Chapter 13.2, Problem 24PSC

To determine

To show: That the medians of $\Delta ABC$ is intersects at $\left(2,2\right)$ .

Answer to Problem 24PSC

It is proved that medians of $\Delta ABC$ is intersects at $\left(2,2\right)$ .

Explanation of Solution

Given:

The points are $A=\left(0,0\right)$ , $B=\left(4,0\right)$ and $C=\left(2,6\right)$ .

Calculation:

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

  $\begin{array}{c}\text{Midpoint}=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ =\left(\frac{4+2}{2},\frac{0+6}{2}\right)\\ =\left(3,3\right)\end{array}$

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

  $\begin{array}{c}\text{Midpoint}=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ =\left(\frac{0+2}{2},\frac{0+6}{2}\right)\\ =\left(1,3\right)\end{array}$

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

  $\begin{array}{c}\text{Midpoint}=\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)\\ =\left(\frac{0+4}{2},\frac{0+0}{2}\right)\\ =\left(2,0\right)\end{array}$

Point-slope formula is given by $m=\frac{y_2-y_1}{x_2-x_1}$ .

Find the slope of line $\overleftrightarrow{\text{BC}}$ .

  $\begin{array}{c}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{6-0}{2-4}\\ =-3\end{array}$

Find the slope of line $\overleftrightarrow{\text{AC}}$ .

  $\begin{array}{c}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{6-0}{2-0}\\ =3\end{array}$

Find the slope of line $\overleftrightarrow{\text{AB}}$ .

  $\begin{array}{c}m=\frac{y_2-y_1}{x_2-x_1}\\ m=\frac{0-0}{4-0}\\ =0\end{array}$

The equation medians will be as follows.

  $\begin{array}{c}y=x\\ y=-x+4\\ x=2\end{array}$

Substitute $2$ for $x$ and 2 for $y$ in the equation $y=x$ .

  $2=2$

Substitute $2$ for $x$ and 2 for $y$ in the equation $y=-x+4$ .

  $\begin{array}{c}2=-2+4\\ 2=2\end{array}$

Substitute $2$ for $x$ and 2 for $y$ in the equation $x=2$ .

  $2=2$

It can be seen that the point $\left(2,2\right)$ is the solution of all the three equations.

Therefore, it is proved that medians of $\Delta ABC$ is intersects at $\left(2,2\right)$ .