Chapter 10.8, Problem 5E

To determine

To prove: that $XYMN$ is a rectangle.

Explanation of Solution

Given Information:

Given diagram:

  

Given key steps of proof:

1. $XYMN$ is a rectangle.
2. The circle circumscribed about $XYMN$ has diameters $\overline{MX}$ and $\overline{NY}$ .
3. Because $\angle XSM$ and $\angle YTN$ are right angles, the circle contains points $S$ and $T$ as well as $X,Y,M$ and $N$ .

Calculation:

Here $\overline{NM}\parallel \overline{AB}$ and $\overline{XY}\parallel \overline{AB}$ , so by theorem, line joining the midpoints of two sides is parallel and of the third side,

Therefore, $NM=\frac{1}{2}AB$ and $XY=\frac{1}{2}AB$

So, $\overline{XY}\parallel \overline{NM}$ and $XY=NM$

Therefore, $XYMN$ is a rectangle by theorem, two sides are parallel and equal then it is a rectangle.

Here $\overline{NX}\parallel \overline{CH}$ by theorem, line joining the midpoint of two sides is parallel and $\overline{NM}\perp \overline{NX}$ .

Therefore, $XYMN$ is a rectangle.

Hence, $XYMN$ is a rectangle.