RT),6)",
"IsCentreOfCircle(S,S)",
"PerpendicularBetweenLine(PT,QT)"
] | Value(LengthOfLine(PR)) | 12 | [
"vertical_angle(1,PTQ,RTS)",
"circle_property_chord_perpendicular_bisect_chord(1,S,ST,RP)",
"line_addition(1,PT,TR)"
] | {"START": ["vertical_angle(1,PTQ,RTS)", "line_addition(1,PT,TR)"], "vertical_angle(1,PTQ,RTS)": ["circle_property_chord_perpendicular_bisect_chord(1,S,ST,RP)"]} | |
860 | NaZhu_2023-04-09 | Geometry3k-889 | 11 | 如图所示，∠BGE=52°，∠GFE=30°，⌒OBA的角度为108，弧OFE的角度为118，圆O的圆心为O。求弧OAC的角度。 | As shown in the diagram, ∠BGE=52°, ∠GFE=30°, the measure of ⌒OBA is 108, the measure of arc OFE is 118, O is the center of ⊙O. Find the measure of ⌒OAC. | 860.png | [
"Shape(OEB,BE)",
"Shape(OE,EB,BO)",
"Shape(OBA,AB)",
"Shape(OAC,CB,BA)",
"Shape(GB,BC,CG)",
"Shape(OCF,FG,GC)",
"Shape(GE,EO,OB,BG)",
"Shape(OFE,EF)",
"Shape(GF,FE,EG)",
"Shape(OAC,AD,DC)",
"Collinear(DAB)",
"Collinear(DCGE)",
"Collinear(FGB)",
"Cocircular(O,ACFEB)"
] | [
"Equal(MeasureOfAngle(BGE),52)",
"Equal(MeasureOfAngle(GFE),30)",
"Equal(MeasureOfArc(OBA),108)",
"Equal(MeasureOfArc(OFE),118)",
"IsCentreOfCircle(O,O)"
] | [
"Equal(MeasureOfAngle(BGE),52)",
"Equal(MeasureOfAngle(GFE),30)",
"Equal(MeasureOfArc(OBA),108)",
"Equal(MeasureOfArc(OFE),118)",
"IsCentreOfCircle(O,O)"
] | Value(MeasureOfArc(OAC)) | 30 | [
"arc_property_circumference_angle_external(1,OEB,C)",
"flat_angle(1,CGE)",
"angle_addition(1,CGB,BGE)",
"triangle_property_angle_sum(1,BCG)",
"arc_property_circumference_angle_external(1,OCF,B)",
"arc_property_center_angle(1,OEB,O)",
"arc_property_center_angle(1,OBE,O)",
"arc_addition_measure(1,OAC,OC... | {"START": ["arc_property_circumference_angle_external(1,OEB,C)", "flat_angle(1,CGE)", "angle_addition(1,CGB,BGE)", "triangle_property_angle_sum(1,BCG)", "arc_property_circumference_angle_external(1,OCF,B)", "arc_property_center_angle(1,OEB,O)", "arc_property_center_angle(1,OBE,O)", "arc_addition_measure(1,OAC,OCE)", "a... | |
861 | NaZhu_2023-04-09 | Geometry3k-890 | 2 | 如图所示，BA=6，CF=7，DA=9，∠BAF=32°，∠CBF=40°，∠FAD=20°，AB和DC是平行四边形ADCB的一组对边。求直线DF的长度。 | As shown in the diagram, BA=6, CF=7, DA=9, ∠BAF=32°, ∠CBF=40°, ∠FAD=20°, DA and CB are opposite sides of the parallelogram ADCB. Find the length of line DF. | 861.png | [
"Shape(AD,DF,FA)",
"Shape(AF,FB,BA)",
"Shape(FD,DC,CF)",
"Shape(BF,FC,CB)",
"Collinear(AFC)",
"Collinear(DFB)"
] | [
"Equal(LengthOfLine(BA),6)",
"Equal(LengthOfLine(CF),7)",
"Equal(LengthOfLine(DA),9)",
"Equal(MeasureOfAngle(BAF),32)",
"Equal(MeasureOfAngle(CBF),40)",
"Equal(MeasureOfAngle(FAD),20)",
"Parallelogram(ADCB)"
] | [
"Equal(LengthOfLine(BA),6)",
"Equal(LengthOfLine(CF),7)",
"Equal(LengthOfLine(DA),9)",
"Equal(MeasureOfAngle(BAF),32)",
"Equal(MeasureOfAngle(CBF),40)",
"Equal(MeasureOfAngle(FAD),20)"
] | Value(LengthOfLine(DF)) | sqrt(130-126*cos(pi/9)) | [
"parallelogram_property_diagonal_bisection(1,ADCB,F)",
"cosine_theorem(1,ADF)"
] | {"START": ["parallelogram_property_diagonal_bisection(1,ADCB,F)", "cosine_theorem(1,ADF)"]} | |
862 | XiaokaiZhang_2023-03-19 | Geometry3k-891 | 3 | 如图所示，JH=4*x+7，JK=25，KH=6*x-2，NQ=12，PN=8，PQ=20，∠JHK=∠QNP，∠KJH=∠PQN。求直线HK的长度。 | As shown in the diagram, JH=4*x+7, JK=25, KH=6*x-2, NQ=12, PN=8, PQ=20, ∠JHK=∠QNP, ∠KJH=∠PQN. Find the length of line HK. | 862.png | [
"Shape(JH,HK,KJ)",
"Shape(PQ,QN,NP)"
] | [
"Equal(LengthOfLine(JH),4*x+7)",
"Equal(LengthOfLine(JK),25)",
"Equal(LengthOfLine(KH),6*x-2)",
"Equal(LengthOfLine(NQ),12)",
"Equal(LengthOfLine(PN),8)",
"Equal(LengthOfLine(PQ),20)",
"Equal(MeasureOfAngle(JHK),MeasureOfAngle(QNP))",
"Equal(MeasureOfAngle(KJH),MeasureOfAngle(PQN))"
] | [
"Equal(LengthOfLine(JH),4*x+7)",
"Equal(LengthOfLine(JK),25)",
"Equal(LengthOfLine(KH),6*x-2)",
"Equal(LengthOfLine(NQ),12)",
"Equal(LengthOfLine(PN),8)",
"Equal(LengthOfLine(PQ),20)",
"Equal(MeasureOfAngle(JHK),MeasureOfAngle(QNP))",
"Equal(MeasureOfAngle(KJH),MeasureOfAngle(PQN))"
] | Value(LengthOfLine(HK)) | 10 | [
"similar_triangle_judgment_aa(1,KJH,PQN)",
"similar_triangle_property_line_ratio(1,JHK,QNP)",
"similar_triangle_property_line_ratio(1,KJH,PQN)"
] | {"START": ["similar_triangle_judgment_aa(1,KJH,PQN)"], "similar_triangle_judgment_aa(1,KJH,PQN)": ["similar_triangle_property_line_ratio(1,KJH,PQN)", "similar_triangle_property_line_ratio(1,JHK,QNP)"]} | |
863 | NaZhu_2023-04-09 | Geometry3k-892 | 3 | 如图所示，AF=c，DF=9，DG=d，DH=11，HE=15，DG是⊙O的切线。求c的值。 | As shown in the diagram, AF=c, DF=9, DG=d, DH=11, HE=15, the tangent to ⊙C is DG. Find the value of c. | 863.png | [
"Shape(CFG,CGA,AF)",
"Shape(CAE,EH,CHF,FA)",
"Shape(CEH,HE)",
"Shape(CFG,FD,DG)",
"Shape(CHF,HD,DF)",
"Collinear(AFD)",
"Collinear(EHD)",
"Cocircular(C,GAEHF)"
] | [
"Equal(LengthOfLine(AF),c)",
"Equal(LengthOfLine(DF),9)",
"Equal(LengthOfLine(DG),d)",
"Equal(LengthOfLine(DH),11)",
"Equal(LengthOfLine(HE),15)",
"IsTangentOfCircle(DG,C)"
] | [
"Equal(LengthOfLine(AF),c)",
"Equal(LengthOfLine(DF),9)",
"Equal(LengthOfLine(DG),d)",
"Equal(LengthOfLine(DH),11)",
"Equal(LengthOfLine(HE),15)"
] | Value(c) | 205/9 | [
"line_addition(1,AF,FD)",
"line_addition(1,EH,HD)",
"circle_property_circular_power_segment_and_segment_line(1,DFA,DHE,C)"
