Chapter 13.3, Problem 9WE

To determine

To show $\Delta RST$ is a right triangle.

Answer to Problem 9WE

It is proved that $\Delta RST$ is a right triangle.

Explanation of Solution

Given:

Coordinates of the vertices of $\Delta RST$ are $R\left(-3,-4\right),S\left(2,2\right),T\left(14,-8\right)$

Calculation:

Slope of a line when two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ are given is

  $m=\frac{y_2-y_1}{x_2-x_1}$

points are $R\left(-3,-4\right),S\left(2,2\right),T\left(14,-8\right)$

  $m_{RS}=\frac{2-\left(-4\right)}{2-\left(-3\right)}=\frac{6}{5}$

  $m_{ST}=\frac{-8-\left(2\right)}{14-\left(2\right)}=\frac{-10}{12}=-\frac{5}{6}$

  $m_{TR}=\frac{-8-\left(-4\right)}{14-\left(-3\right)}=\frac{-4}{17}$

Since , product of slope of $RS$ and $ST$ is

  $\begin{array}{l}{m}_{RS}\times m_{ST}=\frac{6}{5}\times -\frac{5}{6}=-1\\ \therefore RS\perp ST\end{array}$

Hence it is proved that $\Delta RST$ is a right triangle.