Chapter 13.3, Problem 11WE

a.

To determine

To show that opposite sides are parallel.

Explanation of Solution

Given information: The points $A\left(-6,-4\right),B\left(4,2\right),C\left(6,8\right),\text{ and }D\left(-4,2\right)$ .

Proof: The required proof is obtained as,

  

Slope of $AB=\frac{2+4}{4+6}=\frac{3}{5}$

Slope of $CD=\frac{8-2}{6+4}=\frac{3}{5}$

Slope of $AD=\frac{2+4}{-4+6}=3$

Slope of $BC=\frac{8-2}{6-4}=3$

Therefore, AB and CD are parallel.

AD and BC are parallel.

b.

To determine

To show that opposite sides are congruent.

Explanation of Solution

Given information: The points $A\left(-6,-4\right),B\left(4,2\right),C\left(6,8\right),\text{ and }D\left(-4,2\right)$ .

Proof: The required proof is obtained as,

  

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

Distance of $AB=\sqrt{{\left(-6-4\right)}^2+{\left(-4-2\right)}^2}=\sqrt{100+36}=\sqrt{136}$

Distance of $BC=\sqrt{{\left(4-6\right)}^2+{\left(2-8\right)}^2}=\sqrt{4+36}=\sqrt{40}$

Distance of $CD=\sqrt{{\left(6-\left(-4\right)\right)}^2+\left(8-2\right)}=\sqrt{100+36}=\sqrt{136}$

Distance of $DA=\sqrt{{\left(-6-\left(-4\right)\right)}^2+{\left(-4-2\right)}^2}=\sqrt{4+36}=\sqrt{40}$

Therefore,

  $\begin{array}{l}AB=CD\\ BC=DA\end{array}$

Hence, opposite side are congruent.