] | {"START": ["line_addition(1,AF,FD)", "line_addition(1,EH,HD)", "circle_property_circular_power_segment_and_segment_line(1,DFA,DHE,C)"]} | |
864 | XiaokaiZhang_2023-03-19 | Geometry3k-893 | 0 | 如图所示，AD=x+2，AY=AC，CB=3/2*x+11，DB=3*y-9，DB=YD，YD=2*y+6。求y的值。 | As shown in the diagram, AD=x+2, AY=AC, CB=3/2*x+11, DB=3*y-9, DB=YD, YD=2*y+6. Find the value of y. | 864.png | [
"Shape(CA,AD,DB,BC)",
"Shape(AY,YD,DA)"
] | [
"Equal(LengthOfLine(AD),x+2)",
"Equal(LengthOfLine(AY),LengthOfLine(AC))",
"Equal(LengthOfLine(CB),3/2*x+11)",
"Equal(LengthOfLine(DB),3*y-9)",
"Equal(LengthOfLine(DB),LengthOfLine(YD))",
"Equal(LengthOfLine(YD),2*y+6)"
] | [
"Equal(LengthOfLine(AD),x+2)",
"Equal(LengthOfLine(AY),LengthOfLine(AC))",
"Equal(LengthOfLine(CB),3/2*x+11)",
"Equal(LengthOfLine(DB),3*y-9)",
"Equal(LengthOfLine(DB),LengthOfLine(YD))",
"Equal(LengthOfLine(YD),2*y+6)"
] | Value(y) | 15 | [] | {"START": []} | |
865 | NaZhu_2023-04-09 | Geometry3k-894 | 1 | 如图所示，∠BAC=2*x°，∠CAD=3*x°，BA⊥DA。求x的值。 | As shown in the diagram, ∠BAC=2*x°, ∠CAD=3*x°, BA⊥DA. Find the value of x. | 865.png | [
"Shape(BA,AC)",
"Shape(CA,AD)"
] | [
"Equal(MeasureOfAngle(BAC),2*x)",
"Equal(MeasureOfAngle(CAD),3*x)",
"PerpendicularBetweenLine(BA,DA)"
] | [
"Equal(MeasureOfAngle(BAC),2*x)",
"Equal(MeasureOfAngle(CAD),3*x)",
"PerpendicularBetweenLine(BA,DA)"
] | Value(x) | 18 | [
"angle_addition(1,BAC,CAD)"
] | {"START": ["angle_addition(1,BAC,CAD)"]} | |
866 | XiaokaiZhang_2023-03-19 | Geometry3k-895 | 3 | 如图所示，BC=32，BD=15，BE=z，CD=9，∠BDE=∠EDC。求z的值。 | As shown in the diagram, BC=32, BD=15, BE=z, CD=9, ∠BDE=∠EDC. Find the value of z. | 866.png | [
"Shape(DC,CE,ED)",
"Shape(DE,EB,BD)",
"Collinear(CEB)"
] | [
"Equal(LengthOfLine(BC),32)",
"Equal(LengthOfLine(BD),15)",
"Equal(LengthOfLine(BE),z)",
"Equal(LengthOfLine(CD),9)",
"Equal(MeasureOfAngle(BDE),MeasureOfAngle(EDC))"
] | [
"Equal(LengthOfLine(BC),32)",
"Equal(LengthOfLine(BD),15)",
"Equal(LengthOfLine(BE),z)",
"Equal(LengthOfLine(CD),9)",
"Equal(MeasureOfAngle(BDE),MeasureOfAngle(EDC))"
] | Value(z) | 20 | [
"bisector_of_angle_judgment_angle_equal(1,DE,BDC)",
"bisector_of_angle_property_line_ratio(1,DE,BDC)",
"line_addition(1,CE,EB)"
] | {"START": ["bisector_of_angle_judgment_angle_equal(1,DE,BDC)", "line_addition(1,CE,EB)"], "bisector_of_angle_judgment_angle_equal(1,DE,BDC)": ["bisector_of_angle_property_line_ratio(1,DE,BDC)"]} | |
867 | NaZhu_2023-04-09 | Geometry3k-896 | 4 | 如图所示，AB=AD，BT=5，CB=CD，TC=8。求直线CD的长度。 | As shown in the diagram, AB=AD, BT=5, CB=CD, TC=8. Find the length of line CD. | 867.png | [
"Shape(CB,BT,TC)",
"Shape(BA,AT,TB)",
"Shape(CT,TD,DC)",
"Shape(DT,TA,AD)",
"Collinear(BTD)",
"Collinear(CTA)"
] | [
"Equal(LengthOfLine(AB),LengthOfLine(AD))",
"Equal(LengthOfLine(BT),5)",
"Equal(LengthOfLine(CB),LengthOfLine(CD))",
"Equal(LengthOfLine(TC),8)"
] | [
"Equal(LengthOfLine(AB),LengthOfLine(AD))",
"Equal(LengthOfLine(BT),5)",
"Equal(LengthOfLine(CB),LengthOfLine(CD))",
"Equal(LengthOfLine(TC),8)"
] | Value(LengthOfLine(CD)) | sqrt(89) | [
"kite_judgment_equal_and_equal(1,CBAD)",
"kite_property_diagonal_perpendicular_bisection(1,CBAD,T)",
"right_triangle_judgment_angle(1,BTC)",
"right_triangle_property_pythagorean(1,BTC)"
] | {"START": ["kite_judgment_equal_and_equal(1,CBAD)"], "kite_judgment_equal_and_equal(1,CBAD)": ["kite_property_diagonal_perpendicular_bisection(1,CBAD,T)"], "kite_property_diagonal_perpendicular_bisection(1,CBAD,T)": ["right_triangle_judgment_angle(1,BTC)"], "right_triangle_judgment_angle(1,BTC)": ["right_triangle_prope... | |
868 | NaZhu_2023-04-09 | Geometry3k-897 | 3 | 如图所示，∠EAD=42°，A是圆A的圆心。求弧ACD的角度。 | As shown in the diagram, ∠EAD=42°, A is the center of ⊙A. Find the measure of ⌒ACD. | 868.png | [
"Shape(ABC,CA,AB)",
"Shape(ACD,DA,AC)",
"Shape(ADE,EA,AD)",
"Shape(AEB,BA,AE)",
"Collinear(BAD)",
"Collinear(CAE)",
"Cocircular(A,CDEB)"
] | [
"Equal(MeasureOfAngle(EAD),42)",
"IsCentreOfCircle(A,A)"
] | [
"Equal(MeasureOfAngle(EAD),42)",
"IsCentreOfCircle(A,A)"
] | Value(MeasureOfArc(ACD)) | 138 | [
"flat_angle(1,EAC)",
"angle_addition(1,EAD,DAC)",
"arc_property_center_angle(1,ACD,A)"
] | {"START": ["flat_angle(1,EAC)", "angle_addition(1,EAD,DAC)", "arc_property_center_angle(1,ACD,A)"]} | |
869 | NaZhu_2023-04-09 | Geometry3k-898 | 5 | 如图所示，∠CFB=80°，⌒EAD的角度为x，弧EBC的角度为110。求x的值。 | As shown in the diagram, ∠CFB=80°, the measure of arc EAD is x, the measure of ⌒EBC is 110. Find the value of x. | 869.png | [
"Shape(EBC,CE,EB)",
"Shape(ECA,AC)",
"Shape(EAD,DF,FA)",
"Shape(EDB,BD)",
"Shape(CA,AF,FC)",
"Shape(FD,DB,BF)",
"Shape(CF,FB,BE,EC)",
"Collinear(AFB)",
"Collinear(DFC)",
"Cocircular(E,CADB)"
] | [
"Equal(MeasureOfAngle(CFB),80)",
"Equal(MeasureOfArc(EAD),x)",
"Equal(MeasureOfArc(EBC),110)"
] | [
"Equal(MeasureOfAngle(CFB),80)",
"Equal(MeasureOfArc(EAD),x)",
"Equal(MeasureOfArc(EBC),110)"
] | Value(x) | 50 | [
"flat_angle(1,CFD)",
"angle_addition(1,CFB,BFD)",
"arc_property_circumference_angle_external(1,EBC,D)",
"triangle_property_angle_sum(1,BFD)",
"arc_property_circumference_angle_external(1,EAD,B)"
] | {"START": ["flat_angle(1,CFD)", "angle_addition(1,CFB,BFD)", "arc_property_circumference_angle_external(1,EBC,D)", "triangle_property_angle_sum(1,BFD)", "arc_property_circumference_angle_external(1,EAD,B)"]} | |
