HMMT 十一月 2024 · 团队赛 · 第 9 题

HMMT November 2024 — Team Round — Problem 9

[55] Let P be a point inside isosceles trapezoid ABCD with AB ∥ CD such that ◦ ∠ P AD = ∠ P DA = 90 − ∠ BP C. If P A = 14 , AB = 18 , and CD = 28 , compute the area of ABCD .

解析

[55] Let P be a point inside isosceles trapezoid ABCD with AB ∥ CD such that ◦ ∠ P AD = ∠ P DA = 90 − ∠ BP C. If P A = 14, AB = 18, and CD = 28, compute the area of ABCD . Proposed by: Karthik Venkata Vedula, Pitchayut Saengrungkongka √ Answer: 345 3 Solution: A B P Q D C ◦ Let Q be the circumcenter of △ BP C . Thus, ∠ QBC = ∠ QCB = 90 − ∠ BP C , and so △ P AD and △ QBC are congruent. This means that P Q , AB , and CD share the common perpendicular bisector. We now find the area by determining the altitude. Note that we have all four side lengths of isosceles trapezoids P QAB and P QCD . Thus, one can compute their altitudes via Pythagorean theorem: q √ 2 18 − 14 2 distance( P, AB ) = 14 − = 8 3 2 q √ 2 28 − 14 2 distance( P, CD ) = 14 − = 7 3 , 2 √ √ √ 1 so the altitude of trapezoid ABCD is 15 3, so the final answer is · (14 + 18) · 15 3 = 345 3 . 2