Difference between revisions of "2022 AIME I Problems/Problem 3"
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== Diagram == | == Diagram == | ||
<asy> | <asy> | ||
− | + | /* Made by MRENTHUSIASM */ | |
− | pair A = (-250, | + | size(300); |
− | + | pair A, B, C, D, A1, B1, C1, D1, P, Q; | |
− | + | A = (-250,6*sqrt(731)); | |
− | + | B = (250,6*sqrt(731)); | |
− | draw(A--B--C--D--cycle); | + | C = (325,-6*sqrt(731)); |
− | pair W = ( | + | D = (-325,-6*sqrt(731)); |
− | + | A1 = bisectorpoint(B,A,D); | |
− | + | B1 = bisectorpoint(A,B,C); | |
− | + | C1 = bisectorpoint(B,C,D); | |
+ | D1 = bisectorpoint(A,D,C); | ||
+ | P = intersectionpoint(A--300*(A1-A)+A,D--300*(D1-D)+D); | ||
+ | Q = intersectionpoint(B--300*(B1-B)+B,C--300*(C1-C)+C); | ||
+ | draw(anglemark(P,A,B,1000),red); | ||
+ | draw(anglemark(D,A,P,1000),red); | ||
+ | draw(anglemark(A,B,Q,1000),red); | ||
+ | draw(anglemark(Q,B,C,1000),red); | ||
+ | draw(anglemark(P,D,A,1000),red); | ||
+ | draw(anglemark(C,D,P,1000),red); | ||
+ | draw(anglemark(Q,C,D,1000),red); | ||
+ | draw(anglemark(B,C,Q,1000),red); | ||
+ | add(pathticks(anglemark(P,A,B,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(D,A,P,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(A,B,Q,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(Q,B,C,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(P,D,A,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | add(pathticks(anglemark(C,D,P,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | add(pathticks(anglemark(Q,C,D,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | add(pathticks(anglemark(B,C,Q,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | dot("$A$",A,1.5*dir(A),linewidth(4)); | ||
+ | dot("$B$",B,1.5*dir(B),linewidth(4)); | ||
+ | dot("$C$",C,1.5*dir(C),linewidth(4)); | ||
+ | dot("$D$",D,1.5*dir(D),linewidth(4)); | ||
+ | dot("$P$",P,1.5*NE,linewidth(4)); | ||
+ | dot("$Q$",Q,1.5*NW,linewidth(4)); | ||
+ | draw(A--B--C--D--cycle^^A--P--D^^B--Q--C^^P--Q); | ||
+ | label("$500$",midpoint(A--B),1.25N); | ||
+ | label("$650$",midpoint(C--D),1.25S); | ||
+ | label("$333$",midpoint(A--D),1.25W); | ||
+ | label("$333$",midpoint(B--C),1.25E); | ||
+ | </asy> | ||
+ | ~MRENTHUSIASM ~ihatemath123 | ||
+ | |||
+ | == Solution 1 == | ||
+ | We have the following diagram: | ||
+ | <asy> | ||
+ | /* Made by MRENTHUSIASM , modified by Cytronical */ | ||
+ | size(300); | ||
+ | pair A, B, C, D, A1, B1, C1, D1, P, Q, X, Y, Z, W; | ||
+ | A = (-250,6*sqrt(731)); | ||
+ | B = (250,6*sqrt(731)); | ||
+ | C = (325,-6*sqrt(731)); | ||
+ | D = (-325,-6*sqrt(731)); | ||
+ | A1 = bisectorpoint(B,A,D); | ||
+ | B1 = bisectorpoint(A,B,C); | ||
+ | C1 = bisectorpoint(B,C,D); | ||
+ | D1 = bisectorpoint(A,D,C); | ||
+ | P = intersectionpoint(A--300*(A1-A)+A,D--300*(D1-D)+D); | ||
+ | Q = intersectionpoint(B--300*(B1-B)+B,C--300*(C1-C)+C); | ||
+ | X = intersectionpoint(B--5*(Q-B)+B,C--D); | ||
+ | Y = (0,6*sqrt(731)); | ||
+ | Z = intersectionpoint(A--4*(P-A)+A,B--4*(Q-B)+B); | ||
+ | W = intersectionpoint(A--5*(P-A)+A,C--D); | ||
+ | draw(anglemark(P,A,B,1000),red); | ||
+ | draw(anglemark(D,A,P,1000),red); | ||
+ | draw(anglemark(A,B,Q,1000),red); | ||
+ | draw(anglemark(Q,B,C,1000),red); | ||
+ | draw(anglemark(P,D,A,1000),red); | ||
+ | draw(anglemark(C,D,P,1000),red); | ||
+ | draw(anglemark(Q,C,D,1000),red); | ||
+ | draw(anglemark(B,C,Q,1000),red); | ||
+ | add(pathticks(anglemark(P,A,B,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(D,A,P,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(A,B,Q,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(Q,B,C,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(P,D,A,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | add(pathticks(anglemark(C,D,P,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | add(pathticks(anglemark(Q,C,D,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | add(pathticks(anglemark(B,C,Q,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | dot("$A$",A,1.5*dir(A),linewidth(4)); | ||
+ | dot("$B$",B,1.5*dir(B),linewidth(4)); | ||
