Chapter 8.1, Problem 40E

Solution

To write: an equation of the line containing the median that joins the first vertex to the side opposite to it.

  $y=-x+8$

Given information:

The ordered pairs given below represent the vertices of a triangle:

  $\left(2,6\right),\left(3,1\right),\left(7,5\right)$

Formula Used:

The Midpoint Formula

The midpoint formula shown below, gives the midpoint of the line segment joining the points $A\left(x_1,y_1\right)$ and $B\left(x_2,y_2\right)$ .

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

  

Slope of the line segment

The slope of the line segment joining the points $A\left(x_1,y_1\right)$ and $B\left(x_2,y_2\right)$ is:

  $m=\frac{y_2-y_1}{x_2-x_1}$

Equation of the line

The equation of a line passing through $\left(x_0,y_0\right)$ and having the slope m is:

  $y-y_0=m\left(x-x_0\right)$

Explanation:

A median of a triangle is a line segment joining a vertex and the midpoint of the opposite side.

So, first step is to find the midpoint between the two opposite vertices $\left(3,1\right)$ and $\left(7,5\right)$ .

By the midpoint formula,

  $\begin{array}{c}M=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ =\left(\frac{3+7}{2},\frac{1+5}{2}\right)\\ =\left(\frac{10}{2},\frac{6}{2}\right)\\ =\left(5,3\right)\end{array}$

Next step is to find the slope that goes through the above midpoint and the first vertex $\left(2,6\right)$ .

  $\begin{array}{c}m=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{6-3}{2-5}\\ =-\frac{3}{3}\\ =-1\end{array}$

Then, the equation of the line that passes through the midpoint $\left(5,3\right)$ and has the slope $-1$ is:

  $\begin{array}{c}y-y_0=m\left(x-x_0\right)\\ y-3=-1\left(x-5\right)\\ y-3=-x+5\\ y=-x+5+3\\ y=-x+8\end{array}$