Chapter 4.1, Problem 43PPS

To determine

The type of the given triangle.

Answer to Problem 43PPS

Scalene triangle.

Explanation of Solution

Given:

The given vertices are:

  $\begin{array}{l}X\left(-5,9\right)\\ Y\left(2,1\right)\\ Z\left(-8,3\right)\end{array}$

Calculation:

The length of the sides will be:

  $\begin{array}{c}XY=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(2-\left(-5\right)\right)}^2+{\left(1-9\right)}^2}\\ =\sqrt{49+64}\\ =\sqrt{113}\\ YZ=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(-8-\left(2\right)\right)}^2+{\left(3-1\right)}^2}\\ =\sqrt{104}\\ =2\sqrt{26}\\ XZ=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}\\ =\sqrt{{\left(-8-\left(-5\right)\right)}^2+{\left(3-9\right)}^2}\\ =\sqrt{45}\\ =3\sqrt{5}\end{array}$

Equilateral triangle: All sides are equal.

Isosceles triangle: Two sides are equal.

Scalene triangle: All sides are different.

Hence, all the sides are different; the triangle XYZ is a scalene triangle.