Chapter 3.7, Problem 14PSB

Solution

To Prove:Median $\overline{AP}$ is an angular bisector of the vertex angle.

Given information: $\Delta ABC$ is an isosceles triangle is provided in the question which will help to find the solution.

  

Proof: In the $\Delta APBand\Delta APC$

  $\overline{AP}=\overline{AP}$ (Common side)

  $\overline{AB}=\overline{AC}$ (sides of isosceles triangle)

  $\overline{PB}=\overline{PB}$ (P is midpoint of $\overline{BC}$ )

Hence, $\Delta APB\cong \Delta APC$

Since, the triangles are congruent, the corresponding angles and sides are also congruent.

  $\therefore \angle PAB\cong \angle PAC$ , hence the median $\overline{AP}$ is an angular bisector of the vertex angle.