Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.7 AIME Prove it United States

Problem:

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.

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 $(2r)^2 = 100$ .

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