How many triangles have area 10 and vertices at (-5,0), (5,0), and (5cosθ,5sinθ) for some angle θ?

1998 AMC 12 Problem 19

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19.

How many triangles have area

10

and vertices at

(-5,0),

(5,0),

and

(5\cos\theta,5\sin\theta)

for some angle

\theta?

0

2

4

6

8

The fixed base has length

10

, and the third vertex has height

|5\sin\theta|

Count the distinct points on the circle satisfying the resulting sine equation

The area is

\frac12(10)|5\sin\theta|=25|\sin\theta|.

Thus

|\sin\theta|=\frac{2}{5}.

There are four corresponding points on the circle, and each gives a distinct triangle. Hence the answer is

4,

making C correct.