Chapter 13, Problem 6RP

(a)

To determine

To find: The slope of the line $\overline{AC}$ .

Answer to Problem 6RP

The slope of the line is $\frac{4}{3}$ .

Explanation of Solution

Given:

The given diagram is shown in Figure 1

  

Figure 1

Calculation:

The formula to determine the slope of the line is of the form,

  $m=\frac{y_2-y_1}{x_2-x_1}$

Then, from the graph the slope of the line $\overline{AC}$ is,

  $\begin{array}{c}m=\frac{10-2}{4-\left(-2\right)}\\ =\frac{8}{6}\\ =\frac{4}{3}\end{array}$

(b)

To determine

To find: The midpoint of $\overline{AC}$ .

Answer to Problem 6RP

The midpoint of $\overline{AC}$ is $\left(1,6\right)$ .

Explanation of Solution

The formula to determine the mid-point of the line is of the form,

  $M=\left(\frac{x_2+x_1}{2},\frac{y_1+y_2}{2}\right)$

Then, from the figure shown in Figure 1

  $\begin{array}{c}M=\left(\frac{4+\left(-2\right)}{2},\frac{10+2}{2}\right)\\ =\left(1,6\right)\end{array}$

(c)

To determine

To find: The slope of the PQ.

Answer to Problem 6RP

The slope of the median from Bis $-\frac{3}{5}$ .

Explanation of Solution

The formula to determine the slope of the line is of the form,

  $m=\frac{y_2-y_1}{x_2-x_1}$

Then,from the figure 1 the slope of the median from B .

  $\begin{array}{c}m=\frac{3-6}{6-1}\\ =-\frac{3}{5}\end{array}$

(d)

To determine

To find: The length of the median from B .

Answer to Problem 6RP

The length of the median from Bis $\sqrt{34}$ .

Explanation of Solution

The distance between the two points in the coordinate plane is of the form,

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

Then, the length of the median is,

  $\begin{array}{c}PQ=\sqrt{{\left(6-1\right)}^2+{\left(3-6\right)}^2}\\ =\sqrt{34}\end{array}$

(e)

To determine

To find: The slope of the altitude from B is,

Answer to Problem 6RP

The slope of the altitude from B is $\frac{-3}{4}$ .

Explanation of Solution

The formula to determine the slope of the altitude from B is,

  $m_s=-\frac{1}{m}$

As the slope of the altitude is perpendicular to AC then,

  $\begin{array}{c}{m}_s=-\frac{1}{\frac{4}{3}}\\ =\frac{-3}{4}\end{array}$

(f)

To determine

To find: The slope of the line through A and parallel to BC.

Answer to Problem 6RP

The slope of the line is $\frac{1}{8}$ .

Explanation of Solution

The formula to determine the slope of the line is of the form,

  $m=\frac{y_2-y_1}{x_2-x_1}$

Then, from the figure 1 the slope of the line through A and parallel to BC.

  $\begin{array}{c}m=\frac{3-2}{6-\left(-2\right)}\\ =\frac{1}{8}\end{array}$

(g)

To determine

To find: The slope of the perpendicular bisector of AC.

Answer to Problem 6RP

The slope of the perpendicular bisector of AC is $\frac{-3}{4}$ .

Explanation of Solution

The formula to determine the slope of the altitude from B is,

  $m_s=-\frac{1}{m}$

As the slope of the perpendicular bisector of AC then,

  $\begin{array}{c}{m}_s=-\frac{1}{\frac{4}{3}}\\ =\frac{-3}{4}\end{array}$