+ | dot("$C$",C,1.5*dir(C),linewidth(4)); | ||
+ | dot("$D$",D,1.5*dir(D),linewidth(4)); | ||
+ | dot("$P$",P,1.5*(-1,0),linewidth(4)); | ||
+ | dot("$Q$",Q,1.5*E,linewidth(4)); | ||
+ | dot("$X$",X,1.5*dir(-105),linewidth(4)); | ||
+ | dot("$Y$",Y,1.5*N,linewidth(4)); | ||
+ | dot("$Z$",Z,4.5*dir(75),linewidth(4)); | ||
+ | dot("$W$",W,1.5*dir(-75),linewidth(4)); | ||
+ | draw(A--B--C--D--cycle^^A--P--D^^B--Q--C^^P--Q); | ||
+ | draw(P--W^^Q--X^^Y--Z,dashed); | ||
+ | </asy> | ||
− | + | Let <math>X</math> and <math>W</math> be the points where <math>AP</math> and <math>BQ</math> extend to meet <math>CD</math>, and <math>YZ</math> be the height of <math>\triangle AZB</math>. As proven in Solution 2, triangles <math>APD</math> and <math>DPW</math> are congruent right triangles. Therefore, <math>AD = DW = 333</math>. We can apply this logic to triangles <math>BCQ</math> and <math>XCQ</math> as well, giving us <math>BC = CX = 333</math>. Since <math>CD = 650</math>, <math>XW = DW + CX - CD = 16</math>. | |
− | |||
− | |||
− | |||
− | + | Additionally, we can see that <math>\triangle XZW</math> is similar to <math>\triangle PQZ</math> and <math>\triangle AZB</math>. We know that <math>\frac{XW}{AB} = \frac{16}{500}</math>. So, we can say that the height of the triangle <math>AZB</math> is <math>500u</math> while the height of the triangle <math>XZW</math> is <math>16u</math>. After that, we can figure out the distance from <math>Y</math> to <math>PQ: \frac{500+16}{2} = 258u</math> and the height of triangle <math>PZQ: 500-258 = 242u</math>. | |
− | + | ||
− | + | Finally, since the ratio between the height of <math>PZQ</math> to the height of <math>AZB</math> is <math>242:500</math> and <math>AB</math> is <math>500</math>, <math>PQ = \boxed{242}.</math> | |
− | draw(D | + | |
− | + | ~Cytronical | |
− | + | ||
− | + | == Solution 2 == | |
− | + | ||
− | + | Extend line <math>PQ</math> to meet <math>AD</math> at <math>P'</math> and <math>BC</math> at <math>Q'</math>. The diagram looks like this: | |
− | + | <asy> | |
− | + | /* Made by MRENTHUSIASM */ | |
− | + | size(300); | |
− | + | pair A, B, C, D, A1, B1, C1, D1, P, Q, P1, Q1; | |
− | + | A = (-250,6*sqrt(731)); | |
+ | B = (250,6*sqrt(731)); | ||
+ | C = (325,-6*sqrt(731)); | ||
+ | D = (-325,-6*sqrt(731)); | ||
+ | A1 = bisectorpoint(B,A,D); | ||
+ | B1 = bisectorpoint(A,B,C); | ||
+ | C1 = bisectorpoint(B,C,D); | ||
+ | D1 = bisectorpoint(A,D,C); | ||
+ | P = intersectionpoint(A--300*(A1-A)+A,D--300*(D1-D)+D); | ||
+ | Q = intersectionpoint(B--300*(B1-B)+B,C--300*(C1-C)+C); | ||
+ | P1 = intersectionpoint(A--D,P--(-300)*(Q-P)+P); | ||
+ | Q1 = intersectionpoint(B--C,Q--300*(Q-P)+Q); | ||
+ | draw(anglemark(P,A,B,1000),red); | ||
+ | draw(anglemark(D,A,P,1000),red); | ||
+ | draw(anglemark(A,B,Q,1000),red); | ||
+ | draw(anglemark(Q,B,C,1000),red); | ||
+ | draw(anglemark(P,D,A,1000),red); | ||
+ | draw(anglemark(C,D,P,1000),red); | ||
+ | draw(anglemark(Q,C,D,1000),red); | ||
+ | draw(anglemark(B,C,Q,1000),red); | ||
+ | add(pathticks(anglemark(P,A,B,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(D,A,P,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(A,B,Q,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(Q,B,C,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(P,D,A,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | add(pathticks(anglemark(C,D,P,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | add(pathticks(anglemark(Q,C,D,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | add(pathticks(anglemark(B,C,Q,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | dot("$A$",A,1.5*dir(A),linewidth(4)); | ||
+ | dot("$B$",B,1.5*dir(B),linewidth(4)); | ||
+ | dot("$C$",C,1.5*dir(C),linewidth(4)); | ||
+ | dot("$D$",D,1.5*dir(D),linewidth(4)); | ||
+ | dot("$P$",P,1.5*NE,linewidth(4)); | ||
+ | dot("$Q$",Q,1.5*NW,linewidth(4)); | ||
+ | dot("$P'$",P1,1.5*W,linewidth(4)); | ||
+ | dot("$Q'$",Q1,1.5*E,linewidth(4)); | ||
+ | draw(A--B--C--D--cycle^^A--P--D^^B--Q--C^^P--Q); | ||
+ | draw(P--P1^^Q--Q1,dashed); | ||
</asy> | </asy> | ||
