1983 AIME Problems/Problem 4

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. [img]http://www.artofproblemsolving.com/Forum/album_pic.php?pic_id=790&sid=cfd5dae222dd7b8944719b56de7b8bf7[/img]

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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 $1^2 + 5^2 = 26$ .

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1983 AIME Problems/Problem 4

Problem

Solution

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