Question

Name the type of triangle formed by the points $A(-5,6)$, $B(-4,-2)$ and $C(7,5)$.$?$

Answer 1

Identifying the triangle formed.

Given points are $A(-5,6)$, $B(-4,-2)$ and $C(7,5)$

We know that the distance formula is,$d=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}$

Step $\mathbf{1}$ Finding the Distance between $AB$

$AB=\sqrt{{(-4-(-5\left)\right)}^2+{(-2-6)}^2}$

$=\sqrt{{(-4+5)}^2+{(-8)}^2}$

$=\sqrt{1+64}=\sqrt{65}.$

Step $\mathbf{2}$ Finding the Distance between $BC$

$BC=\sqrt{{(7-(-4\left)\right)}^2+{(5-(-2\left)\right)}^2}$

$=\sqrt{{(7+4)}^2+{(5+2)}^2}$

$=\sqrt{{\left(11\right)}^2+{\left(7\right)}^2}$

$=\sqrt{121+49}=\sqrt{170}$

Step $\mathbf{3}$ Finding the Distance between $CA$

$CA=\sqrt{{(7-(-5\left)\right)}^2+{(5-6)}^2}$

$=\sqrt{{(7+5)}^2+{(-1)}^2}$

$=\sqrt{{\left(12\right)}^2+{(-1)}^2}$

$=\sqrt{144+1}=\sqrt{145}$

All the sides are of different length,.$AB\ne BC\ne CA$

Hence, the triangle formed by the points $A(-5,6)$, $B(-4,-2)$ and $C(7,5)$ is scalene triangle.