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Problem 913

AMC 12 late, AIME early

Geometry Difficulty 4.7 Prove it Harvard-MIT Math Tournament · United States

A point on a circle inscribed in a square is $1$ and $2$ units from the two closest sides of the square. Find the area of the square.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Solution:

Call the point in question $A$ , the center of the circle $O$ , and its radius $r$ . Consider a right triangle $BOA$ with hypotenuse $OA$ : $OA$ has length $r$ , and $BO$ and $BA$ have lengths $r-1$ and $r-2$ . By the Pythagorean theorem,
$(r-1)^2 + (r-2)^2 = r^2$
which gives
$r^2 - 6r + 5 = 0$
so $r = 5$ since $r > 4$ . The area of the square is $(2r)^2 = 100$ .

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.