Question

d. if the perpendicular bisector of $overline{ac}$ and the perpendicular bisector of $overline{bc}$ intersect at $p$, what two points determine the perpendicular bisector of $overline{ab}$? justify your answer.

Explanation:

Step1: Recall circum - center property

The point of intersection of the perpendicular bisectors of the sides of a triangle is called the circum - center. Given that the perpendicular bisector of $\overline{AC}$ and the perpendicular bisector of $\overline{BC}$ intersect at $P$, then $P$ is the circum - center of $\triangle ABC$.

Step2: Use circum - center property for perpendicular bisector of third side

Since the circum - center is equidistant from all the vertices of the triangle, $P$ is equidistant from $A$, $B$ and $C$. The perpendicular bisector of a line segment is the set of all points that are equidistant from the endpoints of the line segment. So, $P$ lies on the perpendicular bisector of $\overline{AB}$. Also, the mid - point of $\overline{AB}$, say $M$, lies on the perpendicular bisector of $\overline{AB}$.

Answer:

The mid - point of $\overline{AB}$ and the point $P$ determine the perpendicular bisector of $\overline{AB}$.