2006 iTest Problems/Problem 24
Problem
Points and are chosen on side of triangle such that is between and and , . If , the area of can be expressed as , where and are relatively prime positive integers and is a positive integer not divisible by the square of any prime. Compute .
Solution
Note that if a circle has center and radius , then is the circle's diameter because . Because is a right angle, is also on the circle with center , so . By the Pythagorean Theorem, .
The area of is . Let be the length of the altitude from to , so . Thus, .
Since is on line , the area of is equal to , so .
See Also
2006 iTest (Problems) | ||
Preceded by: Problem 23 |
Followed by: Problem 25 | |
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