2008 iTest Problems/Problem 52
Problem
A triangle has sides of length , and . A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the two vertices not on that side of the square touch the other two sides of the triangle. If and are relatively prime positive integers such that is the length of a side of the square, ﬁnd the value of .
Solution
Let be the altitude to the longest side, and let be the length of the square. Using similar triangles to write a proportion,
Note that , so the sides are part of a right triangle. That means the altitude to the hypotenuse is . Substitute that to get The prime factorization of is . Since both of the numbers are not a factor of , we can confirm that and are relatively prime, so .
See Also
2008 iTest (Problems) | ||
Preceded by: Problem 51 |
Followed by: Problem 53 | |
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