HMMT 十一月 2023 · 冲刺赛 · 第 25 题

HMMT November 2023 — Guts Round — Problem 25

[13] A right triangle and a circle are drawn such that the circle is tangent to the legs of the right triangle. The circle cuts the hypotenuse into three segments of lengths 1, 24, and 3, and the segment of length 24 is a chord of the circle. Compute the area of the triangle.

解析

[13] A right triangle and a circle are drawn such that the circle is tangent to the legs of the right triangle. The circle cuts the hypotenuse into three segments of lengths 1, 24, and 3, and the segment of length 24 is a chord of the circle. Compute the area of the triangle. Proposed by: Karthik Venkata Vedula Answer: 192 Solution 1: C F E G B A D Let the triangle be △ ABC , with AC as the hypotenuse, and let D , E , F , G be on sides AB , BC , AC , AC , respectively, such that they all lie on the circle. We have AG = 1, GF = 24, and F C = 3. By power of a point, we have √ p AD = AG · AF = 1(1 + 24) = 5 √ p CE = CF · CG = 3(3 + 24) = 9 . Now, let BD = BE = x . By the Pythagorean Theorem, we get that 2 2 2 ( x + 5) + ( x + 9) = 28 2 2 2 2 2 ( x + 5) + ( x + 9) − ( x + 9) − ( x + 5) = 28 − 4 2( x + 5)( x + 9) = 768 ( x + 5)( x + 9) = 384 . 1 1 The area of △ ABC is ( x + 5)( x + 9) = · 384 = 192. 2 2