In triangle , side and side . Point lies on such that . A line through drawn parallel to intersects side at point . What is the length of ?

Solution available with step-by-step explanation

SAT Lines, Angles & Triangles Question 6 | Free SAT Prep | Aniko

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Question 6·Hard·Lines, Angles, and Triangles

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In triangle $XYZ$ , side $XY = 15$ and side $YZ = 20$ . Point $W$ lies on $XY$ such that $XW : WY = 3 : 2$ . A line through $W$ drawn parallel to $YZ$ intersects side $XZ$ at point $V$ . What is the length of $WV$ ?

Your answer

(Express the answer as an integer)

For SAT geometry problems where a line inside a triangle is drawn parallel to one side, immediately think of similar triangles. Identify which sides correspond, then use the given segment ratios on one side of the triangle to find the scale factor between the small and large triangles. Apply that scale factor directly to the corresponding side you’re asked about, setting up a simple proportion instead of trying to compute multiple individual segment lengths.

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Use the parallel line

A line drawn through $W$ parallel to $YZ$ creates a smaller triangle inside triangle $XYZ$ . What can you say about the relationship between triangle $XWV$ and triangle $XYZ$ ?

Find the scale factor

Use the ratio $XW : WY = 3 : 2$ to determine what fraction of the whole side $XY$ the segment $XW$ is.

Relate corresponding sides

Once you know the fraction $XW/XY$ , apply the same ratio to the side $YZ$ to find the corresponding side in the smaller triangle, which is $WV$ .

Desmos Guide

Compute the similarity ratio and apply it

First, note from $XW : WY = 3 : 2$ that $XW$ is $3/5$ of $XY$ , so the scale factor from triangle $XYZ$ to triangle $XWV$ is $3/5$ . In Desmos, type (3/5)*20 and read the output; this value is the length of $WV$ , which is the side in the smaller triangle corresponding to $YZ = 20$ .

Step-by-step Explanation

Identify the similar triangles

Since $WV$ is drawn through $W$ parallel to $YZ$ , angle at $X$ is common to both triangles $XYZ$ and $XWV$ , and the angles at $W$ and $Y$ are corresponding because of the parallel lines.

Therefore, triangle $XWV$ is similar to triangle $XYZ$ .

Find the ratio of similarity using the sides on $XY$

We are told that point $W$ is on $XY$ such that $XW : WY = 3 : 2$ .

The whole side $XY$ is split into $3$ parts for $XW$ and $2$ parts for $WY$ , so:

Total parts on $XY$ = $3 + 2 = 5$ parts.
So $XW$ is

XW = \frac{3}{5} \cdot XY

This means the scale factor from triangle $XYZ$ (large) to triangle $XWV$ (small) is $\dfrac{XW}{XY} = \dfrac{3}{5}$ .

Match corresponding sides

In the similar triangles:

Side $XY$ in the big triangle corresponds to $XW$ in the smaller triangle.
Side $YZ$ in the big triangle corresponds to $WV$ in the smaller triangle (they are parallel).

So the ratio of similarity is

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

We already found $\dfrac{XW}{XY} = \dfrac{3}{5}$ , and we know $YZ = 20$ .

Set up and solve the proportion for $WV$

Use the proportion with the known values:

\frac{3}{5} = \frac{WV}{20}.

Solve for $WV$ by multiplying both sides by $20$ :

WV = 20 \cdot \frac{3}{5} = 4 \cdot 3 = 12.

So, the length of $WV$ is $12$ .

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation