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Problem 538

AMC 10/12, early questions

Geometry Difficulty 3.3 Find the answer CEMC Fermat · Canada · 2020

The figure consists of five squares and two right-angled triangles.

Hide/Reveal Alternative Format for the Five Squares and Two Right-Angled Triangles

Together, these squares and triangles form a crooked tower, where each shape that is joined to another shape shares a common side as follows: a large square with area 32 is the base of the tower. Stacked on top of the square is a right-angled triangle whose right angle lines up with the left side of the square. A second large square (unshaded and unmarked) is joined to the hypotenuse of the triangle, and thus leans to the right.
A third smaller square with area 8 is joined to the left side of the triangle. Now a second right-angled triangle is stacked on top of the leaning large square so that its right angle lines up with the right side of the leaning square. A fourth smaller square with area 5 is joined to the right side of the triangle. Finally, a large shaded square is stacked on the hypotenuse of the right-angled triangle.

The areas of three of the squares are 5, 8 and 32, as shown. What is the area of the shaded square?

$35$
$45$
$29$
$19$
$75$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Official solution

The Pythagorean Theorem tells us that if a right-angled triangle has sides of length $a$ , $b$ and $c$ , with $c$ the hypotenuse, then $a^2+b^2=c^2$ .
Since the area of a square of side length $a$ is $a^2$ , the Pythagorean Theorem can be re-phrased to say that the sum of the areas of the squares that can be drawn on the two shorter sides equals the area of the square that can be drawn on the hypotenuse. (In the figure below, this says that $x+y=z$ where $x$ , $y$ and $z$ are the areas of the squares, as shown.)

[[IMAGE0]]

In the given diagram, this means that the area of the unmarked, unshaded square is $8+32=40$ .

[[IMAGE1]]

This means that the area of the shaded square is $40+5=45$ .

Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.