2013 AMC 10A Problem 15

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

Express the same doubled area using each side and its altitude

Use the average-altitude condition to relate the unknown third side to a known altitude

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.

Problem 15 in other years