2021 AMC 12B Fall Problem 8

Question

The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle’s height to the base. What is the measure, in degrees, of the vertex angle of this triangle?

105

120

135

150

165

Answer: D

Concepts:isosceles triangle triangle area trigonometry

Difficulty rating: 1530

Small hint:

Write the area two ways using the legs and the vertex angle

\theta

Big hint:

The area is both

\tfrac12 b h

and

\tfrac12 s^2 \sin\theta

Solution:

Let the congruent sides have length

s,

the base be

b,

and the height to the base be

h.

The given condition is

s^2 = 2bh.

The area equals

\tfrac12 bh

and also

\tfrac12 s^2 \sin\theta,

where

\theta

is the vertex angle. So

bh = s^2 \sin\theta.

Substituting

s^2 = 2bh

gives

bh = 2bh\sin\theta,

so

\sin\theta = \tfrac12.

Since the triangle is obtuse,

\theta = 150^\circ.

Thus, the correct answer is D.