Chapter 5.1, Problem 37PPS

To determine

To prove: $C$ and $D$ are on the perpendicular bisector of $\overline{AB}$ .

Answer to Problem 37PPS

The proof is shown below.

Explanation of Solution

Given information:

  $\overline{CA}\cong \overline{CB}$ and $\overline{AD}\cong \overline{BD}$ .

The given figure is

  

Calculation:

Since, $\overline{CA}\cong \overline{CB}$ , $\overline{AD}\cong \overline{BD}$ and $\overline{CD}\cong \overline{CD}$ , triangle $\Delta ACD\cong \Delta BCD$ by sides side, side property. So $\angle ACD\cong \angle BCD$

Since. $\overline{CA}\cong \overline{CB}$ , $\overline{AD}\cong \overline{BD}$ and $\overline{CE}\cong \overline{CE}$ , triangle $\Delta CEA\cong \Delta CEB$ by sides angle, side property. So $\overline{AE}\cong \overline{BE}$ $E$ d the mid-point of $\overline{AB}$ .

As per corresponding part of congruent triangle $\angle CEA\cong \angle CEB$ .

We can write,

  $\angle CEA+\angle CEB=180$

since $\angle CEA\cong \angle CEB$ it can be written that,

  $\begin{array}{l}2\angle CEA=180\\ \angle CEA=\frac{180}{2}\\ \angle CEA=90\text{or}\angle CEB=90\end{array}$

From above result it is clear that $\overline{CD}\perp \overline{AB}$ .

Thus it is clear that $C$ and $D$ are on the perpendicular bisector of $\overline{AB}$ .

Therefore, $C$ and $D$ are on the perpendicular bisector of $\overline{AB}$ .