Question

1. in the diagram below, line r is the perpendicular bisector of $overline{mn}$

(a) what special point is point p on $overline{mn}$. explain how you know this.
(b) choose any other point on line r and mark it. label it as point q.
(c) draw in $overline{mq}$ and $overline{nq}$. measure each of their lengths to the nearest tenth of a centimeter.
(d) what do you notice about point q relative to points m and n?

Explanation:

Step1: Recall perpendicular - bisector property

A perpendicular bisector of a line segment divides the segment into two equal parts and is perpendicular to it. Since line \(r\) is the perpendicular bisector of \(\overline{MN}\) and point \(P\) lies on \(r\), \(P\) is the mid - point of \(\overline{MN}\).

Step2: Mark a point \(Q\)

You can choose any point on line \(r\) other than \(P\) and label it as \(Q\). This is a construction step and there are infinitely many such points.

Step3: Draw and measure

Draw line segments \(\overline{MQ}\) and \(\overline{NQ}\) using a straight - edge and measure their lengths with a ruler to the nearest tenth of a centimeter.

Step4: Analyze the relationship

Since \(r\) is the perpendicular bisector of \(\overline{MN}\), by the property of the perpendicular bisector, any point \(Q\) on \(r\) is equidistant from \(M\) and \(N\), i.e., \(MQ = NQ\).

Answer:

(a) Point \(P\) is the mid - point of \(\overline{MN}\) because a perpendicular bisector of a line segment intersects the segment at its mid - point.
(b) (This is a construction step. You can mark any point \(Q\) on line \(r\) other than \(P\)).
(c) (After drawing \(\overline{MQ}\) and \(\overline{NQ}\) and measuring, record the lengths).
(d) Point \(Q\) is equidistant from points \(M\) and \(N\), i.e., the length of \(\overline{MQ}\) is equal to the length of \(\overline{NQ}\).