Question

what is the midpoint of $overline{pq}$? what are the segment bisectors of $overline{pq}$?

Explanation:

Step1: Recall mid - point formula

If \(P(x_1,y_1)\) and \(Q(x_2,y_2)\) are two points in a coordinate plane, the mid - point \(M\) of the line segment \(\overline{PQ}\) has coordinates \(M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). Without the coordinates of \(P\) and \(Q\), we can only state the formula.

Step2: Recall segment bisector definition

A segment bisector is a line, ray, or another line segment that divides the line segment \(\overline{PQ}\) into two equal parts. The mid - point of \(\overline{PQ}\) is a point on any segment bisector. A perpendicular bisector of \(\overline{PQ}\) is a line that is perpendicular to \(\overline{PQ}\) and passes through its mid - point. There are infinitely many segment bisectors of \(\overline{PQ}\), including the line passing through the mid - point with any slope, and the perpendicular bisector.

Answer:

Mid - point: If \(P(x_1,y_1)\) and \(Q(x_2,y_2)\), mid - point \(M(\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\). Segment bisectors: Infinitely many lines, rays or line segments that divide \(\overline{PQ}\) into two equal parts, including the perpendicular bisector.