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 |