HMMT 二月 2006 · 几何 · 第 2 题
HMMT February 2006 — Geometry — Problem 2
题目详情
- Suppose ABC is a scalene right triangle, and P is the point on hypotenuse AC such ◦ that ∠ ABP = 45 . Given that AP = 1 and CP = 2, compute the area of ABC .
解析
- Suppose ABC is a scalene right triangle, and P is the point on hypotenuse AC such ◦ that ∠ ABP = 45 . Given that AP = 1 and CP = 2, compute the area of ABC . 9 Answer: 5 Solution: Notice that BP bisects the right angle at B . Thus, we write AB = 2 x , 1 2 2 BC = x . By the Pythagorean theorem, 5 x = 9, from which the area ( x )(2 x ) = x = 2 9 . 5