Question

in exercises 9 and 10, identify the segment bisector of $overline{xy}$. then find $xy$. (see example 2.) 9. 10.

Explanation:

Step1: Recall the property of a segment - bisector

A segment - bisector divides a segment into two equal parts. So, for segment $\overline{XY}$ bisected at $M$, we set the two sub - segments equal to each other. In exercise 9, we have $3x + 1=8x−24$.
$3x + 1=8x−24$

Step2: Solve the equation for $x$

Subtract $3x$ from both sides:
$1 = 8x-3x - 24$
$1=5x - 24$
Add 24 to both sides:
$1 + 24=5x$
$25 = 5x$
Divide both sides by 5:
$x = 5$

Step3: Find the length of $\overline{XY}$

The length of $\overline{XY}$ is $(3x + 1)+(8x - 24)$. Substitute $x = 5$ into the expression:
$3(5)+1+8(5)-24=15 + 1+40-24=32$

In exercise 10:

Step1: Set the two sub - segments equal

Since $n$ is the segment - bisector of $\overline{XY}$, we have $5x + 8=9x+12$.
$5x + 8=9x+12$

Step2: Solve the equation for $x$

Subtract $5x$ from both sides:
$8=9x - 5x+12$
$8 = 4x+12$
Subtract 12 from both sides:
$8-12=4x$
$- 4=4x$
Divide both sides by 4:
$x=-1$

Step3: Find the length of $\overline{XY}$

The length of $\overline{XY}$ is $(5x + 8)+(9x + 12)$. Substitute $x=-1$ into the expression:
$5(-1)+8+9(-1)+12=-5 + 8-9 + 12=6$

Answer:

Exercise 9: The segment - bisector is $\overline{MN}$, $XY = 32$.
Exercise 10: The segment - bisector is line $n$, $XY = 6$.