Chapter 13, Problem 2CR

To determine

To calculate: To show that $\Delta XYZ$ is a right triangle using distance formula.

Answer to Problem 2CR

  $\Delta XYZ$ is a right triangle

Explanation of Solution

Given information: $X\left(-2,-4\right)$ , $Y(2,4)$ and $Z(2,-6)$ are three points

Formula used:

Distance formula: distance between $(x_1,y_1)$ and $(x_2,y_2)$ is given by $\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Pythagoras theorem : if $A$ , $B$ , $C$ are three side of a right triangle where $C$ is hypotonus then,

  $C^2=A^2+B^2$

Calculation:

Finding the length of the three-side using distance formula

  $\begin{array}{l}d(X,Y)=\sqrt{{\left(2-(-2)\right)}^2+{\left(4-(-4)\right)}^2}=\sqrt{16+64}=\sqrt{80}\\ d(Y,Z)=\sqrt{{\left(2-2\right)}^2+{\left((-6)-4\right)}^2}=\sqrt{0+100}=\sqrt{100}\\ d(Z,X)=\sqrt{{\left(2-(-2)\right)}^2+{\left((-6)-(-4)\right)}^2}=\sqrt{16+4}=\sqrt{20}\end{array}$

Now using Pythagoras theorem, we have

  ${\left(\sqrt{80}\right)}^2+{\left(\sqrt{20}\right)}^2=80+20=100={\left(\sqrt{100}\right)}^2$

From the above now we have $X{Y}^2+Z{X}^2=Y{Z}^2$

As the three side of triangle satisfy Pythagoras theorem, we can conclude that the triangle is a right triangle.