Question

identify the segment bisector of $overline{xy}$.
3x + 1 8x - 24
x m y
n
o n
o x
o $overrightarrow{mn}$
o $overline{my}$
the length of $overline{xy}$ is

Explanation:

Step1: Recall definition of bisector

A segment bisector divides a segment into two equal - length parts. So, $XM = MY$. Set up the equation $3x + 1=8x - 24$.

Step2: Solve the equation for x

Subtract $3x$ from both sides: $1 = 8x-3x - 24$, which simplifies to $1 = 5x - 24$. Then add 24 to both sides: $1 + 24=5x$, so $25 = 5x$. Divide both sides by 5, we get $x = 5$.

Step3: Find the length of $XM$ and $MY$

Substitute $x = 5$ into the expression for $XM$: $XM=3x + 1=3\times5 + 1=15 + 1 = 16$. Substitute $x = 5$ into the expression for $MY$: $MY=8x - 24=8\times5 - 24=40 - 24 = 16$.

Step4: Identify the segment bisector

Since a segment bisector passes through the mid - point of the segment, and the line $\overrightarrow{MN}$ passes through the mid - point $M$ of $\overline{XY}$, the segment bisector of $\overline{XY}$ is $\overrightarrow{MN}$.

Step5: Calculate the length of $\overline{XY}$

$XY=XM + MY$. Since $XM = 16$ and $MY = 16$, then $XY=16+16 = 32$.

Answer:

The segment bisector of $\overline{XY}$ is $\overrightarrow{MN}$. The length of $\overline{XY}$ is $32$.