+ | Because the trapezoid is isosceles, by symmetry <math>PQ</math> is parallel to <math>AB</math> and <math>CD</math>. Therefore, <math>\angle PAB \cong \angle APP'</math> by interior angles and <math>\angle PAB \cong \angle PAD</math> by the problem statement. Thus, <math>\triangle P'AP</math> is isosceles with <math>P'P = P'A</math>. By symmetry, <math>P'DP</math> is also isosceles, and thus <math>P'A = \frac{AD}{2}</math>. Similarly, the same thing is happening on the right side of the trapezoid, and thus <math>P'Q'</math> is the midline of the trapezoid. Then, <math>PQ = P'Q' - (P'P + Q'Q)</math>. | ||
− | == | + | Since <math>P'P = P'A = \frac{AD}{2}, Q'Q = Q'B = \frac{BC}{2}</math> and <math>AD = BC = 333</math>, we have <math>P'P + Q'Q = \frac{333}{2} + \frac{333}{2} = 333</math>. The length of the midline of a trapezoid is the average of their bases, so <math>P'Q' = \frac{500+650}{2} = 575</math>. Finally, <math>PQ = 575 - 333 = \boxed{242}</math>. |
+ | ~KingRavi | ||
+ | |||
+ | == Solution 3 == | ||
+ | |||
+ | We have the following diagram: | ||
+ | <asy> | ||
+ | /* Made by MRENTHUSIASM */ | ||
+ | size(300); | ||
+ | pair A, B, C, D, A1, B1, C1, D1, P, Q, X, Y, Z, W; | ||
+ | A = (-250,6*sqrt(731)); | ||
+ | B = (250,6*sqrt(731)); | ||
+ | C = (325,-6*sqrt(731)); | ||
+ | D = (-325,-6*sqrt(731)); | ||
+ | A1 = bisectorpoint(B,A,D); | ||
+ | B1 = bisectorpoint(A,B,C); | ||
+ | C1 = bisectorpoint(B,C,D); | ||
+ | D1 = bisectorpoint(A,D,C); | ||
+ | P = intersectionpoint(A--300*(A1-A)+A,D--300*(D1-D)+D); | ||
+ | Q = intersectionpoint(B--300*(B1-B)+B,C--300*(C1-C)+C); | ||
+ | X = intersectionpoint(B--5*(Q-B)+B,C--D); | ||
+ | Y = intersectionpoint(C--5*(Q-C)+C,A--B); | ||
+ | Z = intersectionpoint(D--5*(P-D)+D,A--B); | ||
+ | W = intersectionpoint(A--5*(P-A)+A,C--D); | ||
+ | draw(anglemark(P,A,B,1000),red); | ||
+ | draw(anglemark(D,A,P,1000),red); | ||
+ | draw(anglemark(A,B,Q,1000),red); | ||
+ | draw(anglemark(Q,B,C,1000),red); | ||
+ | draw(anglemark(P,D,A,1000),red); | ||
+ | draw(anglemark(C,D,P,1000),red); | ||
+ | draw(anglemark(Q,C,D,1000),red); | ||
+ | draw(anglemark(B,C,Q,1000),red); | ||
+ | add(pathticks(anglemark(P,A,B,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(D,A,P,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(A,B,Q,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(Q,B,C,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(P,D,A,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | add(pathticks(anglemark(C,D,P,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | add(pathticks(anglemark(Q,C,D,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | add(pathticks(anglemark(B,C,Q,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | dot("$A$",A,1.5*dir(A),linewidth(4)); | ||
+ | dot("$B$",B,1.5*dir(B),linewidth(4)); | ||
+ | dot("$C$",C,1.5*dir(C),linewidth(4)); | ||
+ | dot("$D$",D,1.5*dir(D),linewidth(4)); | ||
+ | dot("$P$",P,1.5*(-1,0),linewidth(4)); | ||
+ | dot("$Q$",Q,1.5*E,linewidth(4)); | ||
+ | dot("$X$",X,1.5*dir(-105),linewidth(4)); | ||
+ | dot("$Y$",Y,1.5*N,linewidth(4)); | ||
+ | dot("$Z$",Z,1.5*N,linewidth(4)); | ||
+ | dot("$W$",W,1.5*dir(-75),linewidth(4)); | ||
+ | draw(A--B--C--D--cycle^^A--P--D^^B--Q--C^^P--Q); | ||
+ | draw(P--Z^^P--W^^Q--X^^Q--Y,dashed); | ||
+ | </asy> | ||
Extend lines <math>AP</math> and <math>BQ</math> to meet line <math>DC</math> at points <math>W</math> and <math>X</math>, respectively, and extend lines <math>DP</math> and <math>CQ</math> to meet <math>AB</math> at points <math>Z</math> and <math>Y</math>, respectively. | Extend lines <math>AP</math> and <math>BQ</math> to meet line <math>DC</math> at points <math>W</math> and <math>X</math>, respectively, and extend lines <math>DP</math> and <math>CQ</math> to meet <math>AB</math> at points <math>Z</math> and <math>Y</math>, respectively. | ||
− | Claim: quadrilaterals <math>AZWD</math> and <math> | + | Claim: quadrilaterals <math>AZWD</math> and <math>BYXC</math> are rhombuses. |
