Chapter 13, Problem 17RP

(a)

To determine

To find: The length of the median from C to AB .

Answer to Problem 17RP

The length of the median is $2\sqrt{5}$ .

Explanation of Solution

Given:

The coordinates of the triangle $\Delta ABC$ are $A\left(2,3\right),B\left(12,5\right)$ and $C\left(9,8\right)$ .

Calculation:

The median of the point A and Bis ,

  $\begin{array}{c}M=\left(\frac{2+12}{2},\frac{5+3}{2}\right)\\ =\left(7,4\right)\end{array}$

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 from AB to C is,

  $\begin{array}{c}AB=\sqrt{{\left(9-7\right)}^2+{\left(8-4\right)}^2}\\ =\sqrt{20}\\ =2\sqrt{5}\end{array}$

(b)

To determine

To find: The equation of the median to AB.

Answer to Problem 17RP

The equation of the median is $2y-4x=-20$ .

Explanation of Solution

Given:

The coordinates of the triangle $\Delta ABC$ are $A\left(2,3\right),B\left(12,5\right)$ and $C\left(9,8\right)$ .

Calculation:

The general form for the equation of the line is,

  $y-y_1=\frac{y_2}{x_2}\left(x-x_2\right)$

Then, the equation of the median to AB is,

  $\begin{array}{l}y-7=\frac{4}{2}\left(x-7\right)\\ 2y-4x=-20\end{array}$

(b)

To determine

To find: The equation of the median to AB.

Answer to Problem 17RP

The equation of the median is $2y-4x=-20$ .

Explanation of Solution

Given:

The coordinates of the triangle $\Delta ABC$ are $A\left(2,3\right),B\left(12,5\right)$ and $C\left(9,8\right)$ .

Calculation:

The general form for the equation of the line is,

  $y-y_1=\frac{y_2}{x_2}\left(x-x_2\right)$

Then, the equation of the median to AB is,

  $\begin{array}{l}y-7=\frac{4}{2}\left(x-7\right)\\ 2y-4x=-20\end{array}$

(c)

To determine

To find: The equation of the perpendicular bisector of AB in the point slope form.

Answer to Problem 17RP

The equation of the perpendicular bisector is $y-4=-5\left(x-7\right)$ .

Explanation of Solution

Given:

The coordinates of the triangle $\Delta ABC$ are and $C\left(9,8\right)$ .

Calculation:

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

  $m=\frac{y_2-y_1}{x_2-x_1}$

Consider the slope of the line AB is,

  $\begin{array}{c}m=\frac{5-3}{12-2}\\ =\frac{1}{5}\end{array}$

Then, the slope of the perpendicular bisector of AB is,

  $m=-5$

The general form for the equation of the line is,

  $y-y_1=m\left(x-x_2\right)$

Then, the equation for the perpendicular bisector of AB is,

  $y-4=-5\left(x-7\right)$

(d)

To determine

To find: The point slope form for the equation of the line that contains C and parallel to AB.

Answer to Problem 17RP

The equation of the perpendicular bisector is $y-8=-5\left(x-9\right)$ .

Explanation of Solution

Given:

The coordinates of the triangle $\Delta ABC$ are $A\left(2,3\right),B\left(12,5\right)$ and $C\left(9,8\right)$ .

Calculation:

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

  $m=\frac{y_2-y_1}{x_2-x_1}$

Consider the slope of the line AB is,

  $\begin{array}{c}m=\frac{5-3}{12-2}\\ =\frac{1}{5}\end{array}$

The general form for the equation of the line is,

  $y-y_1=m\left(x-x_2\right)$

Then, the point slope form for the equation of the line that contains C and parallel to AB is,

  $y-8=-5\left(x-9\right)$

(e)

To determine

To find: The point slope form for the equation of the line that containing C and parallel to AB.

Answer to Problem 17RP

The equation of the perpendicular bisector is $y-8=\frac{1}{5}\left(x-9\right)$ .

Explanation of Solution

Given:

The coordinates of the triangle $\Delta ABC$ are $A\left(2,3\right),B\left(12,5\right)$ and $C\left(9,8\right)$ .

Calculation:

The general form for the equation of the line is,

  $y-y_1=m\left(x-x_2\right)$

Then, the point slope form for the equation of the line that containing C and parallel to AB is,$y-8=\frac{1}{5}\left(x-9\right)$