Chapter 5, Problem 11CUR

To determine

To Complete: the statement “ the median to the base of an isosceles triangle $\underset{\_}{?}$ the vertex angle and is $\underset{\_}{?}$ to the base.

Answer to Problem 11CUR

Bisects and Perpendicular.

Explanation of Solution

Given:

An isosceles triangle with median to the base.

In an isosceles triangle median to the base bisects the vertex angle is perpendicular to the base.

Proof is shown below.

Diagram can be represented as follows

  

Using two −column proof.

  $\begin{array}{cc}\text{Statements }& \text{Reasons}\\ AB=AC& \text{Isosceles Triangle}\\ \angle BAD=\angle CAD& \text{Bisects}\\ \begin{array}{l}AD=AD\\ \Delta BAD\cong \Delta CAD\\ BD=DC\text{ and }\angle BDA\cong CDA=90\\ \text{Since,}BD=DC\text{therefore it is a median}\end{array}& \begin{array}{l}\text{Common}\\ \text{By SAS}\\ \text{By CPCT}\end{array}\end{array}$

Therefore, the answer is bisecting and perpendicular.