In right triangle ABC, the measure of angle C is 90^\circ. Point D lies on side AC and poi… — SAT

In right triangle $ABC$ , the measure of angle $C$ is $90^\circ$ . Point $D$ lies on side $AC$ and point $E$ lies on hypotenuse $AB$ such that segment $DE$ is perpendicular to $AB$ . The length of segment $AE$ is $x$ , the length of segment $AD$ is $x + 1$ , the length of segment $CD$ is $2$ , and the length of segment $BE$ is $5$ . What is the value of $x$ ?

Answer: 3

Answer

By identifying that triangle

AED

is similar to triangle

ACB

(due to shared angle

A

and right angles at

E

and

C

), we can set up the proportion

\frac{AE}{AC} = \frac{AD}{AB}

. Expressing the segments as

AC = x + 3

and

AB = x + 5

leads to the equation

\frac{x}{x + 3} = \frac{x + 1}{x + 5}

. Solving this equation yields

x = 3

Step-by-Step Solution

Establish similarity between triangles

AED

and

ACB

\triangle AED \sim \triangle ACB

Both triangles share the acute angle

A

(

\angle EAD = \angle CAB

) and both have a right angle (

\angle AED = \angle ACB = 90^\circ

), satisfying the Angle-Angle (AA) similarity criterion.

Write the proportion of corresponding sides.

\frac{AE}{AC} = \frac{AD}{AB}

In similar triangles, the ratios of the lengths of corresponding sides are equal.

Express the total side lengths of triangle

ABC

using segment addition.

AC = x + 3

and

AB = x + 5

Since

D

is on

AC

AC = AD + CD = (x + 1) + 2 = x + 3

. Since

E

is on

AB

AB = AE + BE = x + 5

Substitute the algebraic expressions into the side ratio proportion.

\frac{x}{x + 3} = \frac{x + 1}{x + 5}

Substituting

AE = x

AD = x + 1

AC = x + 3

, and

AB = x + 5

into the similarity proportion.

Solve the proportion for

x

x = 3

Cross-multiplying gives

x(x + 5) = (x + 1)(x + 3) \implies x^2 + 5x = x^2 + 4x + 3

. Subtracting

x^2

and

4x

from both sides results in

x = 3

Key Concept

Identifying similar right triangles via the AA similarity criterion and solving resulting algebraic proportions.

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic