For triangle ABC, points D and E are on sides AB and AC, respectively, such that segment D… — SAT

For triangle $ABC$ , points $D$ and $E$ are on sides $AB$ and $AC$ , respectively, such that segment $DE$ is parallel to segment $BC$ . If the length of segment $AD$ is $x + 5$ , the length of segment $DB$ is $6$ , the length of segment $AE$ is $x - 2$ , and the length of segment $EC$ is $3$ . What is the length of segment $AE$ ?

A
2
B
5
7Answer
D
9

Answer

According to the Triangle Proportionality Theorem, since segment

DE

is parallel to segment

BC

, the sides of triangle

ABC

are divided proportionally:

\frac{AD}{DB} = \frac{AE}{EC}

. Substituting the given expressions, we get

\frac{x+5}{6} = \frac{x-2}{3}

. Multiplying both sides of the equation by 6 gives

x+5 = 2(x-2)

. Distributing the 2 to both terms inside the parentheses yields

x+5 = 2x-4

. Subtracting

x

from both sides gives

5 = x-4

, and adding 4 to both sides gives

x = 9

. The question asks for the length of segment

AE

, which is defined as

x-2

. Substituting

x = 9

into this expression gives

9 - 2 = 7

Step-by-Step Solution

Identify the relationship between the triangles and set up the proportion.

Since

DE \parallel BC

, triangle

ADE

is similar to triangle

ABC

(

\triangle ADE \sim \triangle ABC

). By the Triangle Proportionality Theorem, this gives the proportion

\frac{AD}{DB} = \frac{AE}{EC}

A line parallel to one side of a triangle divides the other two sides proportionally.

Substitute the given side lengths into the proportion.

\frac{x+5}{6} = \frac{x-2}{3}

Using the algebraic expressions provided in the problem statement.

Solve the equation for

x

Multiplying both sides by 6 gives

x+5 = 2(x-2)

. Distributing the 2 gives

x+5 = 2x-4

. Subtracting

x

and adding 4 to both sides yields

x = 9

To determine the value of the variable

x

Calculate the length of segment

AE

AE = x - 2 = 9 - 2 = 7

The question asks for the length of segment

AE

, so we substitute

x = 9

back into the expression for

AE

Key Concept

Using the Triangle Proportionality Theorem and properties of similar triangles to solve for unknown side lengths using algebraic expressions.

Estimated Time:1m 30s

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