Chapter 7.6, Problem 39SPR

To determine

Give the explanation for the given statement using coordinate.

Explanation of Solution

Given:

The given statement is that the midpoints of two sides of a tringle is parallel to the third side.

Calculation:

Take a triangle $\Delta PQR$ .

  

The vertices of the triangle are $P\left(0,0\right),Q\left(4,0\right)$ and $R\left(2,4\right)$ .

Find the midpoint of the side $\overline{PR}$ .

  $S=\left(\frac{0+2}{2},\frac{0+4}{2}\right)=\left(1,2\right)$

Find the midpoint of the side $\overline{QR}$ .

  $T=\left(\frac{4+2}{2},\frac{0+4}{2}\right)=\left(3,2\right)$

Slope of the line segment $=\frac{y_2-y_1}{x_2-x_1}$

Find the slope of the mid segment $\overline{ST}$ .

  $x_1=1$ , $y_1=2$ , $x_2=3$ , $y_2=2$

Slope $=\frac{2-2}{3-1}=0$

Slope of the third line segment $\overline{PQ}$ .

  $x_1=0$ , $y_1=0$ , $x_2=4$ , $y_2=0$

Slope $=\frac{0-0}{4-0}=0$

Now,

Slope of $\overline{ST}$

  $=$ Slope of $\overline{PQ}$

Hence $\overline{PQ}\parallel \overline{ST}$ .