Question

Two sides of a triangle have lengths $10$ and $15.$ The length of the altitude to the third side is the average of the lengths of the altitudes to the two given sides. How long is the third side?

$6$

$8$

$9$

$12$

$18$

Answer: D

Concepts:triangle area altitude ratio and proportion

Difficulty rating: 1660

Solution:

Let $h_1$ be the length of the altitude to the side of length $10$ and similarly define $h_2$ for the other given side.

We have that $10h_1 = 15h_2$ $h_1 = \dfrac{3}{2}h_2.$

The third altitude is the average of the other two, which makes its length $\dfrac{h_2 + \frac{3}{2}h_2}{2} = \dfrac{5}{4}h_2.$

Let the third side have length $x.$ Then $\dfrac{5}{4}h_2x = 15h_2$ $x = 12.$

Thus, D is the correct answer.

2013 AMC 10A Problem 15

Answer: D

Difficulty rating: 1660

Solution:

Problem 15 in Other Years