Q. 42 - Geometry

Question

Discover and prove two things about the triangle with vertices K(-3, 4), M(3, 1), and J(-6, -2)

Step-by-Step Solution

Answer

Triangle KMJ is an isosceles triangle.

Step-1 – Given

Given that the triangle has vertices $K\left(-3,4\right),M\left(3,1\right)\text{ and }J\left(-6,-2\right)$.

Step-2 – To determine

We have to discover and prove something about the triangle.

Step-3 – Calculation

We’ll use the distance formula to find the lengths of$KM,KJ\text{ and }MJ$ .

The distance formula is given by:$$\begin{gathered} d=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\left[d=\text{distance}\right] \\ \text{So} \end{gathered}$$

$\begin{array}{l}KM=\sqrt{{\left(3-\left(-3\right)\right)}^2+{\left(1-4\right)}^2}\left[\text{since, }K\left(-3,4\right)\text{ and }M\left(3,1\right)\right]\\ KM=\sqrt{{\left(3+3\right)}^2+{\left(1-4\right)}^2}\\ KM=\sqrt{{\left(6\right)}^2+{\left(-3\right)}^2}\\ KM=\sqrt{36+9}\\ KM=\sqrt{45}\\ KM=3\sqrt{5}\end{array}$

$\begin{array}{l}KJ=\sqrt{{\left(-6-\left(-3\right)\right)}^2+{\left(-2-4\right)}^2}\left[\text{since, }K\left(-3,4\right)\text{ and }J\left(-6,-2\right)\right]\\ KJ=\sqrt{{\left(-6+3\right)}^2+{\left(-2-4\right)}^2}\\ KJ=\sqrt{{\left(-3\right)}^2+{\left(-6\right)}^2}\\ KJ=\sqrt{9+36}\\ KJ=\sqrt{45}\\ KJ=3\sqrt{5}\end{array}$

$\begin{array}{l}MJ=\sqrt{{\left(-6-3\right)}^2+{\left(-2-1\right)}^2}\left[\text{since, }M\left(3,1\right)\text{ and }J\left(-6,-2\right)\right]\\ MJ=\sqrt{{\left(-9\right)}^2+{\left(-3\right)}^2}\\ MJ=\sqrt{81+9}\\ MJ=\sqrt{90}\\ MJ=3\sqrt{10}\end{array}$

We see that the pairs of two sides are of equal lengths but the length of one side is different.

It means, triangle KMJ is an isosceles triangle.