870 | NaZhu_2023-04-09 | Geometry3k-899 | 2 | 如图所示，∠DCA=40°，∠ECB=35°。求∠BCD的大小。 | As shown in the diagram, ∠DCA=40°, ∠ECB=35°. Find the measure of ∠BCD. | 870.png | [
"Shape(CEA,AC,CE)",
"Shape(CAD,DC,CA)",
"Shape(CDB,BC,CD)",
"Shape(CBE,EC,CB)",
"Collinear(ACB)",
"Cocircular(C,ADBE)"
] | [
"Equal(MeasureOfAngle(DCA),40)",
"Equal(MeasureOfAngle(ECB),35)"
] | [
"Equal(MeasureOfAngle(DCA),40)",
"Equal(MeasureOfAngle(ECB),35)"
] | Value(MeasureOfAngle(BCD)) | 140 | [
"flat_angle(1,BCA)",
"angle_addition(1,BCD,DCA)"
] | {"START": ["flat_angle(1,BCA)", "angle_addition(1,BCD,DCA)"]} | |
871 | XiaokaiZhang_2023-03-19 | Geometry3k-900 | 1 | 如图所示，AB=c，CA=b，CB=a，a=8，b=15，c=17，BC垂直于AC。求cos(AB)的值。 | As shown in the diagram, AB=c, CA=b, CB=a, a=8, b=15, c=17, BC⊥AC. Find the value of cos(AB). | 871.png | [
"Shape(BC,CA,AB)"
] | [
"Equal(LengthOfLine(AB),c)",
"Equal(LengthOfLine(CA),b)",
"Equal(LengthOfLine(CB),a)",
"Equal(a,8)",
"Equal(b,15)",
"Equal(c,17)",
"PerpendicularBetweenLine(BC,AC)"
] | [
"Equal(LengthOfLine(AB),c)",
"Equal(LengthOfLine(CA),b)",
"Equal(LengthOfLine(CB),a)",
"PerpendicularBetweenLine(BC,AC)"
] | Value(Cos(MeasureOfAngle(ABC))) | 8/17 | [
"cosine_theorem(1,BCA)"
] | {"START": ["cosine_theorem(1,BCA)"]} | |
872 | NaZhu_2023-04-09 | Geometry3k-901 | 2 | 如图所示，BC=DB，BC=DE，EC=6，EC=BC，CB垂直于DB。求CBDE的面积。 | As shown in the diagram, BC=DB, BC=DE, EC=6, EC=BC, CB⊥DB. Find the area of quadrilateral CBDE. | 872.png | [
"Shape(CB,BD,DE,EC)"
] | [
"Equal(LengthOfLine(BC),LengthOfLine(DB))",
"Equal(LengthOfLine(BC),LengthOfLine(DE))",
"Equal(LengthOfLine(EC),6)",
"Equal(LengthOfLine(EC),LengthOfLine(BC))",
"PerpendicularBetweenLine(CB,DB)"
] | [
"Equal(LengthOfLine(BC),LengthOfLine(DB))",
"Equal(LengthOfLine(BC),LengthOfLine(DE))",
"Equal(LengthOfLine(EC),6)",
"Equal(LengthOfLine(EC),LengthOfLine(BC))",
"PerpendicularBetweenLine(CB,DB)"
] | Value(AreaOfQuadrilateral(CBDE)) | 36 | [
"kite_judgment_equal_and_equal(1,CBDE)",
"kite_area_formula_sine(1,CBDE)"
] | {"START": ["kite_judgment_equal_and_equal(1,CBDE)"], "kite_judgment_equal_and_equal(1,CBDE)": ["kite_area_formula_sine(1,CBDE)"]} | |
873 | NaZhu_2023-04-09 | Geometry3k-902 | 3 | 如图所示，∠AEC=63°，∠DAB=47°，∠DFB=136°，∠EAD=69°。求∠CAE的大小。 | As shown in the diagram, ∠AEC=63°, ∠DAB=47°, ∠DFB=136°, ∠EAD=69°. Find the measure of ∠CAE. | 873.png | [
"Shape(EC,CA,AE)",
"Shape(CG,GA,AC)",
"Shape(DA,AF,FD)",
"Shape(EA,AD)",
"Shape(EA,AF)",
"Shape(DF,FB)",
"Collinear(CAFB)",
"Collinear(EAG)"
] | [
"Equal(MeasureOfAngle(AEC),63)",
"Equal(MeasureOfAngle(DAB),47)",
"Equal(MeasureOfAngle(DFB),136)",
"Equal(MeasureOfAngle(EAD),69)"
] | [
"Equal(MeasureOfAngle(AEC),63)",
"Equal(MeasureOfAngle(DAB),47)",
"Equal(MeasureOfAngle(DFB),136)",
"Equal(MeasureOfAngle(EAD),69)"
] | Value(MeasureOfAngle(CAE)) | 64 | [
"angle_addition(1,EAD,DAF)",
"angle_addition(1,CAE,EAF)",
"flat_angle(1,CAF)"
] | {"START": ["angle_addition(1,EAD,DAF)", "angle_addition(1,CAE,EAF)", "flat_angle(1,CAF)"]} | |
874 | XiaokaiZhang_2023-03-19 | Geometry3k-903 | 4 | 如图所示，AB=4*z，AT=2*x-5，AU=2.9，DA=6，EA=y，FA=4.6，S是线段DE的中点，T平分线段EF，U是线段DF的中点。求x的值。 | As shown in the diagram, AB=4*z, AT=2*x-5, AU=2.9, DA=6, EA=y, FA=4.6, S is the midpoint of segment DE, T is the midpoint of segment EF, U is the midpoint of segment DF. Find the value of x. | 874.png | [
"Shape(ES,SA,AE)",
"Shape(SD,DA,AS)",
"Shape(AD,DU,UA)",
"Shape(AU,UF,FA)",
"Shape(AF,FT,TA)",
"Shape(AT,TE,EA)",
"Collinear(ESD)",
"Collinear(DUF)",
"Collinear(FTE)",
"Collinear(EAU)",
"Collinear(SAF)",
"Collinear(DAT)"
] | [
"Equal(LengthOfLine(AB),4*z)",
"Equal(LengthOfLine(AT),2*x-5)",
"Equal(LengthOfLine(AU),2.9)",
"Equal(LengthOfLine(DA),6)",
"Equal(LengthOfLine(EA),y)",
"Equal(LengthOfLine(FA),4.6)",
"IsMidpointOfLine(S,DE)",
"IsMidpointOfLine(T,EF)",
"IsMidpointOfLine(U,DF)"
] | [
"Equal(LengthOfLine(AB),4*z)",
"Equal(LengthOfLine(AT),2*x-5)",
"Equal(LengthOfLine(AU),2.9)",
"Equal(LengthOfLine(DA),6)",
"Equal(LengthOfLine(EA),y)",
"Equal(LengthOfLine(FA),4.6)"
] | Value(x) | 4 | [
"median_of_triangle_judgment(1,DT,DFE)",
"median_of_triangle_judgment(1,EU,EDF)",
"centroid_of_triangle_judgment_intersection(1,A,FED,T,U)",
"centroid_of_triangle_property_line_ratio(1,A,DFE,T)"
] | {"START": ["median_of_triangle_judgment(1,DT,DFE)", "median_of_triangle_judgment(1,EU,EDF)"], "centroid_of_triangle_judgment_intersection(1,A,FED,T,U)": ["centroid_of_triangle_property_line_ratio(1,A,DFE,T)"], "median_of_triangle_judgment(1,DT,DFE)": ["centroid_of_triangle_judgment_intersection(1,A,FED,T,U)"], "median_... | |
875 | XiaokaiZhang_2023-03-19 | Geometry3k-904 | 1 | 如图所示，PY=2*c-1，YR=4*c-11，ZP=a+5，ZQ=3*a-11，∠PRZ=4*b-17°，∠QYR=7*b+6°，∠RXP=2*a+10°，∠ZRQ=3*b-4°，QY是△QPR的中线。求直线PR的长度。 | As shown in the diagram, PY=2*c-1, YR=4*c-11, ZP=a+5, ZQ=3*a-11, ∠PRZ=4*b-17°, ∠QYR=7*b+6°, ∠RXP=2*a+10°, ∠ZRQ=3*b-4°, QY is the median of △ QPR. Find the length of line PR. | 875.png | [
"Shape(QZ,ZC,CB,BQ)",
"Shape(ZP,PC,CZ)",
"Shape(CP,PY,YA,AC)",
"Shape(BC,CA,AB)",
"Shape(QB,BX,XQ)",
"Shape(XB,BA,AR,RX)",
"Shape(AY,YR,RA)",
"Collinear(QZP)",
"Collinear(PYR)",
"Collinear(RXQ)",
"Collinear(QBAY)",
"Collinear(PCBX)",
"Collinear(ZCAR)"
] | [
