Triangle ABC is similar to triangle DEF, where the ratio of the length of side AB to the l… — SAT

Triangle $ABC$ is similar to triangle $DEF$ , where the ratio of the length of side $AB$ to the length of side $DE$ is $3$ to $5$ . If the area of triangle $ABC$ is $18$ , what is the area of triangle $DEF$ ?

Answer: 50

Answer

The area of triangle

DEF

50

Since triangle

ABC

is similar to triangle

DEF

, the ratio of their areas is the square of the ratio of their corresponding side lengths. The ratio of side

AB

to side

DE

3/5

, so the ratio of the area of triangle

ABC

to the area of triangle

DEF

(3/5)^2 = 9/25

. Given that the area of triangle

ABC

18

, we can set up the proportion

18 / x = 9 / 25

, where

x

represents the area of triangle

DEF

. Solving for

x

gives

x = 18 \times (25/9) = 2 \times 25 = 50

Step-by-Step Solution

Determine the ratio of the areas of the two similar triangles using their side length ratio.

The ratio of the area of triangle

ABC

to the area of triangle

DEF

(3/5)^2 = 9/25

For similar triangles, the ratio of their areas is equal to the square of the ratio of their corresponding side lengths.

Set up a proportion to solve for the unknown area of triangle

DEF

\frac{18}{x} = \frac{9}{25}

, where

x

is the area of triangle

DEF

We equate the ratio of the actual areas to the theoretical ratio of areas derived from the side lengths.

Solve the proportion for

x

x = 18 \times \frac{25}{9} = 2 \times 25 = 50

Multiplying both sides by the reciprocal isolates the variable and yields the area.

Key Concept

The ratio of the areas of two similar triangles is the square of the ratio of their corresponding side lengths.

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