Chapter 13.3, Problem 5WE

a.

To determine

To find the slope of $\overline{LM}$ and $\overline{PN}$

Answer to Problem 5WE

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 $L\left(-3,-1\right)\text{,M}\left(-4,2\right),N\left(2,4\right)\text{ and P}\left(4,-2\right)$

Slope of line of $\overline{LM}$

  $m_{LM}=\frac{2-\left(-1\right)}{-4-\left(-3\right)}=\frac{3}{-1}=-3$

Slope of line of $\overline{PN}$

  $m_{PN}=\frac{4-\left(-2\right)}{2-\left(4\right)}=\frac{6}{-2}=-3$

Therefore, the slope of line $\overline{LM}$ and $\overline{PN}$ is $-3$ .

b.

To determine

To find why $\overline{LM}\Vert \overline{PN}$ ?

Answer to Problem 5WE

the slope of $\overline{LM}$ and $\overline{PN}$ is $-3$ so, lines are parallel.

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 $L\left(-3,-1\right)\text{,M}\left(-4,2\right),N\left(2,4\right)\text{ and P}\left(4,-2\right)$

Slope of line of $\overline{LM}$

  $m_{LM}=\frac{2-\left(-1\right)}{-4-\left(-3\right)}=\frac{3}{-1}=-3$

Slope of line of $\overline{PN}$

  $m_{PN}=\frac{4-\left(-2\right)}{2-\left(4\right)}=\frac{6}{-2}=-3$

Since, the slope of $\overline{LM}$ and $\overline{PN}$ is $-3$ so, lines are parallel.

c.

To determine

To find the slope of $\overline{MN}$ and $\overline{LP}$

Answer to Problem 5WE

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 $L\left(-3,-1\right)\text{,M}\left(-4,2\right),N\left(2,4\right)\text{ and P}\left(4,-2\right)$

Slope of line of $\overline{MN}$

  $m_{MN}=\frac{4-\left(2\right)}{2-\left(-4\right)}=\frac{2}{6}=\frac{1}{3}$

Slope of line of $\overline{LP}$

  $m_{LP}=\frac{-2-\left(-1\right)}{4-\left(-3\right)}=\frac{-1}{7}$

Therefore, the slope of line $\overline{MN}$ is $\frac{1}{3}$ and $\overline{LP}$ is $-\frac{1}{7}$ .

d.

To determine

To find why $\overline{MN}$ is not parallel to $\overline{LP}$ ?

Answer to Problem 5WE

The slope of $\overline{MN}$ and $\overline{LP}$ are not equal so, lines are not parallel.

Explanation of Solution

Given:

Below is the figure

  

Calculation:

Slope of line of $\overline{MN}$ is $\frac{1}{3}$

Slope of line of $\overline{LP}$ is $-\frac{1}{7}$

Since, the slope of $\overline{MN}$ and $\overline{LP}$ are not equal so, lines are not parallel.

e.

To determine

To tell what special kind of quadrilateral is $LMNP$ .

Answer to Problem 5WE

The quadrilateral $LMNP$ is trapezium.

Explanation of Solution

Given:

Below is the figure

  

Calculation:

Since,

the slope of line $\overline{LM}$ and $\overline{PN}$ are equal , so line $\overline{LM}$ and $\overline{PN}$ are parallel

the slope of $\overline{MN}$ and $\overline{LP}$ are not equal so, lines $\overline{MN}$ and $\overline{LP}$ are not parallel.

One pair of opposite sides are parallel to each other therefore quadrilateral $LMNP$ is trapezium.