Difference between revisions of "2022 AIME I Problems/Problem 3"
Line 36: | Line 36: | ||
</asy> | </asy> | ||
− | == Solution 1== | + | == Solution 1 == |
+ | |||
+ | 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> | ||
+ | unitsize(0.016cm); | ||
+ | pair A = (-250,324.4); | ||
+ | pair B = (250, 324.4); | ||
+ | pair C = (325, 0); | ||
+ | pair D = (-325, 0); | ||
+ | draw(A--B--C--D--cycle); | ||
+ | |||
+ | |||
+ | pair P = (-121, 162.2); | ||
+ | pair P' =(-412.5,162.2); | ||
+ | pair Q = (121, 162.2); | ||
+ | pair Q' = (412.5,162.5); | ||
+ | dot(P); | ||
+ | dot(Q); | ||
+ | |||
+ | draw(P'--Q'); | ||
+ | |||
+ | label("$A$", A, N); | ||
+ | label("$B$", B, N); | ||
+ | |||
+ | label("$C$", C, S); | ||
+ | label("$D$", D, S); | ||
+ | |||
+ | label("$P$", P, N); | ||
+ | label("$Q$", Q, N); | ||
+ | |||
+ | label("$P'$", P', W); | ||
+ | label($"Q'$", Q', E); | ||
+ | </asy> | ||
+ | |||
+ | |||
+ | == Solution 2== | ||
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. |
Revision as of 19:10, 17 February 2022
Contents
[hide]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
Solution 1
Extend line to meet at and at . The diagram looks like this:
unitsize(0.016cm); pair A = (-250,324.4); pair B = (250, 324.4); pair C = (325, 0); pair D = (-325, 0); draw(A--B--C--D--cycle); pair P = (-121, 162.2); pair P' =(-412.5,162.2); pair Q = (121, 162.2); pair Q' = (412.5,162.5); dot(P); dot(Q); draw(P'--Q'); label("$A$", A, N); label("$B$", B, N); label("$C$", C, S); label("$D$", D, S); label("$P$", P, N); label("$Q$", Q, N); label("$P'$", P', W); label($"Q'$", Q', E); (Error making remote request. Unknown error_msg)
Solution 2
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
Video Solution (Mathematical Dexterity)
https://www.youtube.com/watch?v=fNAvxXnvAxs
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.