2018 AMC 10A Problem 9

Problem 9 of 25EasierGeometry

All of the triangles in the diagram below are similar to isosceles triangle

ABC,

in which

AB=AC.

Each of the

7

smallest triangles has area

1,

and

\triangle ABC

has area

40.

What is the area of trapezoid

DBCE?

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Solution

We know that the side length of the smaller triangles is

\sqrt{\frac{1}{40}}

times the length of the larger triangle from similar triangles. Then the side length of

\triangle ADE

4\sqrt{\frac{1}{40}}

times the length of the side length of the larger triangle. This makes the ratio of the areas

\left(4\sqrt{\dfrac{1}{40}}\right)^2 = 16 \cdot \dfrac{1}{40} = \dfrac{2}{5}.

Therefore, the area of

\triangle ADE

\frac{2}{5} \cdot 40 = 16.

The area of the trapezoid is then

40 - 16 = 24.

Thus, E is the correct answer.