Chapter 4.3, Problem 14WE

To determine

To prove: O is the midpoint of $\overline{PQ}$

Explanation of Solution

Given information: $\overline{PA}\perp \text{plane}X,\overline{QB}\perp \text{plane}X$ , O is the midpoint of $\overline{AB}$ .

Proof:

  

First prove $\Delta POA\cong \Delta QOB$

As $\overline{PA}\perp \text{plane}X,\overline{QB}\perp \text{plane}X$ ,

  $\begin{array}{l}\angle PAO\cong 90°\\ \angle QBO\cong 90°\end{array}$

So,

  $\angle PAO\cong \angle QBO$ (two right angles are congruent to each other)

As O is the midpoint of $\overline{AB}$ ,

  $\overline{AO}\cong \overline{OB}$

(Midpoint is a point on line segment that divides it into two congruent line segments)

  $\angle POA\cong \angle QOB$ (Vertically opposite angles)

So,

  $\Delta POA\cong \Delta QOB$ (ASA congruence criteria)

Therefore,

  $\overline{PO}\cong \overline{QO}$ (Corresponding sides of congruent triangles are congruent)

So,

O is the midpoint of $\overline{PQ}$ .

(Midpoint is a point on line segment that divides it into two congruent line segments)