"Equal(LengthOfLine(PY),2*c-1)",
"Equal(LengthOfLine(YR),4*c-11)",
"Equal(LengthOfLine(ZP),a+5)",
"Equal(LengthOfLine(ZQ),3*a-11)",
"Equal(MeasureOfAngle(PRZ),4*b-17)",
"Equal(MeasureOfAngle(QYR),7*b+6)",
"Equal(MeasureOfAngle(RXP),2*a+10)",
"Equal(MeasureOfAngle(ZRQ),3*b-4)",
"IsMedianOfTriangle(QY... | [] | Value(LengthOfLine(PR)) | 18 | [
"line_addition(1,PY,YR)"
] | {"START": ["line_addition(1,PY,YR)"]} | |
876 | NaZhu_2023-04-09 | Geometry3k-905 | 1 | 如图所示，JO=12，KN=7，LN=x+3，LO=4*x-9，JM是⊙O的切线，圆O的切线为JO，KM是⊙O的切线，KN是⊙O的切线，LN是圆O的切线，圆O的切线为LO。求x的值。 | As shown in the diagram, JO=12, KN=7, LN=x+3, LO=4*x-9, the tangent to circle R is JM, the tangent to circle R is JO, KM is the tangent to ⊙R, KN is the tangent to ⊙R, LN is the tangent to circle R, the tangent to ⊙R is LO. Find the value of x. | 876.png | [
"Shape(RNM,RMO,RON)",
"Shape(RNM,NK,KM)",
"Shape(RMO,MJ,JO)",
"Shape(RON,OL,LN)",
"Collinear(JMK)",
"Collinear(JOL)",
"Collinear(KNL)",
"Cocircular(R,NMO)"
] | [
"Equal(LengthOfLine(JO),12)",
"Equal(LengthOfLine(KN),7)",
"Equal(LengthOfLine(LN),x+3)",
"Equal(LengthOfLine(LO),4*x-9)",
"IsTangentOfCircle(JM,R)",
"IsTangentOfCircle(JO,R)",
"IsTangentOfCircle(KM,R)",
"IsTangentOfCircle(KN,R)",
"IsTangentOfCircle(LN,R)",
"IsTangentOfCircle(LO,R)"
] | [
"Equal(LengthOfLine(JO),12)",
"Equal(LengthOfLine(KN),7)",
"Equal(LengthOfLine(LN),x+3)",
"Equal(LengthOfLine(LO),4*x-9)"
] | Value(x) | 4 | [
"tangent_of_circle_property_length_equal(1,LO,LN,R)"
] | {"START": ["tangent_of_circle_property_length_equal(1,LO,LN,R)"]} | |
877 | NaZhu_2023-04-09 | Geometry3k-906 | 1 | 如图所示，EG=x+5，FD=3*x-7，DG⊥FG，DGFE是长方形。求直线EG的长度。 | As shown in the diagram, EG=x+5, FD=3*x-7, DG is perpendicular to FG, DGFE is a rectangle. Find the length of line EG. | 877.png | [
"Shape(AD,DG,GA)",
"Shape(AG,GF,FA)",
"Shape(AF,FE,EA)",
"Shape(ED,DA,AE)",
"Collinear(DAF)",
"Collinear(GAE)"
] | [
"Equal(LengthOfLine(EG),x+5)",
"Equal(LengthOfLine(FD),3*x-7)",
"PerpendicularBetweenLine(DG,FG)",
"Rectangle(DGFE)"
] | [
"PerpendicularBetweenLine(DG,FG)"
] | Value(LengthOfLine(EG)) | 11 | [
"rectangle_property_diagonal_equal(1,DGFE)"
] | {"START": ["rectangle_property_diagonal_equal(1,DGFE)"]} | |
878 | XiaokaiZhang_2023-03-19 | Geometry3k-907 | 4 | 如图所示，BA=16，DA=9，DC=15，CA垂直于DA。求三角形CBA的面积。 | As shown in the diagram, BA=16, DA=9, DC=15, CA is perpendicular to DA. Find the area of △CBA. | 878.png | [
"Shape(CB,BA,AC)",
"Shape(CA,AD,DC)",
"Collinear(BAD)"
] | [
"Equal(LengthOfLine(BA),16)",
"Equal(LengthOfLine(DA),9)",
"Equal(LengthOfLine(DC),15)",
"PerpendicularBetweenLine(CA,DA)"
] | [
"Equal(LengthOfLine(BA),16)",
"Equal(LengthOfLine(DA),9)",
"Equal(LengthOfLine(DC),15)",
"PerpendicularBetweenLine(CA,DA)"
] | Value(AreaOfTriangle(CBA)) | 96 | [
"adjacent_complementary_angle(1,BAC,CAD)",
"right_triangle_judgment_angle(1,CAD)",
"right_triangle_property_pythagorean(1,CAD)",
"triangle_area_formula_sine(1,ACB)"
] | {"START": ["adjacent_complementary_angle(1,BAC,CAD)", "right_triangle_judgment_angle(1,CAD)", "triangle_area_formula_sine(1,ACB)"], "right_triangle_judgment_angle(1,CAD)": ["right_triangle_property_pythagorean(1,CAD)"]} | |
879 | XiaokaiZhang_2023-03-19 | Geometry3k-908 | 2 | 如图所示，AB=x，AC=9*sqrt(2)，BC=9*sqrt(2)，∠CBA=45°，AC⊥BC。求x的值。 | As shown in the diagram, AB=x, AC=9*sqrt(2), BC=9*sqrt(2), ∠CBA=45°, AC is perpendicular to BC. Find the value of x. | 879.png | [
"Shape(BA,AC,CB)"
] | [
"Equal(LengthOfLine(AB),x)",
"Equal(LengthOfLine(AC),9*sqrt(2))",
"Equal(LengthOfLine(BC),9*sqrt(2))",
"Equal(MeasureOfAngle(CBA),45)",
"PerpendicularBetweenLine(AC,BC)"
] | [
"Equal(LengthOfLine(AB),x)",
"Equal(LengthOfLine(AC),9*sqrt(2))",
"Equal(LengthOfLine(BC),9*sqrt(2))",
"Equal(MeasureOfAngle(CBA),45)",
"PerpendicularBetweenLine(AC,BC)"
] | Value(x) | 18 | [
"right_triangle_judgment_angle(1,ACB)",
"right_triangle_property_pythagorean(1,ACB)"
] | {"START": ["right_triangle_judgment_angle(1,ACB)"], "right_triangle_judgment_angle(1,ACB)": ["right_triangle_property_pythagorean(1,ACB)"]} | |
880 | NaZhu_2023-04-09 | Geometry3k-909 | 13 | 如图所示，AB=BI，AC=4，AC=ID，BE=8，CD=9，AC垂直于EC，CE垂直于FE，ED垂直于ID。求三角形BAI的面积与ACDI的面积之和。 | As shown in the diagram, AB=BI, AC=4, AC=ID, BE=8, CD=9, AC⊥EC, CE is perpendicular to FE, ED⊥ID. Find the sum of the area of triangle BAI and the area of ACDI. | 880.png | [
"Shape(BA,AF,FB)",
"Shape(BF,FI,IB)",
"Shape(FE,ED,DI,IF)",
"Shape(AC,CE,EF,FA)",
"Collinear(AFI)",
"Collinear(BFE)",
"Collinear(CED)"
] | [
"Equal(LengthOfLine(AB),LengthOfLine(BI))",
"Equal(LengthOfLine(AC),4)",
"Equal(LengthOfLine(AC),LengthOfLine(ID))",
"Equal(LengthOfLine(BE),8)",
"Equal(LengthOfLine(CD),9)",
"PerpendicularBetweenLine(AC,EC)",
"PerpendicularBetweenLine(CE,FE)",
"PerpendicularBetweenLine(ED,ID)"
] | [
"Equal(LengthOfLine(AB),LengthOfLine(BI))",
"Equal(LengthOfLine(AC),4)",
"Equal(LengthOfLine(AC),LengthOfLine(ID))",
"Equal(LengthOfLine(BE),8)",
"Equal(LengthOfLine(CD),9)",
"PerpendicularBetweenLine(CE,FE)"
] | Value(Add(AreaOfTriangle(BAI),AreaOfQuadrilateral(ACDI))) | 54 | [
"parallel_judgment_ipsilateral_internal_angle(1,CA,EF)",
"parallel_judgment_ipsilateral_internal_angle(1,CA,DI)",
"parallelogram_judgment_parallel_and_equal(1,DIAC)",
