Chapter 13.3, Problem 6WE

a.

To determine

To find the slope of $\overline{RV}$ and $\overline{TV}$

Answer to Problem 6WE

Explanation of Solution

Given:

Below is the figure

  

Calculation:

Slope of a line when two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ are given is

  $m=\frac{y_2-y_1}{x_2-x_1}$

points are $R\left(-2,-6\right)\text{,V}\left(0,-8\right)\text{ and T}\left(5,-3\right)$

Slope of line of $\overline{RV}$

  $m_{RV}=\frac{-8-\left(-6\right)}{0-\left(-2\right)}=\frac{-2}{2}=-1$

Slope of line of $\overline{TV}$

  $m_{TV}=\frac{-8-\left(-3\right)}{0-5}=\frac{-5}{-5}=1$

Therefore, the slope of line $\overline{RV}$ is $-1$ and $\overline{TV}$ is $1$ .

b.

To determine

To find why $\overline{RV}$ is perpendicular to $\overline{TV}$

Answer to Problem 6WE

The product of slope of $\overline{RV}$ and $\overline{TV}$ is $-1$ so, $\overline{RV}\perp \overline{TV}$

Explanation of Solution

Given:

Below is the figure

  

Calculation:

Slope of a line when two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ are given is

  $m=\frac{y_2-y_1}{x_2-x_1}$

points are $R\left(-2,-6\right)\text{,V}\left(0,-8\right)\text{ and T}\left(5,-3\right)$

Slope of line of $\overline{RV}$

  $m_{RV}=\frac{-8-\left(-6\right)}{0-\left(-2\right)}=\frac{-2}{2}=-1$

Slope of line of $\overline{TV}$

  $m_{TV}=\frac{-8-\left(-3\right)}{0-5}=\frac{-5}{-5}=1$

Since the product of slope of $\overline{RV}$ and $\overline{TV}$ is $-1$

  $m_{RV}\times m_{TV}=-1\times 1=-1$

Therefore , $\overline{RV}$ is perpendicular to $\overline{TV}$ .

c.

To determine

To find why $▱RSTV$ is a rectangle.

Answer to Problem 6WE

It is proved that the $▱RSTV$ is a rectangle.

Explanation of Solution

Given:

Below is the figure

  

Calculation:

Slope of a line when two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ are given is

  $m=\frac{y_2-y_1}{x_2-x_1}$

points are $R\left(-2,-6\right)\text{,V}\left(0,-8\right)\text{ and T}\left(5,-3\right)$

also, from part (d) coordinates of point $S$ is $\left(3,-1\right)$ .

Slope of line of $\overline{RV}$

  $m_{RV}=\frac{-8-\left(-6\right)}{0-\left(-2\right)}=\frac{-2}{2}=-1$

Slope of line of $\overline{TV}$

  $m_{TV}=\frac{-8-\left(-3\right)}{0-5}=\frac{-5}{-5}=1$

Slope of line of $\overline{RS}$

  $m_{RS}=\frac{-1-\left(-6\right)}{3-\left(-2\right)}=\frac{5}{5}=1$

Slope of line of $\overline{ST}$

  $m_{ST}=\frac{-3-\left(-1\right)}{5-\left(3\right)}=\frac{-2}{2}=-1$

the product of slope of $\overline{RV}$ and $\overline{TV}$ is

  $m_{RV}\times m_{TV}=-1\times 1=-1$

the product of slope of $\overline{ST}$ and $\overline{TV}$ is

  $m_{ST}\times m_{TV}=-1\times 1=-1$

the product of slope of $\overline{RV}$ and $\overline{RS}$ is

  $m_{RV}\times m_{RS}=-1\times 1=-1$

the product of slope of $\overline{ST}$ and $\overline{RS}$ is

  $m_{ST}\times m_{RS}=-1\times 1=-1$

It means

  $\overline{RV}$ is perpendicular to $\overline{TV}$ .

  $\overline{ST}$ is perpendicular to $\overline{TV}$ .

  $\overline{RV}$ is perpendicular to $\overline{RS}$ .

  $\overline{ST}$ is perpendicular to $\overline{RS}$ .

Therefore, all the angles are of ${90}^{\circ }$

Hence, it is proved that the $▱RSTV$ is a rectangle.

d.

To determine

To find the coordinates of $S$ .

Answer to Problem 6WE

Coordinates of point $S$ is $\left(3,-1\right)$ .

Explanation of Solution

Given:

Below is the figure

  

Calculation:

Since, $▱RSTV$ is a rectangle and diagonals of rectangle bisect each other .

Let $O$ be the point where diagonals bisect each other.

Using midpoint formula

  $O$ is the midpoint of diagonal $RT$

  $O\left(x,y\right)=O\left(\frac{-2+5}{2},\frac{-6+\left(-3\right)}{2}\right)=O\left(\frac{3}{2},-\frac{9}{2}\right)$

  $O$ is the midpoint of diagonal $SV$

Let the coordinates of point $S$ be $\left(x_1,y_1\right)$ and $V\left(0,-8\right)$

Using midpoint formula

  $\begin{array}{l}\frac{3}{2}=\frac{x_1+0}{2}\\ x_1=3\end{array}$

  $\begin{array}{l}-\frac{9}{2}=\frac{y_1+\left(-8\right)}{2}\\ y_1=-9+8=-1\end{array}$

Therefore, coordinates of point $S$ is $\left(3,-1\right)$ .