HMMT 二月 2022 · 几何 · 第 5 题

HMMT February 2022 — Geometry — Problem 5

Let triangle ABC be such that AB = AC = 22 and BC = 11. Point D is chosen in the interior of the ◦ 2 2 triangle such that AD = 19 and ∠ ABD + ∠ ACD = 90 . The value of BD + CD can be expressed a as , where a and b are relatively prime positive integers. Compute 100 a + b . b

解析

Let triangle ABC be such that AB = AC = 22 and BC = 11. Point D is chosen in the interior of the ◦ 2 2 triangle such that AD = 19 and ∠ ABD + ∠ ACD = 90 . The value of BD + CD can be expressed a as , where a and b are relatively prime positive integers. Compute 100 a + b . b Proposed by: Akash Das Answer: 36104 ′ Solution: Rotate triangle ABD about A so that B coincides with C . Let D map to D under this. ′ ′ Note that CDD is a right triangle with right angle at C . Also, note that ADD is similar to ABC . ′ AD 19 Thus, we have DD = = . Finally, note that 2 2 361 2 2 ′ 2 2 ′ 2 BD + CD = CD + CD = DD = . 4