Chapter 6, Problem 17CR

To determine

To prove : The $\Delta ABC$ is isosceles

Explanation of Solution

Given information : Here, we have given that segments drawn from midpoint of one side of triangle perpendicular to other two sides are equal.

  

Proof: In $\Delta EDB$ and $\Delta FDC$

  $\overline{BD}\cong \overline{CD}$ (D is midpoint of $\overline{BC}$ )

  $\overline{ED}\cong \overline{FD}$ (Given)

  $\angle DEB\cong \angle DFC$ (segments are perpendicular to the sides)

  $\Rightarrow \Delta EDB\cong \Delta FDC$ (By RHS Congruency Rule)

  $\therefore \angle B\cong \angle C$ (By Corresponding Part of Congruent Triangles)

As, $\angle B\cong \angle C$ ,

  $\Rightarrow \overline{AB}\cong \overline{AC}$ (Sides opposite to equal angles are equal)

  $\therefore \Delta ABC$ is isosceles.