Chapter 13, Problem 18CT

To determine

To show: The triangle JKL is an isosceles triangle.

Explanation of Solution

Given information: The points are $J\left(-12,0\right)$ , $K\left(0,6\right)$ and $L\left(-3,-3\right)$ .

Formula used: The distance between the two points $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$ is given by:

  $\text{distance}=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$

Calculation:

Use distance formula to calculate the length of three sides of the triangle $JK$ , $KL$ and $LJ$ .

The length of side $JK$ is:

  $\begin{array}{l}JK=\sqrt{{\left(0-\left(-12\right)\right)}^2+{\left(6-0\right)}^2}\\ =\sqrt{144+36}\\ =\sqrt{180}\text{ units}\end{array}$

The length of side $KL$ is:

  $\begin{array}{l}KL=\sqrt{{\left(0-\left(-3\right)\right)}^2+{\left(6-\left(-3\right)\right)}^2}\\ =\sqrt{9+81}\\ =\sqrt{90}\text{ units}\end{array}$

The length of side $LJ$ is:

  $\begin{array}{l}LJ=\sqrt{{\left(\left(-3\right)-\left(-12\right)\right)}^2+{\left(\left(-3\right)-0\right)}^2}\\ =\sqrt{9^2+9}\\ =\sqrt{81+9}\\ =\sqrt{90}\text{ units}\end{array}$

Since, the length of two sides are equal that is $KL=LJ$ .

Hence, it is proved that $\Delta JKL$ is an isosceles triangle.