Proof: Since <math>\angle DAB + \angle ADC = 180^{\circ}</math>, <math>\angle ADP + \angle PAD = 90^{\circ}</math>. Therefore, triangles <math>APD</math>, <math>APZ</math>, <math>DPW</math> and <math>PZW</math> are all right triangles. By SAA congruence, the first three triangles are congruent; by SAS congruence, <math>\triangle PZW</math> is congruent to the other three. Therefore, <math>AD = DW = WZ = AZ</math>, so <math>AZWD</math> is a rhombus. By symmetry, <math>BYXC</math> is also a rhombus. | Proof: Since <math>\angle DAB + \angle ADC = 180^{\circ}</math>, <math>\angle ADP + \angle PAD = 90^{\circ}</math>. Therefore, triangles <math>APD</math>, <math>APZ</math>, <math>DPW</math> and <math>PZW</math> are all right triangles. By SAA congruence, the first three triangles are congruent; by SAS congruence, <math>\triangle PZW</math> is congruent to the other three. Therefore, <math>AD = DW = WZ = AZ</math>, so <math>AZWD</math> is a rhombus. By symmetry, <math>BYXC</math> is also a rhombus. | ||
Line 47: | Line 216: | ||
~ihatemath123 | ~ihatemath123 | ||
+ | |||
+ | == Solution 4 == | ||
+ | |||
+ | <asy> | ||
+ | /* Made by MRENTHUSIASM */ | ||
+ | size(300); | ||
+ | pair A, B, C, D, A1, B1, C1, D1, P, Q, X, Y, Z, W; | ||
+ | A = (-250,6*sqrt(731)); | ||
+ | B = (250,6*sqrt(731)); | ||
+ | C = (325,-6*sqrt(731)); | ||
+ | D = (-325,-6*sqrt(731)); | ||
+ | A1 = bisectorpoint(B,A,D); | ||
+ | B1 = bisectorpoint(A,B,C); | ||
+ | C1 = bisectorpoint(B,C,D); | ||
+ | D1 = bisectorpoint(A,D,C); | ||
+ | P = intersectionpoint(A--300*(A1-A)+A,D--300*(D1-D)+D); | ||
+ | Q = intersectionpoint(B--300*(B1-B)+B,C--300*(C1-C)+C); | ||
+ | draw(anglemark(P,A,B,1000),red); | ||
+ | draw(anglemark(D,A,P,1000),red); | ||
+ | draw(anglemark(A,B,Q,1000),red); | ||
+ | draw(anglemark(Q,B,C,1000),red); | ||
+ | draw(anglemark(P,D,A,1000),red); | ||
+ | draw(anglemark(C,D,P,1000),red); | ||
+ | draw(anglemark(Q,C,D,1000),red); | ||
+ | draw(anglemark(B,C,Q,1000),red); | ||
+ | add(pathticks(anglemark(P,A,B,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(D,A,P,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(A,B,Q,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(Q,B,C,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(P,D,A,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | add(pathticks(anglemark(C,D,P,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | add(pathticks(anglemark(Q,C,D,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | add(pathticks(anglemark(B,C,Q,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | dot("$A$",A,1.5*dir(A),linewidth(4)); | ||
+ | dot("$B$",B,1.5*dir(B),linewidth(4)); | ||
+ | dot("$C$",C,1.5*dir(C),linewidth(4)); | ||
+ | dot("$D$",D,1.5*dir(D),linewidth(4)); | ||
+ | dot("$P$",P,1.5*NE,linewidth(4)); | ||
+ | dot("$Q$",Q,1.5*NW,linewidth(4)); | ||
+ | draw(A--B--C--D--cycle^^A--P--D^^B--Q--C^^P--Q); | ||
+ | X = (-121,6*sqrt(731)); | ||
+ | Y = (121,6*sqrt(731)); | ||
+ | Z = (-121,-6*sqrt(731)); | ||
+ | W = (121,-6*sqrt(731)); | ||
+ | draw(X--Z^^Y--W,dashed); | ||
+ | draw(rightanglemark(A,X,Z,500),red); | ||
+ | draw(rightanglemark(B,Y,W,500),red); | ||
+ | draw(rightanglemark(C,W,Y,500),red); | ||
+ | draw(rightanglemark(D,Z,X,500),red); | ||
+ | dot("$X$",X,1.5*N,linewidth(4)); | ||
+ | dot("$Y$",Y,1.5*N,linewidth(4)); | ||
+ | dot("$Z$",Z,1.5*S,linewidth(4)); | ||
+ | dot("$W$",W,1.5*S,linewidth(4)); | ||
+ | </asy> | ||
+ | |||
+ | Let <math>X</math> and <math>Y</math> be the feet of the altitudes from <math>P</math> and <math>Q</math>, respectively, to <math>AB</math>, and let <math>Z</math> and <math>W</math> be the feet of the altitudes from <math>P</math> and <math>Q</math>, respectively, to <math>CD</math>. Side <math>AB</math> is parallel to side <math>CD</math>, so <math>XYWZ</math> is a rectangle with width <math>PQ</math>. Furthermore, because <math>CD - AB = 650-500 = 150</math> and trapezoid <math>ABCD</math> is isosceles, <math>WC - YB = ZD - XA = 75</math>. | ||
+ | |||
