In the figure shown, and intersect at point . The length of is , the length of is , , and . Point is closer to than to , and . Which choice is the length of ?

Solution available with step-by-step explanation

SAT Lines, Angles & Triangles Question 30 | Free SAT Prep | Aniko

00:00

Question 30·Hard·Lines, Angles, and Triangles

0 / 50

In the figure shown, $\overline{WZ}$ and $\overline{XY}$ intersect at point $Q$ . The length of $\overline{WZ}$ is $120$ , the length of $\overline{XY}$ is $118$ , $XQ = 64$ , and $WX = 88$ . Point $Q$ is closer to $Z$ than to $W$ , and $\angle QWX = \angle QYZ$ .

Which choice is the length of $\overline{YZ}$ ?

When intersecting segments create two triangles and you are told one non-vertical angle pair is equal, immediately check for triangle similarity using that angle plus the vertical angles at the intersection. Then express missing pieces (like $QY$ or $WQ$ ) using whole-segment lengths, set up a proportion with an unknown, and solve. If the algebra gives two candidates, use any geometric condition (such as “closer to”) to eliminate the value that contradicts the diagram.

Hints

Click me!

Look for similar triangles

The given equality $\angle QWX = \angle QYZ$ and the fact that the segments intersect at $Q$ should give you two equal angles in two different triangles.

Use the whole segment lengths

Compute $QY$ from $XY$ and $XQ$ . Also, if you let $QZ=z$ , then $WQ$ can be written using $WZ=120$ .

Use the “closer to” condition

Your equation for $QZ$ may produce two solutions. Use “ $Q$ is closer to $Z$ than to $W$ ” to decide which solution fits the diagram.

Desmos Guide

Compute known segment pieces

Compute $QY$ as $118-64$ , and let $z$ represent $QZ$ so that $WQ=120-z$ .

Graph the equation for $z$

Enter $y=z(120-z)$ and enter $y=54\cdot 64$ as a second equation. The $x$ -coordinates of the intersection points are the possible values of $z=QZ$ .

Choose the intersection that matches the condition

Use the condition “ $Q$ is closer to $Z$ than to $W$ ” to pick the smaller value of $z$ (so $QZ < WQ$ ).

Use a similarity ratio to match an answer choice

Compute the ratio $\frac{WQ}{QY}$ and then compute $YZ=WX\cdot \frac{QY}{WQ}$ . Choose the answer option that equals that result.

Step-by-step Explanation

Find $QY$ from the whole segment $XY$

Because $Q$ is on $\overline{XY}$ ,

$QY = XY - XQ = 118 - 64 = 54$ .

Use similar triangles to set up a ratio with an unknown

Triangles $\triangle WQX$ and $\triangle YQZ$ are similar:

$\angle QWX = \angle QYZ$ (given)
$\angle WQX = \angle YQZ$ (vertical angles)

So corresponding side ratios are equal:

\frac{WQ}{QY} = \frac{XQ}{QZ}.

Let $QZ = z$ . Since $WZ = 120$ and $Q$ lies on $\overline{WZ}$ ,

WQ = 120 - z.

Substitute $QY=54$ and $XQ=64$ :

\frac{120-z}{54} = \frac{64}{z}.

Solve for $z=QZ$

Cross-multiply:

z(120-z) = 54\cdot 64 = 3456.

Rearrange:

\begin{aligned} z^2 - 120z + 3456 &= 0 \\ (z-48)(z-72) &= 0. \end{aligned}

So $z=48$ or $z=72$ .

Choose the value that matches the geometric condition

If $z=48$ , then $WQ = 120-48=72$ , so $QZ < WQ$ .

If $z=72$ , then $WQ = 120-72=48$ , so $QZ > WQ$ .

Since point $Q$ is closer to $Z$ than to $W$ , we need $QZ < WQ$ , so $QZ=48$ and $WQ=72$ .

Use the similarity ratio to find $YZ$

From similarity,

\frac{WX}{YZ} = \frac{WQ}{QY}.

Substitute $WX=88$ , $WQ=72$ , and $QY=54$ :

\frac{88}{YZ} = \frac{72}{54} = \frac{4}{3}.

So $YZ = 88\cdot \frac{3}{4} = 66$ .

Therefore, the length of $\overline{YZ}$ is $66$ .

Strategy

Hints

Desmos Guide

Step-by-step Explanation

Study Hints

Strategy

Hints

Desmos Guide

Step-by-step Explanation