2008 iTest Problems/Problem 6

Problem

Let $L$ be the length of the altitude to the hypotenuse of a right triangle with legs 5 and 12. Find the least integer greater than $L$ .

Solution

By the Pythagorean Theorem, the hypotenuse will be $\sqrt{5^2 + 12^2} = \sqrt{25+144} = \sqrt{169} = 13$ . The area of this triangle can be expressed in 2 ways: half the product of the legs, and half the product of the hypotenuse and its altitude. We can easily find the area using the first method: $\frac{1}{2}\cdot 5 \cdot 12 = 5\cdot 6 = 30$ . Therefore, we have $\frac{1}{2}L \cdot 13 = 30$ . Multiplying both sides by $2$ and dividing both sides by $13$ , we get $L = \frac{60}{13}$ . The least integer greater than $\frac{60}{13}$ is $\boxed{5}$

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2008 iTest Problems/Problem 6

Problem

Solution