Chapter 13, Problem 15CUR

To determine

To prove: If the ray that bisects an angle of a triangle is perpendicular to the side that it intersects, then the triangle is an isosceles triangle.

Explanation of Solution

Given information: A ray that bisects an angle of a triangle is perpendicular to the side that it intersects.

Proof:

Draw the figure of a triangle ABC according to the given conditions as shown below.

  

Here, $\text{AD}\perp \text{BC}$ and $\text{BD}=\text{DC}$ .

In $\Delta \text{ABD}$ and $\Delta \text{ADC}$ ,

  $\begin{array}{l}\text{BD = DC}\left(\text{given}\right)\\ \text{AD}=\text{AD}\left(\text{common}\right)\\ \angle \text{BAD}=\angle \text{DAC}\left(\text{ray bisects angle}\right)\end{array}$

It can be observed that two sides and an angle of one triangle is equal to the two sides and an angle of other triangle. So, by SAS theorem both triangles are congruent.

  $\Delta \text{ABD}\cong \Delta \text{ADC}$

By corresponding part of congruent triangle, $\text{AB}=\text{AC}$ .

Hence, if the ray that bisects an angle of a triangle is perpendicular to the side that it intersects, then the triangle is an isosceles triangle.