In the figure below, AD is the angle bisector of \angle BAC in triangle ABC. The length of… — SAT

In the figure below, $AD$ is the angle bisector of $\angle BAC$ in triangle $ABC$ . The length of segment $AB$ is $2x$ , the length of segment $AC$ is $3x-3$ , the length of segment $BD$ is $6$ , and the length of segment $CD$ is $8$ . What is the value of $x$ ?

A
2
B
3
C
6
9Answer

Answer

The value of

x

is 9.

According to the Angle Bisector Theorem, the bisector of an angle in a triangle divides the opposite side into segments that are proportional to the adjacent sides. This gives the proportion

\frac{AB}{AC} = \frac{BD}{CD}

. Substituting the given values, we get

\frac{2x}{3x-3} = \frac{6}{8}

. Simplifying the right side to

\frac{3}{4}

and cross-multiplying yields

2x(4) = 3(3x-3)

, which simplifies to

8x = 9x - 9

. Solving for

x

gives

x = 9

Step-by-Step Solution

Apply the Angle Bisector Theorem, which states that an angle bisector in a triangle divides the opposite side into two segments that are proportional to the other two sides.

\frac{AB}{AC} = \frac{BD}{CD}

Because

AD

is the angle bisector of

\angle BAC

, it splits

BC

D

proportionally to the adjacent sides

AB

and

AC

Substitute the given side lengths into the proportion.

\frac{2x}{3x-3} = \frac{6}{8}

To set up the algebraic equation for

x

using the given lengths

AB = 2x

AC = 3x-3

BD = 6

, and

CD = 8

Simplify the ratio on the right side of the equation and cross-multiply to solve for

x

8x = 9x - 9

, which simplifies to

x = 9

First simplify

\frac{6}{8}

\frac{3}{4}

, then cross-multiply:

2x(4) = 3(3x-3)

, which gives

8x = 9x - 9

. Subtracting

8x

and adding

9

yields

x = 9

Key Concept

Angle Bisector Theorem in Triangles

Question

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