2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 8
Let be a right triangle with right angle at . Suppose and and is the diameter of a semicircle, where lies on and the semicircle is tangent to side . Find the radius of the semicircle.
We can compute the area in two ways: or . Setting the two areas equal we obtain .
Method 2: Place the point C on the origin of the xy plane, at and at . Point lies at cartesian coordinate . The line AB has formula . The vector has coordinates since it has length in the unit direction which is orthogonal to the line AB.
Then point Y has coordinates and lies on the line . Substituting for these equations gives .
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