+ | Also because <math>ABCD</math> is isosceles, <math>\angle ABC + \angle BCD</math> is half the total sum of angles in <math>ABCD</math>, or <math>180^{\circ}</math>. Since <math>BQ</math> and <math>CQ</math> bisect <math>\angle ABC</math> and <math>\angle BCD</math>, respectively, we have <math>\angle QBC + \angle QCB = 90^{\circ}</math>, so <math>\angle BQC = 90^{\circ}</math>. | ||
+ | |||
+ | Letting <math>BQ = 333k</math>, applying Pythagoras to <math>\triangle BQC</math> yields <math>QC = 333\sqrt{1-k^2}</math>. We then proceed using similar triangles: <math>\angle BYQ = \angle BQC = 90^{\circ}</math> and <math>\angle YBQ = \angle QBC</math>, so by AA similarity <math>YB = 333k^2</math>. Likewise, <math>\angle CWQ = \angle BQC = 90^{\circ}</math> and <math>\angle WCQ = \angle QCB</math>, so by AA similarity <math>WC = 333(1 - k^2)</math>. Thus <math>WC + YB = 333</math>. | ||
+ | |||
+ | Adding our two equations for <math>WC</math> and <math>YB</math> gives <math>2WC = 75 + 333 = 408</math>. Therefore, the answer is <math>PQ = ZW = CD - 2WC = 650 - 408 = \boxed{242}</math>. | ||
+ | |||
+ | ~Orange_Quail_9 | ||
+ | |||
+ | == Solution 5 == | ||
+ | |||
+ | This will be my first solution on AoPS. My apologies in advance for any errors. | ||
+ | |||
+ | Angle bisectors can be thought of as the locus of all points equidistant from the lines whose angle they bisect. It can thus be seen that <math>P</math> is equidistant from <math>AB, AD,</math> and <math>CD</math> and <math>Q</math> is equidistant from <math>AB, BC,</math> and <math>CD.</math> If we let the feet of the altitudes from <math>P</math> to <math>AB, AD,</math> and <math>CD</math> be called <math>E, F,</math> and <math>G</math> respectively, we can say that <math>PE = PF = PG.</math> Analogously, we let the feet of the altitudes from <math>Q</math> to <math>AB, BC,</math> and <math>CD</math> be <math>H, I,</math> and <math>J</math> respectively. Thus, <math>QH = QI = QJ.</math> Because <math>ABCD</math> is an isosceles trapezoid, we can say that all of the altitudes are equal to each other. | ||
+ | |||
+ | By SA as well as SS congruence for right triangles, we find that triangles <math>AEP, AFP, BHQ,</math> and <math>BIQ</math> are congruent. Similarly, <math>DFP, DGP, CJQ,</math> and <math>CIQ</math> by the same reasoning. Additionally, <math>EH = GJ = PQ</math> since <math>EHQP</math> and <math>GJQP</math> are congruent rectangles. | ||
+ | |||
+ | If we then let <math>x = AE = AF = BH = BI,</math> let <math>y = CI = CJ = DG = DF,</math> and let <math>z = EH = GJ = PQ,</math> we can create the following system of equations with the given side length information: | ||
+ | <cmath>\begin{align*} | ||
+ | 2x + z &= 500, \\ | ||
+ | 2y + z &= 650, \\ | ||
+ | x + y &= 333. | ||
+ | \end{align*}</cmath> | ||
+ | Adding the first two equations, subtracting by twice the second, and dividing by <math>2</math> yields <math>z = PQ = \boxed{242}.</math> | ||
+ | |||
+ | ~regular | ||
+ | |||
+ | == Solution 6 == | ||
+ | Extend line <math>PQ</math> to meet <math>AD</math> at <math>P'</math> and <math>BC</math> at <math>Q'</math>. The diagram looks like this: | ||
+ | <asy> | ||
+ | /* Made by MRENTHUSIASM */ | ||
+ | size(300); | ||
+ | pair A, B, C, D, A1, B1, C1, D1, P, Q, P1, Q1; | ||
+ | A = (-250,6*sqrt(731)); | ||
+ | B = (250,6*sqrt(731)); | ||
+ | C = (325,-6*sqrt(731)); | ||
+ | D = (-325,-6*sqrt(731)); | ||
+ | A1 = bisectorpoint(B,A,D); | ||
+ | B1 = bisectorpoint(A,B,C); | ||
+ | C1 = bisectorpoint(B,C,D); | ||
+ | D1 = bisectorpoint(A,D,C); | ||
+ | P = intersectionpoint(A--300*(A1-A)+A,D--300*(D1-D)+D); | ||
+ | Q = intersectionpoint(B--300*(B1-B)+B,C--300*(C1-C)+C); | ||
+ | P1 = intersectionpoint(A--D,P--(-300)*(Q-P)+P); | ||
+ | Q1 = intersectionpoint(B--C,Q--300*(Q-P)+Q); | ||
+ | draw(anglemark(P,A,B,1000),red); | ||
+ | draw(anglemark(D,A,P,1000),red); | ||
+ | draw(anglemark(A,B,Q,1000),red); | ||
+ | draw(anglemark(Q,B,C,1000),red); | ||
+ | draw(anglemark(P,D,A,1000),red); | ||
+ | draw(anglemark(C,D,P,1000),red); | ||
+ | draw(anglemark(Q,C,D,1000),red); | ||
+ | draw(anglemark(B,C,Q,1000),red); | ||
