In triangle XYZ, point W lies on side XY and point V lies on side XZ such that line segmen… — SAT

In triangle $XYZ$ , point $W$ lies on side $XY$ and point $V$ lies on side $XZ$ such that line segment $WV$ is parallel to side $YZ$ . If $XW = 4$ , $WY = 2$ , and $WV = 6$ , what is the length of side $YZ$ ?

Answer: 9

Answer

Because segment

WV

is parallel to segment

YZ

, triangle

XWV

is similar to triangle

XYZ

by the Angle-Angle (AA) similarity theorem. The ratio of the corresponding side lengths is constant, which gives the proportion

\frac{YZ}{WV} = \frac{XY}{XW}

. The length of side

XY

XW + WY = 4 + 2 = 6

. Substituting the values into the proportion yields

\frac{YZ}{6} = \frac{6}{4}

. Solving for

YZ

gives

YZ = 9

Step-by-Step Solution

Calculate the length of side

XY

XY = 6

The length of side

XY

is the sum of segment lengths

XW

and

WY

XY = XW + WY = 4 + 2 = 6

Determine that triangle

XWV

is similar to triangle

XYZ

\triangle XWV \sim \triangle XYZ

Because segment

WV

is parallel to segment

YZ

, corresponding angles

\angle XWV

and

\angle XYZ

are congruent, and corresponding angles

\angle XVW

and

\angle XZY

are congruent. Since they also share

\angle X

, the two triangles are similar by the Angle-Angle (AA) similarity criterion.

Set up a proportion and solve for

YZ

YZ = 9

Corresponding sides of similar triangles are proportional:

\frac{YZ}{WV} = \frac{XY}{XW}

. Substituting the known values gives

\frac{YZ}{6} = \frac{6}{4}

. Multiplying both sides by 6 yields

YZ = \frac{36}{4} = 9

Key Concept

Finding side lengths in similar triangles using the Angle-Angle similarity theorem when a line is parallel to one side of a triangle.

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