"parallel_property_collinear_extend(3,DC,IA,E)",
"parallel_property_collinear_extend(3,AI,CE,F)",
"parallel_property_corresponding_angle(2,... | {"START": ["parallel_judgment_ipsilateral_internal_angle(1,CA,EF)", "parallel_judgment_ipsilateral_internal_angle(1,CA,DI)", "line_addition(1,BF,FE)", "triangle_area_formula_common(1,BAI)"], "parallel_judgment_ipsilateral_internal_angle(1,CA,DI)": ["parallelogram_judgment_parallel_and_equal(1,DIAC)"], "parallel_judgmen... | |
881 | NaZhu_2023-04-09 | Geometry3k-910 | 2 | 如图所示，AD=x-3，AF=y+1，DB=12，FB=AF，LM=8，LO=10，LO=NO，MN=16，∠AFB=∠NOL，∠DAF=∠MNO，BD⊥AD，FB垂直于DB，LM垂直于NM，OL垂直于ML，四边形BDAF与四边形LMNO相似。求y的值。 | As shown in the diagram, AD=x-3, AF=y+1, DB=12, FB=AF, LM=8, LO=10, LO=NO, MN=16, ∠AFB=∠NOL, ∠DAF=∠MNO, BD⊥AD, FB⊥DB, LM is perpendicular to NM, OL is perpendicular to ML, quadrilateral BDAF is similar to quadrilateral LMNO. Find the value of y. | 881.png | [
"Shape(BD,DA,AF,FB)",
"Shape(LM,MN,NO,OL)"
] | [
"Equal(LengthOfLine(AD),x-3)",
"Equal(LengthOfLine(AF),y+1)",
"Equal(LengthOfLine(DB),12)",
"Equal(LengthOfLine(FB),LengthOfLine(AF))",
"Equal(LengthOfLine(LM),8)",
"Equal(LengthOfLine(LO),10)",
"Equal(LengthOfLine(LO),LengthOfLine(NO))",
"Equal(LengthOfLine(MN),16)",
"Equal(MeasureOfAngle(AFB),Meas... | [
"Equal(LengthOfLine(AD),x-3)",
"Equal(LengthOfLine(AF),y+1)",
"Equal(LengthOfLine(DB),12)",
"Equal(LengthOfLine(FB),LengthOfLine(AF))",
"Equal(LengthOfLine(LM),8)",
"Equal(LengthOfLine(LO),10)",
"Equal(LengthOfLine(LO),LengthOfLine(NO))",
"Equal(LengthOfLine(MN),16)",
"Equal(MeasureOfAngle(AFB),Meas... | Value(y) | 14 | [
"similar_quadrilateral_property_line_ratio(1,BDAF,LMNO)",
"similar_quadrilateral_property_line_ratio(1,AFBD,NOLM)"
] | {"START": ["similar_quadrilateral_property_line_ratio(1,BDAF,LMNO)", "similar_quadrilateral_property_line_ratio(1,AFBD,NOLM)"]} | |
882 | NaZhu_2023-04-09 | Geometry3k-911 | 1 | 如图所示，WZ=8，XY=20，∠XYZ=70°，WXYZ是梯形。求∠YZW的大小。 | As shown in the diagram, WZ=8, XY=20, ∠XYZ=70°, WZ and XY are the parallel sides of trapezoid WXYZ. Find the measure of ∠YZW. | 882.png | [
"Shape(WX,XY,YZ,ZW)"
] | [
"Equal(LengthOfLine(WZ),8)",
"Equal(LengthOfLine(XY),20)",
"Equal(MeasureOfAngle(XYZ),70)",
"Trapezoid(WXYZ)"
] | [
"Equal(LengthOfLine(WZ),8)",
"Equal(LengthOfLine(XY),20)",
"Equal(MeasureOfAngle(XYZ),70)"
] | Value(MeasureOfAngle(YZW)) | 110 | [
"parallel_property_ipsilateral_internal_angle(1,YX,ZW)"
] | {"START": ["parallel_property_ipsilateral_internal_angle(1,YX,ZW)"]} | |
883 | NaZhu_2023-04-09 | Geometry3k-912 | 3 | 如图所示，WT=3，WX⊥YX，XY垂直于ZY，YZ⊥WZ，ZW垂直于XW，WXYZ是正方形。求直线ZX的长度。 | As shown in the diagram, WT=3, WX is perpendicular to YX, XY is perpendicular to ZY, YZ⊥WZ, ZW is perpendicular to XW, WXYZ is a square. Find the length of line ZX. | 883.png | [
"Shape(XY,YT,TX)",
"Shape(TY,YZ,ZT)",
"Shape(TZ,ZW,WT)",
"Shape(XT,TW,WX)",
"Collinear(XTZ)",
"Collinear(YTW)"
] | [
"Equal(LengthOfLine(WT),3)",
"PerpendicularBetweenLine(WX,YX)",
"PerpendicularBetweenLine(XY,ZY)",
"PerpendicularBetweenLine(YZ,WZ)",
"PerpendicularBetweenLine(ZW,XW)",
"Square(WXYZ)"
] | [
"PerpendicularBetweenLine(WX,YX)",
"PerpendicularBetweenLine(XY,ZY)",
"PerpendicularBetweenLine(YZ,WZ)",
"PerpendicularBetweenLine(ZW,XW)"
] | Value(LengthOfLine(ZX)) | 6 | [
"line_addition(1,YT,TW)",
"parallelogram_property_diagonal_bisection(1,WXYZ,T)",
"rectangle_property_diagonal_equal(1,WXYZ)"
] | {"START": ["line_addition(1,YT,TW)", "parallelogram_property_diagonal_bisection(1,WXYZ,T)", "rectangle_property_diagonal_equal(1,WXYZ)"]} | |
884 | XiaokaiZhang_2023-03-19 | Geometry3k-913 | 2 | 如图所示，∠ACB=80°，∠CBA=∠GHF，∠FGH=2*x°，BA⊥CA，HF垂直于GF。求x的值。 | As shown in the diagram, ∠ACB=80°, ∠CBA=∠GHF, ∠FGH=2*x°, BA is perpendicular to CA, HF is perpendicular to GF. Find the value of x. | 884.png | [
"Shape(BA,AC,CB)",
"Shape(GH,HF,FG)"
] | [
"Equal(MeasureOfAngle(ACB),80)",
"Equal(MeasureOfAngle(CBA),MeasureOfAngle(GHF))",
"Equal(MeasureOfAngle(FGH),2*x)",
"PerpendicularBetweenLine(BA,CA)",
"PerpendicularBetweenLine(HF,GF)"
] | [
"Equal(MeasureOfAngle(ACB),80)",
"Equal(MeasureOfAngle(CBA),MeasureOfAngle(GHF))",
"Equal(MeasureOfAngle(FGH),2*x)",
"PerpendicularBetweenLine(BA,CA)",
"PerpendicularBetweenLine(HF,GF)"
] | Value(x) | 40 | [
"similar_triangle_judgment_aa(1,CBA,GHF)",
"similar_triangle_property_angle_equal(1,CBA,GHF)"
] | {"START": ["similar_triangle_judgment_aa(1,CBA,GHF)"], "similar_triangle_judgment_aa(1,CBA,GHF)": ["similar_triangle_property_angle_equal(1,CBA,GHF)"]} | |
885 | NaZhu_2023-04-09 | Geometry3k-914 | 3 | 如图所示，GF=2，∠FGH=99°，圆G的圆心为G。求扇形GHF的面积。 | As shown in the diagram, GF=2, ∠FGH=99°, G is the center of ⊙G. Find the area of the sector GHF. | 885.png | [
"Shape(GFH,HG,GF)",
"Shape(FG,GH,GHF)",
"Cocircular(G,FH)"
] | [
"Equal(LengthOfLine(GF),2)",
"Equal(MeasureOfAngle(FGH),99)",
"IsCentreOfCircle(G,G)"
] | [
"Equal(LengthOfLine(GF),2)",
"Equal(MeasureOfAngle(FGH),99)",
"IsCentreOfCircle(G,G)"
] | Value(AreaOfSector(GHF)) | 11*pi/10 | [
"arc_property_center_angle(1,GHF,G)",
"radius_of_circle_property_length_equal(1,GF,G)",
"sector_area_formula(1,GHF)"
] | {"START": ["arc_property_center_angle(1,GHF,G)", "radius_of_circle_property_length_equal(1,GF,G)", "sector_area_formula(1,GHF)"]} | |
