We are given a diagram and asked to find the length of $\overline{X Z} .$

As we can see, $5 y - 6$ represents the length of $\overline{X Z} .$ Therefore, to find $X Z,$ we first need to find the value of $y .$

From the diagram, we see that $\overline{W X}$ intersects $\overline{Z Y}$ at the midpoint $W .$ Additionally, it is perpendicular to $\overline{Z Y} .$ Therefore, $\overline{W X}$ is the perpendicular bisector of $\overline{Z Y} .$ Let's use the Perpendicular Bisector Theorem.

Perpendicular Bisector Theorem
If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

By this theorem,

X

is equidistant from the endpoints of

\overline{Z Y} .

This means that

\overline{X Z}

and

\overline{Y X}

have the same measures. From the diagram, we know that

Z X = 5 y - 6

and

Y X = 18 + 2 y .

Let's set these expressions equal.

5 y - 6 = 18 + 2 y

Now, we can solve this equation for

y .

5 y - 6 = 18 + 2 y

▼

Solve for

y

$LHS - 2 y = RHS - 2 y$

3 y - 6 = 18

$LHS + 6 = RHS + 6$

3 y = 24

$LHS / 3 = RHS / 3$

y = 8

Now that we know the value of

y,

we can evaluate the expression

5 y - 6

and find the measure of

\overline{X Z} .

X Z = 5 y - 6

$y = 8$

X Z = 5 (8) - 6

X Z = 40 - 6

Subtract term

X Z = 34

The length of

\overline{X Z}

is

34 .

Exercise 13 Page 382

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