2018 AMC 12A Problem 8

Attempt Problem 8 of the 2018 AMC 12A below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2018 AMC 12A solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

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

The base

DE

of triangle

ADE

spans

4

small-triangle bases

Area scales as the square of length, so

[ADE] = 4^2 \cdot 1;

subtract this from

[ABC] = 40

The base

DE

\triangle ADE

4

times the base of a smallest triangle, so by the square scaling of similar areas,

[ADE] = 4^2 \cdot 1 = 16.

The trapezoid

DBCE

is what remains of

\triangle ABC,

so its area is

40 - 16 = 24.

Thus, the correct answer is E.

Problem 8 in other years