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Problem 547

Geometry Difficulty 2.5 Find the answer CEMC Cayley · Canada · 2020

In the diagram, the circle has centre $O$ and square $OPQR$ has vertex $Q$ on the circle.

If the area of the circle is 72 $\pi$ , the area of the square is

$38$
$48$
$25$
$12$
$36$

Your answer

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

Official solution

Let $s$ be the side length of the square. Therefore, $OR=RQ=s$ .

Let $r$ be the radius of the circle. Therefore, $OQ=r$ since $O$ is the centre of the circle and $Q$ is on the circumference of the circle.

Since the square has a right-angle at each of its vertices, then $\triangle ORQ$ is right-angled at $R$ .

By the Pythagorean Theorem, $OR^2 + RQ^2 = OQ^2$ and so $s^2 + s^2 = r^2$ or $2s^2 = r^2$ .

In terms of $r$ , the area of the circle is $\pi r^2$ .

Since we are given that the area of the circle is $72\pi$ , then $\pi r^2 = 72\pi$ or $r^2 = 72$ .

Since $2s^2 = r^2 = 72$ , then $s^2 = 36$ .

In terms of $s$ , the area of the square is $s^2$ , so the area of the square is 36.

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