Chapter 4.3, Problem 29E

(a)

To determine

To find: quadrilateral ABCD a parallelogram.

Answer to Problem 29E

Hence, the quadrilateral is a parallelogram.

Explanation of Solution

Given:

  $A\left(2,2\right),B\left(6,4\right),C\left(8,10\right)\text{ and D}\left(4,8\right)$ .

Calculation:

Consider a quadrilateral with vertices $A\left(2,2\right),B\left(6,4\right),C\left(8,10\right)\text{ and D}\left(4,8\right)$ .

The objective is to determine whether the quadrilateral is parallelogram or not.

Find the slopes of the lines joining the vertices $AB$ .

  $m=\frac{4-2}{\begin{array}{l}6-2\\ =\frac{1}{2}\end{array}}$

Now, the slopes of the lines joining the vertices $BC$ is,

  $\begin{array}{l}m=\frac{10-4}{8-6}\\ \text{ =}\frac{6}{2}\\ =3\\ \end{array}$

Again, the slopes of the lines joining the vertices $\text{CD}$ is

  $\begin{array}{l}m=\frac{8-10}{4-8}\\ =\frac{1}{2}\end{array}$

The slopes of the lines joining the vertices $AD$ is,

  $\begin{array}{l}m=\frac{8-2}{4-2}\\ =3\end{array}$

The slopes of the opposite side $AB\text{ and C}D$ are equal and $BC\text{ and AD}$ is equal.

Hence, the quadrilateral is a parallelogram.

Conclusion:

Hence, the quadrilateral is a parallelogram.

(b)

To determine

To find: quadrilateral ABCD a rectangle.

Answer to Problem 29E

Hence, the quadrilateral is not a rectangle.

Explanation of Solution

Given:

  $A\left(2,2\right),B\left(6,4\right),C\left(8,10\right)\text{ and D}\left(4,8\right)$

Calculation:

The objective is to determine whether the quadrilateral is rectangle or not.

Since the adjacent sides $\left(AB\text{ and }AD\right)$ and $\left(BC\text{ and CD}\right)$ have slopes which are not negative reciprocals to each other, therefore, the angle between the adjacent sides is not right angle.

Hence, the quadrilateral is not a rectangle.

Conclusion:

Hence, the quadrilateral is not a rectangle.