Maths Olympiad Prep

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Geometry Difficulty 4.7 AIME Prove it Italy

Problem:

Given a rectangular sheet with sides aa and bb, with a>ba > 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).

Figure 1

Solution

Solution:

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

Setting DB=aDB = a, AD=bAD = b, CB=AC=xCB = AC = x and DC=a−xDC = a - x, the Pythagorean theorem applied to the right triangle ACDACD gives
b2+(a−x)2=x2 b^2 + (a - x)^2 = x^2
from which
x=a2+b22a x = \frac{a^2 + b^2}{2a}
and finally
S=b(a2+b2)4a S = \frac{b\left(a^2 + b^2\right)}{4a}

Figure 2

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