Question

identify the segment bisector of (overline{xy}).
diagram: x---m---y (xm labeled (3x + 1), my labeled (8x - 24)), segment from m to n (point n on the segment). options: (n), (x), (overrightarrow{mn}), (overline{my}).
the length of (overline{xy}) is (square).

Explanation:

Part 1: Identify the segment bisector of \(\overline{XY}\)

A segment bisector is a line, ray, or segment that divides another segment into two equal parts. Since \(M\) is the midpoint (as the segments \(XM = 3x + 1\) and \(MY = 8x - 24\) should be equal for \(M\) to be the midpoint), the ray \(\overrightarrow{MN}\) passes through \(M\), the midpoint of \(\overline{XY}\), so \(\overrightarrow{MN}\) is the segment bisector.

Part 2: Find the length of \(\overline{XY}\)

Step 1: Set \(XM = MY\) (since \(M\) is the midpoint)

Since \(XM = 3x + 1\) and \(MY = 8x - 24\), we set them equal:
\[
3x + 1 = 8x - 24
\]

Step 2: Solve for \(x\)

Subtract \(3x\) from both sides:
\[
1 = 5x - 24
\]
Add 24 to both sides:
\[
25 = 5x
\]
Divide both sides by 5:
\[
x = 5
\]

Step 3: Find the length of \(XM\) and \(MY\)

Substitute \(x = 5\) into \(XM = 3x + 1\):
\[
XM = 3(5) + 1 = 15 + 1 = 16
\]
Substitute \(x = 5\) into \(MY = 8x - 24\):
\[
MY = 8(5) - 24 = 40 - 24 = 16
\]

Step 4: Find the length of \(XY\)

Since \(XY = XM + MY\):
\[
XY = 16 + 16 = 32
\]

Answer:

s:

• The segment bisector of \(\overline{XY}\) is \(\boldsymbol{\overrightarrow{MN}}\).
• The length of \(\overline{XY}\) is \(\boldsymbol{32}\).