1983 AIME Problems/Problem 4

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Problem

A machine shop cutting tool is in the shape of a notched circle, as shown. The radius of the circle is 50 cm, the length of $AB$ is 6 cm, and that of $BC$ is 2 cm. The angle $ABC$ is a right angle. Find the square of the distance (in centimeters) from $B$ to the center of the circle. AIME 83 -4.JPG

Solution

Because we are given a right angle, we look for ways to apply the Pythagorean Theorem. Extend a perpendicular from $O$ to $AB$ and label it $D$. Additionally, extend a perpendicular from $O$ to the line $BC$, and label it $E$. Let $OE=x$ and $OD=y$. We're trying to find $x^2+y^2$.

Applying the Pythagorean Theorem, $OA^2 = OD^2 + AD^2$, and $OC^2 = EC^2 + EO^2$.

Thus, $(\sqrt{50})^2 = y^2 + (6-x)^2$, and $(\sqrt{50})^2 = x^2 + (y+2)^2$. We solve this system to get $x = 1$ and $y = 5$, resulting in an answer of $1^2 + 5^2 = 26$.


See also