2017 UNM-PNM Statewide High School Mathematics Contest II Problems/Problem 8
Problem
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.
Solution
Method 1:
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 .
See also
2017 UNM-PNM Contest II (Problems • Answer Key • Resources) | ||
Preceded by Problem 7 |
Followed by Problem 9 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
All UNM-PNM Problems and Solutions |