In triangle ABC, point D lies on side AB such that AD : DB = 3 : 1, and point E lies on si… — GRE General Test

In triangle $ABC$ , point $D$ lies on side $AB$ such that $AD : DB = 3 : 1$ , and point $E$ lies on side $AC$ such that segment $DE$ is parallel to side $BC$ . If the area of triangle $ABC$ is $64$ , what is the area of triangle $ADE$ ?

A
16
B
27
36Answer
D
48
E
52

Answer

Because segment

DE

is parallel to

BC

, triangle

ADE

is similar to triangle

ABC

. The ratio of side

AD

to side

AB

3 : (3 + 1) = 3 : 4

. For similar triangles, the ratio of their areas equals the square of the ratio of their corresponding side lengths, which is

(3/4)^2 = 9/16

. Multiplying the area of triangle

ABC

(

64

) by

9/16

yields an area of

36

for triangle

ADE

Step-by-Step Solution

Determine the scale factor between similar triangles

ADE

and

ABC

Since

DE \parallel BC

, triangle

ADE

is similar to triangle

ABC

. The side length ratio is

AD / AB = AD / (AD + DB) = 3 / (3 + 1) = 3/4

Parallel lines create corresponding equal angles, making triangles

ADE

and

ABC

similar.

Calculate the ratio of the areas of the similar triangles.

The area ratio is the square of the side scale factor:

(3/4)^2 = 9/16

The ratio of the areas of two similar figures is equal to the square of their scale factor.

Compute the area of triangle

ADE

\text{Area}(\triangle ADE) = \frac{9}{16} \times 64 = 36.

Multiply the total area of triangle

ABC

by the area ratio

9/16

Key Concept

Area Ratio of Similar Triangles

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