Chapter 13.2, Problem 17PSB

To determine

To find: The slope of the line passing through the midpoints of $\overleftrightarrow{\text{AC}}$ and $\overleftrightarrow{\text{BC}}$ .

Answer to Problem 17PSB

The equation of the median from $\text{C}$ to $\overleftrightarrow{\text{AB}}$ is $y=\frac{1}{7}x+\frac{5}{7}$ .

Explanation of Solution

Given:

The given figure is as follows.

  

Calculation:

Find the mid-point of the line $\overleftrightarrow{\text{AC}}$ .

  $\begin{array}{c}\text{Midpoint}=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ =\left(\frac{2+4}{2},\frac{1+12}{2}\right)\\ =\left(3,\frac{13}{2}\right)\end{array}$

Find the mid-point of the line $\overleftrightarrow{\text{BC}}$ .

  $\begin{array}{c}\text{Midpoint}=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ =\left(\frac{4+16}{2},\frac{12+3}{2}\right)\\ =\left(10,\frac{15}{2}\right)\end{array}$

Find the slope.

Substitute $3$ for $x_1$ , $10$ for $x_2$ , $\frac{15}{2}$ for $y_2$ and $\frac{13}{2}$ for $y_1$ in the equation $m=\frac{y_2-y_1}{x_2-x_1}$ .

  $\begin{array}{c}m=\frac{\frac{15}{2}-\frac{13}{2}}{10-3}\\ =\frac{\frac{2}{2}}{7}\\ =\frac{1}{7}\end{array}$

Therefore, the slope of the line passing through the midpoints of $\overleftrightarrow{\text{AC}}$ and $\overleftrightarrow{\text{BC}}$ is $\frac{1}{7}$ .