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Gara Nazionale di Matematica

Geometry Difficulty 4.7 AIME Prove it Italy

Problem:

Given a rectangular sheet with sides $a$ and $b$ , with $a > b$ , determine the area of the triangle that results from the overlap of the two flaps obtained by folding the sheet along a diagonal (the triangle colored gray in the figure).

Solution

Solution:

The triangle $ABC$ is isosceles (the angles $\widehat{BAC}$ and $\widehat{ABC}$ correspond to two alternate interior angles in the rectangle divided by a diagonal) and its area is $S = CB \cdot AD / 2$ ( $AD$ is the height relative to $CB$ ).

Setting $DB = a$ , $AD = b$ , $CB = AC = x$ and $DC = a - x$ , the Pythagorean theorem applied to the right triangle $ACD$ gives
$b^2 + (a - x)^2 = x^2$
from which
$x = \frac{a^2 + b^2}{2a}$
and finally
$S = \frac{b\left(a^2 + b^2\right)}{4a}$

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.