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Harvard-MIT Mathematics Tournament

Geometry Difficulty 4.7 AIME Prove it United States

Problem:

Suppose $ABC$ is a scalene right triangle, and $P$ is the point on hypotenuse $\overline{AC}$ such that $\angle ABP = 45^{\circ}$ . Given that $AP = 1$ and $CP = 2$ , compute the area of $ABC$ .

Solution

Solution:

Notice that $\overline{BP}$ bisects the right angle at $B$ . Thus, we write $AB = 2x$ , $BC = x$ . By the Pythagorean theorem, $5x^{2} = 9$ , from which the area $\frac{1}{2}(x)(2x) = x^{2} = \frac{9}{5}$ .

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