In this exercise we are asked to find lengths of three segments. Let's take a look at the given picture.

First, let's notice that, since $\overline{P R}$ is parallel to $\overline{W X},$ $△ Y P S$ and $△ W Q S$ are similar by Angle-Angle Similarity Theorem. Using the same theorem, we can prove that $△ W Q S$ and $△ W X Y$ are similar.

By the Transitive Property of Similarity,

△ Y P S

is similar to

△ W X Y .

This means that corresponding sides of these triangles are proportional.

\frac{P Y}{W X} = \frac{P S}{X Y} = \frac{S Y}{W Y}

Let's solve the above proportion by substituting appropriate side lengths. We will start with the left and the middle ratio.

\frac{P Y}{W X} = \frac{P S}{X Y}

SubstituteValues

Substitute values

\frac{P Y}{1 0} = \frac{3}{6}

▼

Solve for

P Y

ReduceFrac

$\frac{a}{b} = \frac{a / 3}{b / 3}$

\frac{P Y}{1 0} = \frac{1}{2}

MultEqn

$LHS \cdot 10 = RHS \cdot 10$

P Y = \frac{1}{2} \cdot 10

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

P Y = \frac{1 0}{2}

CalcQuot

Calculate quotient

P Y = 5

The length of

\overline{P Y}

is

5 .

Next, we will find the length of

\overline{S Y}

by evaluating the middle and the right ratio.

\frac{P S}{X Y} = \frac{S Y}{W Y}

SubstituteValues

Substitute values

\frac{3}{6} = \frac{S Y}{8}

▼

Solve for

S Y

ReduceFrac

$\frac{a}{b} = \frac{a / 3}{b / 3}$

\frac{1}{2} = \frac{S Y}{8}

MultEqn

$LHS \cdot 8 = RHS \cdot 8$

\frac{1}{2} \cdot 8 = S Y

RearrangeEqn

Rearrange equation

S Y = \frac{1}{2} \cdot 8

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

S Y = \frac{8}{2}

CalcQuot

Calculate quotient

S Y = 4

The length of

\overline{S Y}

is

4 .

Let's add this information to our picture.

Having in mind the Transitive Property of Similarity, we can see that

△ P Q R

is similar to

△ X Y W .

Let's use the fact that in similar triangles corresponding sides are proportional.

\frac{P Q}{X Y} = \frac{P R}{X W} ⇓ \frac{P Q}{6} = \frac{5 + 5}{1 0}

Finally, we will solve the above proportion to find the length of

\overline{P Q} .

\frac{P Q}{6} = \frac{5 + 5}{1 0}

AddTerms

Add terms

\frac{P Q}{6} = \frac{1 0}{1 0}

CalcQuot

Calculate quotient

\frac{P Q}{6} = 1

MultEqn

$LHS \cdot 6 = RHS \cdot 6$

P Q = 6

The length of

\overline{P Q}

is

6 .

We can present our answers in a table.

Segment	Length
$\overline{P Y}$	$5$
$\overline{S Y}$	$4$
$\overline{P Q}$	$6$

Exercise

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