2018 AMC 10A Problem 9

Question

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?

16

18

20

22

24

Area scales as the square of side length for similar triangles

Find the area of the top triangle and subtract from

40

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

is

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

is

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

The area of the trapezoid is then

40 - 16 = 24.

Thus, E is the correct answer.

Problem 9 in other years