Chapter 4.8, Problem 27PPS

To determine

To prove: The figure formed is a right triangle.

Explanation of Solution

Given:

The given vertices are:

  $\begin{array}{l}\mathrm{X}\left(0,0\right)\\ \mathrm{Y}\left(12,9\right)\\ \mathrm{Z}\left(0,25\right)\end{array}$

Calculation:

The figure can be drawn as:

  

The slope of line XY will be:

  $\begin{array}{c}{\mathrm{m}}_{\mathrm{X}\mathrm{Y}}=\frac{{\mathrm{y}}_2-{\mathrm{y}}_1}{{\mathrm{x}}_2-{\mathrm{x}}_1}\\ =\frac{9-0}{12-0}\\ =\frac{3}{4}\end{array}$

The slope of line YZ will be:

  $\begin{array}{c}{\mathrm{m}}_{\mathrm{Y}\mathrm{Z}}=\frac{{\mathrm{y}}_2-{\mathrm{y}}_1}{{\mathrm{x}}_2-{\mathrm{x}}_1}\\ =\frac{25-9}{0-12}\\ =-\frac{4}{3}\end{array}$

As $\left({\mathrm{m}}_{\mathrm{X}\mathrm{Y}}\right)\left({\mathrm{m}}_{\mathrm{Y}\mathrm{Z}}\right)=\frac{3}{4}\left(-\frac{4}{3}\right)=-1$

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.