Track / Stage 4 / 290 of 340 #550 of 1964

Problem 550

AMC 12 late, AIME early

Geometry Difficulty 5.0 Find the answer

$17 \cdot 177$ Let the two legs of a right triangle be $a, b$ , the hypotenuse be $c$ , and the altitude to the hypotenuse be $h$ . Then, the shape of the triangle formed by $c+h, a+b, h$ as sides is

Pick one

Official solution

[Solution] According to the Pythagorean theorem, we have $a^{2}+b^{2}=c^{2}$ , and the area of the triangle is
$S=\frac{1}{2} a b=\frac{1}{2} c h .$

Then
$(c+h)^{2}=c^{2}+2 c h+h^{2} \text {. }$

Also, $(a+b)^{2}=a^{2}+2 a b+b^{2}=a^{2}+2 c h+c^{2}-a^{2}=c^{2}+2 c h$ .
$\therefore$
$(c+h)^{2}=(a+b)^{2}+h^{2}$ .

Thus, the newly formed triangle is a right triangle.
Therefore, the answer is $(A)$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.