886 | NaZhu_2023-04-09 | Geometry3k-915 | 2 | 如图所示，UT=12，VS=9，S是线段RT的中点，V是线段QU的中点，四边形QUTR是梯形。求直线QR的长度。 | As shown in the diagram, UT=12, VS=9, S bisects segment RT, V bisects segment QU, QR and UT are the parallel sides of trapezoid QUTR. Find the length of line QR. | 886.png | [
"Shape(QV,VS,SR,RQ)",
"Shape(VU,UT,TS,SV)",
"Collinear(QVU)",
"Collinear(RST)"
] | [
"Equal(LengthOfLine(UT),12)",
"Equal(LengthOfLine(VS),9)",
"IsMidpointOfLine(S,RT)",
"IsMidpointOfLine(V,QU)",
"Trapezoid(QUTR)"
] | [] | Value(LengthOfLine(QR)) | 6 | [
"midsegment_of_quadrilateral_judgment_midpoint(1,VS,QUTR)",
"midsegment_of_quadrilateral_property_length(1,VS,QUTR)"
] | {"START": ["midsegment_of_quadrilateral_judgment_midpoint(1,VS,QUTR)"], "midsegment_of_quadrilateral_judgment_midpoint(1,VS,QUTR)": ["midsegment_of_quadrilateral_property_length(1,VS,QUTR)"]} | |
887 | NaZhu_2023-04-09 | Geometry3k-916 | 7 | 如图所示，AC=AB，BA=5*sqrt(2)，BC=y，BE=x，AE垂直于BE，BC⊥DC，CA⊥BA，CD垂直于AD，EB⊥CB。求x的值。 | As shown in the diagram, AC=AB, BA=5*sqrt(2), BC=y, BE=x, AE⊥BE, BC is perpendicular to DC, CA is perpendicular to BA, CD is perpendicular to AD, EB is perpendicular to CB. Find the value of x. | 887.png | [
"Shape(CD,DA,AC)",
"Shape(AE,EB,BA)",
"Shape(CA,AB,BC)",
"Collinear(DAE)"
] | [
"Equal(LengthOfLine(AC),LengthOfLine(AB))",
"Equal(LengthOfLine(BA),5*sqrt(2))",
"Equal(LengthOfLine(BC),y)",
"Equal(LengthOfLine(BE),x)",
"PerpendicularBetweenLine(AE,BE)",
"PerpendicularBetweenLine(BC,DC)",
"PerpendicularBetweenLine(CA,BA)",
"PerpendicularBetweenLine(CD,AD)",
"PerpendicularBetween... | [
"Equal(LengthOfLine(AC),LengthOfLine(AB))",
"Equal(LengthOfLine(BA),5*sqrt(2))",
"Equal(LengthOfLine(BC),y)",
"Equal(LengthOfLine(BE),x)",
"PerpendicularBetweenLine(AE,BE)",
"PerpendicularBetweenLine(BC,DC)",
"PerpendicularBetweenLine(CA,BA)",
"PerpendicularBetweenLine(CD,AD)",
"PerpendicularBetween... | Value(x) | 5 | [
"isosceles_triangle_judgment_line_equal(1,ABC)",
"isosceles_triangle_property_angle_equal(1,ABC)",
"triangle_property_angle_sum(1,AEB)",
"triangle_property_angle_sum(1,CAB)",
"angle_addition(1,EBA,ABC)",
"sine_theorem(1,AEB)",
"sine_theorem(1,EBA)"
] | {"START": ["isosceles_triangle_judgment_line_equal(1,ABC)", "triangle_property_angle_sum(1,AEB)", "triangle_property_angle_sum(1,CAB)", "angle_addition(1,EBA,ABC)", "sine_theorem(1,AEB)", "sine_theorem(1,EBA)"], "isosceles_triangle_judgment_line_equal(1,ABC)": ["isosceles_triangle_property_angle_equal(1,ABC)"]} | |
888 | NaZhu_2023-04-09 | Geometry3k-917 | 4 | 如图所示，圆A的直径为8，圆B的直径为18，⊙C的直径为11，⊙A的圆心为A，⊙B的圆心为B，C是⊙C的圆心，AG是⊙B的直径。求直线FG的长度。 | As shown in the diagram, the diameter of circle A is 8, the diameter of ⊙B is 18, the diameter of ⊙C is 11, the center of circle A is A, B is the center of circle B, C is the center of ⊙C, the diameter of circle B is AG. Find the length of line FG. | 888.png | [
"Shape(BAM,AMN,BNA)",
"Shape(AFM,BMA,AF)",
"Shape(FA,BAN,ANF)",
"Shape(BXM,AMF,AFN,BNY,CYE,CEX)",
"Shape(CXE,EG,BGX)",
"Shape(CEY,BYG,GE)",
"Shape(CYX,BXG,BGY)",
"Collinear(AFBEGC)",
"Cocircular(A,NFM)",
"Cocircular(B,MANYGX)",
"Cocircular(C,XEY)"
] | [
"Equal(DiameterOfCircle(A),8)",
"Equal(DiameterOfCircle(B),18)",
"Equal(DiameterOfCircle(C),11)",
"IsCentreOfCircle(A,A)",
"IsCentreOfCircle(B,B)",
"IsCentreOfCircle(C,C)",
"IsDiameterOfCircle(AG,B)"
] | [
"Equal(DiameterOfCircle(A),8)",
"Equal(DiameterOfCircle(B),18)",
"Equal(DiameterOfCircle(C),11)",
"IsCentreOfCircle(A,A)",
"IsCentreOfCircle(B,B)",
"IsCentreOfCircle(C,C)",
"IsDiameterOfCircle(AG,B)"
] | Value(LengthOfLine(FG)) | 14 | [
"diameter_of_circle_property_length_equal(1,AG,B)",
"circle_property_length_of_radius_and_diameter(1,A)",
"radius_of_circle_property_length_equal(1,AF,A)",
"line_addition(1,AF,FG)"
] | {"START": ["diameter_of_circle_property_length_equal(1,AG,B)", "circle_property_length_of_radius_and_diameter(1,A)", "radius_of_circle_property_length_equal(1,AF,A)", "line_addition(1,AF,FG)"]} | |
889 | NaZhu_2023-04-09 | Geometry3k-918 | 1 | 如图所示，∠EDH=x°，∠HDA=3*x°。求x的值。 | As shown in the diagram, ∠EDH=x°, ∠HDA=3*x°. Find the value of x. | 889.png | [
"Shape(DA)",
"Shape(ED)",
"Shape(DH)",
"Shape(HD,DA)",
"Shape(ED,DH)",
"Collinear(ADE)"
] | [
"Equal(MeasureOfAngle(EDH),x)",
"Equal(MeasureOfAngle(HDA),3*x)"
] | [
"Equal(MeasureOfAngle(EDH),x)",
"Equal(MeasureOfAngle(HDA),3*x)"
] | Value(x) | 45 | [
"adjacent_complementary_angle(1,EDH,HDA)"
] | {"START": ["adjacent_complementary_angle(1,EDH,HDA)"]} | |
890 | XiaokaiZhang_2023-04-09 | Geometry3k-919 | 3 | 如图所示，弧ANX的长度为x，XA=3，∠XAN=78°，⊙A的圆心为A。求x的值。 | As shown in the diagram, the length of ⌒ANX is x, XA=3, ∠XAN=78°, A is the center of circle A. Find the value of x. | 890.png | [
"Shape(AN,ANX,XA)",
"Shape(AX,AXN,NA)",
"Cocircular(A,XN)"
] | [
"Equal(LengthOfArc(ANX),x)",
"Equal(LengthOfLine(XA),3)",
"Equal(MeasureOfAngle(XAN),78)",
"IsCentreOfCircle(A,A)"
] | [
"Equal(LengthOfArc(ANX),x)",
"Equal(LengthOfLine(XA),3)",
"Equal(MeasureOfAngle(XAN),78)",
"IsCentreOfCircle(A,A)"
] | Value(x) | 13*pi/10 | [
"radius_of_circle_property_length_equal(1,AX,A)",
"arc_property_center_angle(1,ANX,A)",
"arc_length_formula(1,ANX)"
