Chapter 3, Problem 5RP

To determine

To prove : $\overline{PR}$ is a median.

Explanation of Solution

Given information : Here, we have given a circle with center O and $\overrightarrow{PR}$ bisects $\angle KPM$ .

Proof: As, $\overrightarrow{PR}$ bisects $\angle KPM$ .

  $\Rightarrow \angle MPR\cong \angle KPR$

Now, In $\Delta PRM$ and $\Delta PRK$ ,

  $\overline{PM}\cong \overline{PK}$ (Radius of circle)

  $\angle MPR\cong \angle KPR$ (Proved above)

  $\overline{PR}\cong \overline{PR}$ (Common)

  $\Rightarrow \Delta PRM\cong \Delta PRK$ (By SAS Congruency Rule)

  $\Rightarrow \overline{KR}\cong \overline{MR}$ (By Corresponding Part of Congruent Triangles)

  $\therefore \overline{PR}$ is a median.