Chapter 13, Problem 25CR

To determine

To prove: that ONQM is a parallelogram.

Explanation of Solution

Given information:

M is the middle point of $\overline{RQ}$ and N is the middle point of $\overline{OP}$ .

Formula used:

Distance between two points

  $=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Proof:

Since coordinates of points O, R, Q, and P are $\left(0,0\right),\left(b,c\right),\left(b+a,c\right)\text{ and }\left(a,0\right)$ respectively.

Coordinates of M

  $=\left(\frac{b+b+a}{2},\frac{c+c}{2}\right)$

Coordinates of N

  $=\left(\frac{0+a}{2},0\right)$

Length of $ON$

  $\begin{array}{l}=\sqrt{{\left(a-\frac{a}{2}\right)}^2+\left(0\right)}\\ =\frac{a}{2}\end{array}$

Length of NQ

  $\begin{array}{l}=\sqrt{{\left(\frac{a}{2}-\left(a+b\right)\right)}^2+{\left(0-c\right)}^2}\\ =\sqrt{{\left(b+\frac{a}{2}\right)}^2+c^2}\end{array}$

Length of OM

  $\begin{array}{l}=\sqrt{{\left(b+\frac{a}{2}-0\right)}^2+{\left(c-0\right)}^2}\\ =\sqrt{{\left(b+\frac{a}{2}\right)}^2+c^2}\end{array}$

Length of MQ

  $\begin{array}{l}=\sqrt{{\left(b+\frac{a}{2}-\left(a+b\right)\right)}^2+{\left(c-c\right)}^2}\\ =\sqrt{\frac{a}{2}}\end{array}$

Here $ON=MQ\text{ and }NQ=OM$ that means opposite sides of the quadrilateral ONQM are congruent.

Therefore,

Thequadrilateral ONQM is a parallelogram.