] | {"START": ["radius_of_circle_property_length_equal(1,AX,A)", "arc_property_center_angle(1,ANX,A)", "arc_length_formula(1,ANX)"]} | |
891 | XiaokaiZhang_2023-04-09 | Geometry3k-920 | 1 | 如图所示，∠JQR=131°，GT∥RQ，HR∥TC。求∠RQT的大小。 | As shown in the diagram, ∠JQR=131°, GT is parallel to RQ, HR∥TC. Find the measure of ∠RQT. | 891.png | [
"Shape(JQ,QR)",
"Shape(QR,RH)",
"Shape(BG,GT)",
"Shape(GT,TC)",
"Shape(QT,TG,GR,RQ)",
"Collinear(JQTC)",
"Collinear(HRGB)"
] | [
"Equal(MeasureOfAngle(JQR),131)",
"ParallelBetweenLine(GT,RQ)",
"ParallelBetweenLine(HR,TC)"
] | [
"ParallelBetweenLine(GT,RQ)",
"ParallelBetweenLine(HR,TC)"
] | Value(MeasureOfAngle(RQT)) | 49 | [
"adjacent_complementary_angle(1,JQR,RQT)"
] | {"START": ["adjacent_complementary_angle(1,JQR,RQT)"]} | |
892 | XiaokaiZhang_2023-04-09 | Geometry3k-921 | 3 | 如图所示，AF=4，GA=3，FG和FD是风筝形CGFD的一组临边。求直线GF的长度。 | As shown in the diagram, AF=4, GA=3, CG and CD are one pair of adjacent sides of the kite CGFD. Find the length of line GF. | 892.png | [
"Shape(CG,GA,AC)",
"Shape(CA,AD,DC)",
"Shape(AG,GF,FA)",
"Shape(AF,FD,DA)",
"Collinear(GAD)",
"Collinear(CAF)"
] | [
"Equal(LengthOfLine(AF),4)",
"Equal(LengthOfLine(GA),3)",
"Kite(CGFD)"
] | [
"Equal(LengthOfLine(AF),4)",
"Equal(LengthOfLine(GA),3)"
] | Value(LengthOfLine(GF)) | 5 | [
"kite_property_diagonal_perpendicular_bisection(1,FDCG,A)",
"right_triangle_judgment_angle(1,FAG)",
"right_triangle_property_pythagorean(1,FAG)"
] | {"START": ["kite_property_diagonal_perpendicular_bisection(1,FDCG,A)"], "kite_property_diagonal_perpendicular_bisection(1,FDCG,A)": ["right_triangle_judgment_angle(1,FAG)"], "right_triangle_judgment_angle(1,FAG)": ["right_triangle_property_pythagorean(1,FAG)"]} | |
893 | XiaokaiZhang_2023-03-19 | Geometry3k-922 | 6 | 如图所示，△BAC的面积为875，△DEF的面积为315，BM=x，DN=21，AM垂直于BM，EN垂直于DN，三角形BAC与三角形DEF是相似三角形。求x的值。 | As shown in the diagram, the area of triangle BAC is 875, the area of triangle DEF is 315, BM=x, DN=21, AM is perpendicular to BM, EN⊥DN, △BAC is similar to △DEF.. Find the value of x. | 893.png | [
"Shape(BA,AM,MB)",
"Shape(BM,MC,CB)",
"Shape(DE,EN,ND)",
"Shape(DN,NF,FD)",
"Collinear(AMC)",
"Collinear(ENF)"
] | [
"Equal(AreaOfTriangle(BAC),875)",
"Equal(AreaOfTriangle(DEF),315)",
"Equal(LengthOfLine(BM),x)",
"Equal(LengthOfLine(DN),21)",
"PerpendicularBetweenLine(AM,BM)",
"PerpendicularBetweenLine(EN,DN)",
"SimilarBetweenTriangle(BAC,DEF)"
] | [
"Equal(AreaOfTriangle(BAC),875)",
"Equal(AreaOfTriangle(DEF),315)",
"Equal(LengthOfLine(BM),x)",
"Equal(LengthOfLine(DN),21)",
"PerpendicularBetweenLine(AM,BM)",
"PerpendicularBetweenLine(EN,DN)"
] | Value(x) | 35 | [
"altitude_of_triangle_judgment(1,BM,BAC)",
"altitude_of_triangle_judgment(1,DN,DEF)",
"triangle_area_formula_common(1,BAC)",
"triangle_area_formula_common(1,DEF)",
"similar_triangle_property_line_ratio(1,BAC,DEF)",
"similar_triangle_property_area_square_ratio(1,BAC,DEF)"
] | {"START": ["altitude_of_triangle_judgment(1,BM,BAC)", "altitude_of_triangle_judgment(1,DN,DEF)", "triangle_area_formula_common(1,BAC)", "triangle_area_formula_common(1,DEF)", "similar_triangle_property_line_ratio(1,BAC,DEF)", "similar_triangle_property_area_square_ratio(1,BAC,DEF)"]} | |
894 | XiaokaiZhang_2023-04-09 | Geometry3k-923 | 1 | 如图所示，BD=5，DF=4，EB=DF，EF=DB，EBDF是平行四边形，EB垂直于DB。求EBDF的面积。 | As shown in the diagram, BD=5, DF=4, EB=DF, EF=DB, BE and DF are opposite sides of the parallelogram EBDF, EB is perpendicular to DB. Find the area of quadrilateral EBDF. | 894.png | [
"Shape(EB,BD,DF,FE)"
] | [
"Equal(LengthOfLine(BD),5)",
"Equal(LengthOfLine(DF),4)",
"Equal(LengthOfLine(EB),LengthOfLine(DF))",
"Equal(LengthOfLine(EF),LengthOfLine(DB))",
"Parallelogram(EBDF)",
"PerpendicularBetweenLine(EB,DB)"
] | [
"Equal(LengthOfLine(BD),5)",
"Equal(LengthOfLine(DF),4)",
"Equal(LengthOfLine(EB),LengthOfLine(DF))",
"Equal(LengthOfLine(EF),LengthOfLine(DB))",
"PerpendicularBetweenLine(EB,DB)"
] | Value(AreaOfQuadrilateral(EBDF)) | 20 | [
"parallelogram_area_formula_sine(1,EBDF)"
] | {"START": ["parallelogram_area_formula_sine(1,EBDF)"]} | |
895 | XiaokaiZhang_2023-03-19 | Geometry3k-924 | 2 | 如图所示，AB=3，AC=x，BC=y，∠ACB=45°，BA垂直于CA。求x的值。 | As shown in the diagram, AB=3, AC=x, BC=y, ∠ACB=45°, BA⊥CA. Find the value of x. | 895.png | [
"Shape(AC,CB,BA)"
] | [
"Equal(LengthOfLine(AB),3)",
"Equal(LengthOfLine(AC),x)",
"Equal(LengthOfLine(BC),y)",
"Equal(MeasureOfAngle(ACB),45)",
"PerpendicularBetweenLine(BA,CA)"
] | [
"Equal(LengthOfLine(AB),3)",
"Equal(LengthOfLine(AC),x)",
"Equal(LengthOfLine(BC),y)",
"Equal(MeasureOfAngle(ACB),45)",
"PerpendicularBetweenLine(BA,CA)"
] | Value(x) | 3 | [
"triangle_property_angle_sum(1,ACB)",
"isosceles_triangle_judgment_angle_equal(1,ACB)"
] | {"START": ["triangle_property_angle_sum(1,ACB)"], "triangle_property_angle_sum(1,ACB)": ["isosceles_triangle_judgment_angle_equal(1,ACB)"]} | |
896 | XiaokaiZhang_2023-04-09 | Geometry3k-925 | 0 | 如图所示，HF=4*x-22，JO=17+5*y，OP=13+6*y，RH=3*x-9，RH=HF。求x的值。 | As shown in the diagram, HF=4*x-22, JO=17+5*y, OP=13+6*y, RH=3*x-9, RH=HF. Find the value of x. | 896.png | [
"Shape(NJ,JC)",
"Shape(CJ,JR)",
"Shape(JR,RL)",
"Shape(LR,RD)",
"Shape(DR,RH)",
"Shape(RH,HK)",
"Shape(KH,HF)",
"Shape(HF,FA)",