+ | add(pathticks(anglemark(P,A,B,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(D,A,P,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(A,B,Q,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(Q,B,C,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(P,D,A,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | add(pathticks(anglemark(C,D,P,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | add(pathticks(anglemark(Q,C,D,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | add(pathticks(anglemark(B,C,Q,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | dot("$A$",A,1.5*dir(A),linewidth(4)); | ||
+ | dot("$B$",B,1.5*dir(B),linewidth(4)); | ||
+ | dot("$C$",C,1.5*dir(C),linewidth(4)); | ||
+ | dot("$D$",D,1.5*dir(D),linewidth(4)); | ||
+ | dot("$P$",P,1.5*NE,linewidth(4)); | ||
+ | dot("$Q$",Q,1.5*NW,linewidth(4)); | ||
+ | dot("$P'$",P1,1.5*W,linewidth(4)); | ||
+ | dot("$Q'$",Q1,1.5*E,linewidth(4)); | ||
+ | draw(A--B--C--D--cycle^^A--P--D^^B--Q--C^^P--Q); | ||
+ | draw(P--P1^^Q--Q1,dashed); | ||
+ | </asy> | ||
+ | Since <math>\angle A + \angle D=\angle B + \angle C = 180^{\circ}</math>, it follows that <math>\angle P'AP+\angle P'DP = \angle Q'BQ + \angle Q'CQ = 90^{\circ}</math>. Thus, <math>\angle APD = \angle BQC = 90^{\circ}</math>, implying that <math>\triangle APD</math> and <math>\triangle BQC</math> are right triangles. Since <math>P'P</math> and <math>Q'Q</math> are medians, <math>P'P+Q'Q=\frac{333\times2}{2}=333</math>. Since <math>P'Q'=\frac{500+650}{2}=575</math>, we have <math>PQ+P'P+Q'Q=575</math>, or <math>PQ=575-333=\boxed{242}</math>. | ||
+ | |||
+ | ~sigma | ||
+ | |||
+ | == Solution 7 (Trigonometry) == | ||
+ | |||
+ | Let <math>PQ = x</math>. Note that since <math>AP</math> bisects <math>\angle{A}</math> and <math>DP</math> bisects <math>\angle{D}</math>, we have <cmath>\angle{APD} = 180^{\circ}-\tfrac12 \angle{A}-\tfrac12 \angle{D}=90^{\circ}.</cmath> Let <math>\angle{ADP}=\theta</math>. We have that <math>\angle{ADC} = 2\theta.</math> Now, drop an altitude from <math>A</math> to <math>CD</math> at <math>E</math>. Notice that <math>DE=\tfrac{650-500}{2}=75</math>. By the definition of cosine, we have <cmath>\cos{2\theta}=1-2\cos^2{\theta}=\tfrac{75}{333}=\tfrac{25}{111} \implies \cos{\theta}=\tfrac{2\sqrt{1887}}{111}.</cmath> Notice, however, that we can also apply this to <math>\triangle{APD}</math>; we have <cmath>\cos{\theta}=\tfrac{DP}{333} \implies DP=6\sqrt{1887}.</cmath> By the Pythagorean Theorem, we get <cmath>AP=\sqrt{333^2-(6\sqrt{1887})^2}=3\sqrt{4773}.</cmath> Then, drop an altitude from <math>P</math> to <math>AB</math> at <math>F</math>; if <math>AF=y</math>, then <math>PQ=x=500-2y</math>. Because <math>AP</math> is an angle bisector, we see that <math>\angle{BAP}=\angle{DAP}=90^{\circ}-\theta</math>. Again, by the definition of cosine, we have <cmath>\cos{(90^{\circ}-\theta)}=\sin{\theta}=\tfrac{\sqrt{4773}}{111}=\tfrac{y}{3\sqrt{4773}} \implies y=129.</cmath> Finally, <math>PQ=500-2y=\boxed{242}</math>. | ||
+ | |||
+ | ~pqr. | ||
+ | |||
+ | == Solution 8 (Pythagoras + Similar Triangles) == | ||
+ | <asy> | ||
+ | /* Made by MRENTHUSIASM */ | ||
+ | size(300); | ||
+ | pair A, B, C, D, A1, B1, C1, D1, P, Q, X, Y, Z, W; | ||
+ | A = (-250,6*sqrt(731)); | ||
+ | B = (250,6*sqrt(731)); | ||
+ | C = (325,-6*sqrt(731)); | ||
+ | D = (-325,-6*sqrt(731)); | ||
+ | A1 = bisectorpoint(B,A,D); | ||
+ | B1 = bisectorpoint(A,B,C); | ||
+ | C1 = bisectorpoint(B,C,D); | ||
+ | D1 = bisectorpoint(A,D,C); | ||
+ | P = intersectionpoint(A--300*(A1-A)+A,D--300*(D1-D)+D); | ||
+ | Q = intersectionpoint(B--300*(B1-B)+B,C--300*(C1-C)+C); | ||
+ | draw(anglemark(P,A,B,1000),red); | ||
+ | draw(anglemark(D,A,P,1000),red); | ||
+ | draw(anglemark(A,B,Q,1000),red); | ||
+ | draw(anglemark(Q,B,C,1000),red); | ||
+ | draw(anglemark(P,D,A,1000),red); | ||
+ | draw(anglemark(C,D,P,1000),red); | ||
+ | draw(anglemark(Q,C,D,1000),red); | ||
+ | draw(anglemark(B,C,Q,1000),red); | ||
+ | add(pathticks(anglemark(P,A,B,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(D,A,P,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(A,B,Q,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(Q,B,C,1000), n = 1, r = 0.15, s = 750, red)); | ||
