Chapter 4.8, Problem 25PPS

To determine

The slope of each side of the triangle.

Whether the triangle is right triangle or not.

Answer to Problem 25PPS

  $\begin{array}{l}{\mathrm{m}}_{\mathrm{X}\mathrm{Y}}=1\\ {\mathrm{m}}_{\mathrm{Y}\mathrm{Z}}=-1\\ {\mathrm{m}}_{\mathrm{Z}\mathrm{X}}=0\end{array}$

Yes

Explanation of Solution

Given:

The given vertices are:

  $\begin{array}{l}\mathrm{X}\left(0,0\right)\\ \mathrm{Y}\left(2\mathrm{h},2\mathrm{h}\right)\\ \mathrm{Z}\left(4\mathrm{h},0\right)\end{array}$

Calculation:

The slope of line XY will be:

  $\begin{array}{c}{\mathrm{m}}_{\mathrm{X}\mathrm{Y}}=\frac{{\mathrm{y}}_2-{\mathrm{y}}_1}{{\mathrm{x}}_2-{\mathrm{x}}_1}\\ =\frac{2\mathrm{h}-0}{2\mathrm{h}-0}\\ =1\end{array}$

The slope of line YZ will be:

  $\begin{array}{c}{\mathrm{m}}_{\mathrm{Y}\mathrm{Z}}=\frac{{\mathrm{y}}_2-{\mathrm{y}}_1}{{\mathrm{x}}_2-{\mathrm{x}}_1}\\ =\frac{0-2\mathrm{h}}{4\mathrm{h}-2\mathrm{h}}\\ =-1\end{array}$

The slope of line ZX will be:

  $\begin{array}{c}{\mathrm{m}}_{\mathrm{Z}\mathrm{X}}=\frac{{\mathrm{y}}_2-{\mathrm{y}}_1}{{\mathrm{x}}_2-{\mathrm{x}}_1}\\ =\frac{0-0}{4\mathrm{h}-0}\\ =0\end{array}$

As $\left({\mathrm{m}}_{\mathrm{X}\mathrm{Y}}\right)\left({\mathrm{m}}_{\mathrm{Y}\mathrm{Z}}\right)=1\left(-1\right)=-1$

Therefore,

  $\begin{array}{l}\mathrm{X}\mathrm{Y}\perp \mathrm{Y}\mathrm{Z}\\ \text{that is,}\\ \angle \mathrm{X}\mathrm{Y}\mathrm{Z}\text{ is a right angle}\text{.}\end{array}$

Hence, the triangle formed is a right triangle.