Chapter 4.6, Problem 13WE

To determine

To draw: two line segments $\overline{KL}$ and $\overline{MN}$ . To prove that $O$ is the midpoint of $\overline{PQ}$ .

Explanation of Solution

Given information:

  $\overline{KL}$ and $\overline{MN}$ , that bisect each other at $O$ . Mark a point $P$ on $\overline{KN}$ and let $Q$ be the point where $\overline{PQ}$ intersects $\overline{ML}$ .

Calculation:

Draw two line segments, $\overline{KL}$ and $\overline{MN}$ , that bisect each other at $O$ . Mark a point $P$ on $\overline{KN}$ and let $Q$ be the point where $\overline{PQ}$ intersects $\overline{ML}$ .

  

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

Consider the following data:

  $\begin{array}{c}\overline{KO}\cong \overline{ON}\\ \overline{MO}\cong \overline{ON}\\ \angle KON\cong \angle MOL\end{array}$

To prove that $\overline{PO}\cong \overline{OQ}$

Prove this in paragraph form.

The proof is as follows:

By the side-angle side postulate $\Delta KNO\cong \Delta MLO$

Take off from this, $\Delta PON\cong \Delta QOM$ by the angle side angle postulate since

  $MO\cong ON$ , $\angle MOQ\cong \angle PON$ as these are vertically opposite angles

  $\angle PNO\cong \angle OMQ$ Corresponding parts of congruent triangle

Hence, $PO$ and $OQ$ , the corresponding parts of congruent triangles are congruent.

Hence, it is proved that $O$ is the midpoint of $\overline{PQ}$ .