Chapter 4.4, Problem 32WE

To determine

To Draw: An isosceles triangle and join the midpoints of its sides to form another angle.

Explanation of Solution

Given information: An isosceles triangle and their midpoints.

Drawing:

Firstly draw the figure:

  

In the above figure $\text{Δ}ABC$ is isosceles with $\overline{AB}\cong \overline{AC}$ and $D,E,F$ are the mid points of sides $AB$ , AC and $BC$ respectively.

By joining the midpoints of sides of $\text{Δ}ABC$ another $\text{Δ}DEF$ is formed.

The line that joins the center of any two sides of a triangle is similar and half of the third side.

This implies;

  $\begin{array}{l}\begin{array}{cc}\overline{DE}& =\frac{1}{2}\overline{BC}\\ \overline{DF}& =\frac{1}{2}\overline{AC}\\ \overline{EF}& =\frac{1}{2}\overline{AB}\end{array}\\ \overline{AB}\cong \overline{AC}\end{array}$

Multiply both sides by $\frac{1}{2};$

  $\frac{1}{2}\overline{AB}=\frac{1}{2}\overline{AC}$

This implies;

  $\overline{DF}=\overline{EF}$

Therefore, by definition of isosceles triangle, $\text{Δ}DEF$ is also isosceles with $\overline{DF}\cong \overline{EF}$ .