Question

in exercises 9 and 10, identify the segment bisector of $overline{xy}$. then find $xy$. (see example 2.) 9. $3x + 1$ $8x - 24$ $x$ $m$ $y$ $n$

Explanation:

Step1: Use the property of segment - bisector

Since a segment bisector divides a segment into two equal parts, we have $XM = MY$. So, $3x + 1=8x - 24$.

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: Find the length of $XY$

Since $XY=XM + MY$, and $XM = MY = 16$, then $XY=16+16 = 32$. Also, the segment bisector of $\overline{XY}$ is $\overrightarrow{MN}$ (because $M$ is the mid - point of $\overline{XY}$ and $\overrightarrow{MN}$ passes through $M$).

Answer:

The segment bisector of $\overline{XY}$ is $\overrightarrow{MN}$, and $XY = 32$.