Question

identify the segment bisector of (overline{pq}).
diagram: segment (pq) with point (m) on it, line (ell) intersecting (pq) at (m). length (pm = 5x - 3), (mq = 11 - 2x).
options: (overrightarrow{mq}), (overrightarrow{mp}), (p), line (ell)
find (mq).
(mq = )

Explanation:

Step1: Determine the segment bisector

A segment bisector of \(\overline{PQ}\) is a line, ray, or segment that intersects \(\overline{PQ}\) at its midpoint \(M\). From the diagram, line \(l\) intersects \(\overline{PQ}\) at \(M\), so line \(l\) is the segment bisector. Also, since \(M\) is the midpoint, \(PM = MQ\). So we set \(5x - 3=11 - 2x\).

Step2: Solve for \(x\)

Solve the equation \(5x - 3 = 11 - 2x\).
Add \(2x\) to both sides: \(5x+2x - 3=11 - 2x + 2x\), which simplifies to \(7x - 3 = 11\).
Add 3 to both sides: \(7x-3 + 3=11 + 3\), so \(7x = 14\).
Divide both sides by 7: \(x=\frac{14}{7}=2\).

Step3: Find \(MQ\)

Substitute \(x = 2\) into the expression for \(MQ\), which is \(11 - 2x\).
\(MQ=11-2\times2=11 - 4 = 7\).

Answer:

The segment bisector of \(\overline{PQ}\) is line \(l\), and \(MQ = 7\).