"Shape(AF,FY)",
"Shape(YF,FP)",
"Shape(FP,PV)",
"Shape(VP,PM)",
"Shape(MP,PO)",
"Shape(PO,OE)",
"Shape(EO,OJ)",
"Shape(OJ,JN)",
"Shape(JO,OE,ER,RJ)",
"Sha... | [
"Equal(LengthOfLine(HF),4*x-22)",
"Equal(LengthOfLine(JO),17+5*y)",
"Equal(LengthOfLine(OP),13+6*y)",
"Equal(LengthOfLine(RH),3*x-9)",
"Equal(LengthOfLine(RH),LengthOfLine(HF))"
] | [
"Equal(LengthOfLine(HF),4*x-22)",
"Equal(LengthOfLine(JO),17+5*y)",
"Equal(LengthOfLine(OP),13+6*y)",
"Equal(LengthOfLine(RH),3*x-9)",
"Equal(LengthOfLine(RH),LengthOfLine(HF))"
] | Value(x) | 13 | [] | {"START": []} | |
897 | XiaokaiZhang_2023-03-19 | Geometry3k-926 | 6 | 如图所示，PS=3，RY=5，WX=10，WY=8，XY=6，RP∥XW，BQ⊥SQ，SY垂直于BY，YS垂直于PS。求直线SY的长度。 | As shown in the diagram, PS=3, RY=5, WX=10, WY=8, XY=6, RP∥XW, BQ is perpendicular to SQ, SY⊥BY, YS⊥PS. Find the length of line SY. | 897.png | [
"Shape(YR,RB,BY)",
"Shape(YB,BQ,QS,SY)",
"Shape(PY,YS,SP)",
"Shape(BX,XQ,QB)",
"Shape(SQ,QW,WS)",
"Collinear(RYP)",
"Collinear(YSW)",
"Collinear(RBQ)",
"Collinear(XQW)",
"Collinear(YBX)",
"Collinear(PSQ)"
] | [
"Equal(LengthOfLine(PS),3)",
"Equal(LengthOfLine(RY),5)",
"Equal(LengthOfLine(WX),10)",
"Equal(LengthOfLine(WY),8)",
"Equal(LengthOfLine(XY),6)",
"ParallelBetweenLine(RP,XW)",
"PerpendicularBetweenLine(BQ,SQ)",
"PerpendicularBetweenLine(SY,BY)",
"PerpendicularBetweenLine(YS,PS)"
] | [
"PerpendicularBetweenLine(BQ,SQ)",
"PerpendicularBetweenLine(SY,BY)",
"PerpendicularBetweenLine(YS,PS)"
] | Value(LengthOfLine(SY)) | 4 | [
"cosine_theorem(1,XWY)",
"parallel_property_collinear_extend(3,RP,XW,Y)",
"parallel_property_ipsilateral_internal_angle(1,YP,XW)",
"angle_addition(1,PYS,SYB)",
"triangle_property_angle_sum(1,PYS)",
"sine_theorem(1,SPY)"
] | {"START": ["cosine_theorem(1,XWY)", "parallel_property_collinear_extend(3,RP,XW,Y)", "angle_addition(1,PYS,SYB)", "triangle_property_angle_sum(1,PYS)", "sine_theorem(1,SPY)"], "parallel_property_collinear_extend(3,RP,XW,Y)": ["parallel_property_ipsilateral_internal_angle(1,YP,XW)"]} | |
898 | XiaokaiZhang_2023-03-19 | Geometry3k-927 | 4 | 如图所示，HQ=y，JQ=2*z，KQ=7，LQ=3，QM=4，QP=2*x-6，J是线段HK的中点，L是线段KM的中点，P是线段MH的中点。求y的值。 | As shown in the diagram, HQ=y, JQ=2*z, KQ=7, LQ=3, QM=4, QP=2*x-6, J is the midpoint of segment HK, L is the midpoint of segment KM, P is the midpoint of segment MH. Find the value of y. | 898.png | [
"Shape(HJ,JQ,QH)",
"Shape(JK,KQ,QJ)",
"Shape(QK,KL,LQ)",
"Shape(QL,LM,MQ)",
"Shape(QM,MP,PQ)",
"Shape(QP,PH,HQ)",
"Collinear(HJK)",
"Collinear(KLM)",
"Collinear(MPH)",
"Collinear(HQL)",
"Collinear(JQM)",
"Collinear(KQP)"
] | [
"Equal(LengthOfLine(HQ),y)",
"Equal(LengthOfLine(JQ),2*z)",
"Equal(LengthOfLine(KQ),7)",
"Equal(LengthOfLine(LQ),3)",
"Equal(LengthOfLine(QM),4)",
"Equal(LengthOfLine(QP),2*x-6)",
"IsMidpointOfLine(J,HK)",
"IsMidpointOfLine(L,KM)",
"IsMidpointOfLine(P,MH)"
] | [
"Equal(LengthOfLine(HQ),y)",
"Equal(LengthOfLine(JQ),2*z)",
"Equal(LengthOfLine(KQ),7)",
"Equal(LengthOfLine(LQ),3)",
"Equal(LengthOfLine(QM),4)",
"Equal(LengthOfLine(QP),2*x-6)"
] | Value(y) | 6 | [
"median_of_triangle_judgment(1,KP,KMH)",
"median_of_triangle_judgment(1,HL,HKM)",
"centroid_of_triangle_judgment_intersection(1,Q,MHK,P,L)",
"centroid_of_triangle_property_line_ratio(1,Q,HKM,L)"
] | {"START": ["median_of_triangle_judgment(1,KP,KMH)", "median_of_triangle_judgment(1,HL,HKM)"], "centroid_of_triangle_judgment_intersection(1,Q,MHK,P,L)": ["centroid_of_triangle_property_line_ratio(1,Q,HKM,L)"], "median_of_triangle_judgment(1,HL,HKM)": ["centroid_of_triangle_judgment_intersection(1,Q,MHK,P,L)"], "median_... | |
899 | XiaokaiZhang_2023-03-19 | Geometry3k-928 | 0 | 如图所示，FJ=y+12，FK=3*x+7，GK=4*x-1，HJ=2*y-5，HJ=FJ，FK⊥JK，KG垂直于HG。求y的值。 | As shown in the diagram, FJ=y+12, FK=3*x+7, GK=4*x-1, HJ=2*y-5, HJ=FJ, FK⊥JK, KG is perpendicular to HG. Find the value of y. | 899.png | [
"Shape(JF,FK,KJ)",
"Shape(JK,KG,GH,HJ)",
"Collinear(FKG)",
"Collinear(FJH)"
] | [
"Equal(LengthOfLine(FJ),y+12)",
"Equal(LengthOfLine(FK),3*x+7)",
"Equal(LengthOfLine(GK),4*x-1)",
"Equal(LengthOfLine(HJ),2*y-5)",
"Equal(LengthOfLine(HJ),LengthOfLine(FJ))",
"PerpendicularBetweenLine(FK,JK)",
"PerpendicularBetweenLine(KG,HG)"
] | [
"Equal(LengthOfLine(FJ),y+12)",
"Equal(LengthOfLine(FK),3*x+7)",
"Equal(LengthOfLine(GK),4*x-1)",
"Equal(LengthOfLine(HJ),2*y-5)",
"Equal(LengthOfLine(HJ),LengthOfLine(FJ))",
"PerpendicularBetweenLine(FK,JK)",
"PerpendicularBetweenLine(KG,HG)"
] | Value(y) | 17 | [] | {"START": []} | |
900 | XiaokaiZhang_2023-03-19 | Geometry3k-929 | 2 | 如图所示，AB=y，AC=15，BC=x，∠CBA=60°，AC垂直于BC。求x的值。 | As shown in the diagram, AB=y, AC=15, BC=x, ∠CBA=60°, AC⊥BC. Find the value of x. | 900.png | [
"Shape(CB,BA,AC)"
] | [
"Equal(LengthOfLine(AB),y)",
"Equal(LengthOfLine(AC),15)",
"Equal(LengthOfLine(BC),x)",
"Equal(MeasureOfAngle(CBA),60)",
"PerpendicularBetweenLine(AC,BC)"
] | [
"Equal(LengthOfLine(AB),y)",
"Equal(LengthOfLine(AC),15)",
"Equal(LengthOfLine(BC),x)",
"Equal(MeasureOfAngle(CBA),60)",
"PerpendicularBetweenLine(AC,BC)"
] | Value(x) | 5*sqrt(3) | [
"triangle_property_angle_sum(1,CBA)",
"sine_theorem(1,CBA)"
] | {"START": ["triangle_property_angle_sum(1,CBA)", "sine_theorem(1,CBA)"]} |
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