The sides of a triangle have length x, x + 4, and 20. If the length of the longest side is 20, which value of x would make the triangle acute?

8

10

12

14

From the triangle inequality, we get the relation x + (x+4) > 20, or x > 8. Also, since 20 is the longest side, we must have x + 4 < 20, or x < 16. Hence from the initial conditions, we must have 8 < x < 16.

Intersect the preceding interval with the interval x > 12.

Therefore, for the triangle to be acute, we must have 12 < x < 16.