Question 332·Hard·Math

0 / 2000

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

A segment parallel to a triangle’s side creates similar triangles: a smaller triangle with the same shape as the whole one. Match their sides, then turn the given part-to-part ratio into a small-to-whole fraction. You can solve the resulting proportion with a Desmos regression. The trap is treating one part of the ratio as the whole side.

Hints

Hint 1
The small triangle and the whole triangle share an angle at $X$ . Because $WV$ is parallel to $YZ$ , they have another pair of equal angles. What does AA similarity tell you about their sides?
Hint 2
The ratio $XW:WY$ compares two pieces of $XY$ . To compare $XW$ with the whole side $XY$ , how many ratio parts make up $XY$ ?
Hint 3
In similar triangles, matching sides have the same small-to-whole ratio. Match $WV$ with $YZ$ , then use the fraction you found for $XW$ compared with $XY$ .

Step-by-step

Scale the smaller triangle
Step 1
Establish that the triangles are similar
Because $W$ lies on $XY$ and $V$ lies on $XZ$ , the small triangle and the whole triangle share the angle at $X$ . The given $WV\parallel YZ$ makes the angle at $W$ equal to the angle at $Y$ . Two pairs of equal angles establish AA similarity, meaning the triangles have the same shape: $\triangle XWV\sim\triangle XYZ$ .
Step 2
Turn the ratio into a fraction of the whole side
The given $XW:WY=3:2$ means $XW$ has $3$ equal parts and $WY$ has $2$ . Since those pieces make $XY$ , the whole side has $3+2$ parts, not $2$ . So the small-to-whole fraction is $\frac{XW}{XY}=\frac{3}{3+2}$ . You don’t need $XY=15$ to find that fraction.
Step 3
Match the sides and write the proportion
In the similar triangles, $XW$ matches $XY$ , and $WV$ matches $YZ$ . Corresponding sides are sides that match, and they share the same small-to-whole ratio. Since $YZ=20$ , write: $\frac{WV}{20}=\frac{3}{3+2}$ .
Step 4
Solve for the requested length
Let $x_1$ stand for $WV$ . Type $\frac{x_1}{20}\sim\frac{3}{3+2}$ in Desmos. The $\sim$ tells Desmos to find the value of $x_1$ that makes the two sides equal. Under PARAMETERS, it shows $x_1=12$ . So $WV$ is $12$ . Grid in 12.

Question 332·Hard·Math

Strategy

Hints

Step-by-step

Lessons that teach this

Study Hints

Strategy

Hints

Step-by-step

Lessons that teach this