+ | add(pathticks(anglemark(P,D,A,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | add(pathticks(anglemark(C,D,P,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | add(pathticks(anglemark(Q,C,D,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | add(pathticks(anglemark(B,C,Q,1000), n = 2, r = 0.12, spacing = 150, s = 750, red)); | ||
+ | dot("$A$",A,1.5*dir(A),linewidth(4)); | ||
+ | dot("$B$",B,1.5*dir(B),linewidth(4)); | ||
+ | dot("$C$",C,1.5*dir(C),linewidth(4)); | ||
+ | dot("$D$",D,1.5*dir(D),linewidth(4)); | ||
+ | dot("$P$",P,1.5*NE,linewidth(4)); | ||
+ | dot("$Q$",Q,1.5*NW,linewidth(4)); | ||
+ | draw(A--B--C--D--cycle^^A--P--D^^B--Q--C^^P--Q); | ||
+ | X = (-121,6*sqrt(731)); | ||
+ | Y = (121,6*sqrt(731)); | ||
+ | Z = (-121,-6*sqrt(731)); | ||
+ | W = (121,-6*sqrt(731)); | ||
+ | draw(X--Z^^Y--W,dashed); | ||
+ | draw(rightanglemark(A,X,Z,500),red); | ||
+ | draw(rightanglemark(B,Y,W,500),red); | ||
+ | draw(rightanglemark(C,W,Y,500),red); | ||
+ | draw(rightanglemark(D,Z,X,500),red); | ||
+ | dot("$X$",X,1.5*N,linewidth(4)); | ||
+ | dot("$Y$",Y,1.5*N,linewidth(4)); | ||
+ | dot("$Z$",Z,1.5*S,linewidth(4)); | ||
+ | dot("$W$",W,1.5*S,linewidth(4)); | ||
+ | </asy> | ||
+ | |||
+ | As in solution 4, <math>\angle APD = 90^{\circ}</math>. Set <math>k = AX</math> and <math>x = DP</math>. | ||
+ | |||
+ | We know that <math>DZ = AX + \frac{DC-AB}{2}</math>, so <math>DZ = k + \frac{650-500}{2} = k + 75</math>. | ||
+ | |||
+ | <math>\triangle DPZ \sim \triangle APD</math> by AA, so we have <math>\frac{PD}{AD} = \frac{ZD}{PD}</math>, resulting in | ||
+ | |||
+ | <cmath> | ||
+ | \frac{x}{333} = \frac{k+75}{x} \text{ (1)} | ||
+ | </cmath> | ||
+ | |||
+ | <math>\triangle APX \sim \triangle ADP</math> by AA, so we have <math>\frac{AP}{AD} = \frac{AX}{AP}</math>, resulting in | ||
+ | |||
+ | <cmath> | ||
+ | \frac{\sqrt{333^2-x^2}}{333} = \frac{k}{\sqrt{333^2-k^2}} \text{ (2)} | ||
+ | </cmath> | ||
+ | |||
+ | From <math>\text{(1)}</math>, we have <math>x^2 = 333k + 333(75) = 333k + 24975</math>. From <math>\text{(2)}</math>, we have <math>333^2 - x^2 = 333k</math>, or <math>x^2 = 333^2 - 333k</math>. Thus, <math>333k + 24975 = 333^2 - 333k</math>. Solving for <math>k</math> yields <math>k = 129</math>. | ||
+ | |||
+ | By symmetry, <math>YB = AX = 129</math>. Thus, <math>PQ = XY = AB - 2AX = 500 - 2(129) = \boxed{242}</math>. | ||
+ | |||
+ | ~ adam_zheng | ||
==Video Solution (Mathematical Dexterity)== | ==Video Solution (Mathematical Dexterity)== | ||
https://www.youtube.com/watch?v=fNAvxXnvAxs | https://www.youtube.com/watch?v=fNAvxXnvAxs | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://www.youtube.com/watch?v=h_LOT-rwt08 | ||
+ | |||
+ | ~Steven Chen (www.professorchenedu.com) | ||
+ | |||
+ | == Video Solution == | ||
+ | |||
+ | https://youtu.be/MJ_M-xvwHLk?t=545 | ||
+ | |||
+ | ~ThePuzzlr | ||
+ | |||
+ | ==Video Solution by MRENTHUSIASM (English & Chinese)== | ||
+ | https://www.youtube.com/watch?v=dqVVOSCWujo&ab_channel=MRENTHUSIASM | ||
+ | |||
+ | ~MRENTHUSIASM | ||
+ | |||
+ | ==Video Solution== | ||
+ | |||
+ | https://youtu.be/Q_S_VhiLRJE | ||
+ | |||
+ | ~AMC & AIME Training | ||
==See Also== | ==See Also== | ||
{{AIME box|year=2022|n=I|num-b=2|num-a=4}} | {{AIME box|year=2022|n=I|num-b=2|num-a=4}} | ||
+ | |||
+ | [[Category:Intermediate Geometry Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 19:59, 2 March 2024
Contents
- 1 Problem
- 2 Diagram
- 3 Solution 1
- 4 Solution 2
- 5 Solution 3
- 6 Solution 4
- 7 Solution 5
- 8 Solution 6
- 9 Solution 7 (Trigonometry)
- 10 Solution 8 (Pythagoras + Similar Triangles)
- 11 Video Solution (Mathematical Dexterity)
- 12 Video Solution
- 13 Video Solution
- 14 Video Solution by MRENTHUSIASM (English & Chinese)
- 15 Video Solution
- 16 See Also
Problem
In isosceles trapezoid , parallel bases and have lengths and , respectively, and . The angle bisectors of and meet at , and the angle bisectors of and meet at . Find .
Diagram
~MRENTHUSIASM ~ihatemath123
Solution 1
We have the following diagram:
Let and be the points where and extend to meet , and be the height of . As proven in Solution 2, triangles and are congruent right triangles. Therefore, . We can apply this logic to triangles and as well, giving us . Since , .
Additionally, we can see that is similar to and . We know that . So, we can say that the height of the triangle is while the height of the triangle is . After that, we can figure out the distance from to and the height of triangle .
Finally, since the ratio between the height of to the height of is and is ,
~Cytronical
Solution 2
Extend line to meet at and at . The diagram looks like this: Because the trapezoid is isosceles, by symmetry is parallel to and . Therefore, by interior angles and by the problem statement. Thus, is isosceles with . By symmetry, is also isosceles, and thus . Similarly, the same thing is happening on the right side of the trapezoid, and thus is the midline of the trapezoid. Then, .
Since and , we have . The length of the midline of a trapezoid is the average of their bases, so . Finally, .
~KingRavi
Solution 3
We have the following diagram: Extend lines and to meet line at points and , respectively, and extend lines and to meet at points and , respectively.
Claim: quadrilaterals and are rhombuses.
Proof: Since , . Therefore, triangles , , and are all right triangles. By SAA congruence, the first three triangles are congruent; by SAS congruence, is congruent to the other three. Therefore, , so is a rhombus. By symmetry, is also a rhombus.
Extend line to meet and at and , respectively. Because of rhombus properties, . Also, by rhombus properties, and are the midpoints of segments and , respectively; therefore, by trapezoid properties, . Finally, .
~ihatemath123
Solution 4
Let and be the feet of the altitudes from and , respectively, to , and let and be the feet of the altitudes from and , respectively, to . Side is parallel to side , so is a rectangle with width . Furthermore, because and trapezoid is isosceles, .
Also because is isosceles, is half the total sum of angles in , or . Since and bisect and , respectively, we have , so .
Letting , applying Pythagoras to yields . We then proceed using similar triangles: and , so by AA similarity . Likewise, and , so by AA similarity . Thus .
Adding our two equations for and gives . Therefore, the answer is .
~Orange_Quail_9
Solution 5
This will be my first solution on AoPS. My apologies in advance for any errors.
Angle bisectors can be thought of as the locus of all points equidistant from the lines whose angle they bisect. It can thus be seen that is equidistant from and and is equidistant from and If we let the feet of the altitudes from to and be called and respectively, we can say that Analogously, we let the feet of the altitudes from to and be and respectively. Thus, Because is an isosceles trapezoid, we can say that all of the altitudes are equal to each other.
By SA as well as SS congruence for right triangles, we find that triangles and are congruent. Similarly, and by the same reasoning. Additionally, since and are congruent rectangles.
If we then let let and let we can create the following system of equations with the given side length information: Adding the first two equations, subtracting by twice the second, and dividing by yields
~regular
Solution 6
Extend line to meet at and at . The diagram looks like this: Since , it follows that . Thus, , implying that and are right triangles. Since and are medians, . Since , we have , or .
~sigma
Solution 7 (Trigonometry)
Let . Note that since bisects and bisects , we have Let . We have that Now, drop an altitude from to at . Notice that . By the definition of cosine, we have Notice, however, that we can also apply this to ; we have By the Pythagorean Theorem, we get Then, drop an altitude from to at ; if , then . Because is an angle bisector, we see that . Again, by the definition of cosine, we have Finally, .
~pqr.
Solution 8 (Pythagoras + Similar Triangles)
As in solution 4, . Set and .
We know that , so .
by AA, so we have , resulting in
by AA, so we have , resulting in
From , we have . From , we have , or . Thus, . Solving for yields .
By symmetry, . Thus, .
~ adam_zheng
Video Solution (Mathematical Dexterity)
https://www.youtube.com/watch?v=fNAvxXnvAxs
Video Solution
https://www.youtube.com/watch?v=h_LOT-rwt08
~Steven Chen (www.professorchenedu.com)
Video Solution
https://youtu.be/MJ_M-xvwHLk?t=545
~ThePuzzlr
Video Solution by MRENTHUSIASM (English & Chinese)
https://www.youtube.com/watch?v=dqVVOSCWujo&ab_channel=MRENTHUSIASM
~MRENTHUSIASM
Video Solution
~AMC & AIME Training
See Also
2022 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.