Chapter 12.1, Problem 10E

To determine

To find: whether then coordinates form right angle triangle and classify the triangle by its side.

Answer to Problem 10E

It is right scalene triangle.

Explanation of Solution

Given:

  $A\left(-2,3\right),B\left(0,-3\right),C\left(3,-2\right)$ .

Concept used:

Sum of the three angles of triangle is always ${180}^0$ .

The distance formula:

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

Slope of two points form is:

  $m=\frac{y_2-y_1}{x_2-x_1}.$

If two lines are perpendicular to each other than the slope is:

  $m_1\times m_2=-1$ .

Calculation:

A triangle is classified by its sides as scalene, isosceles or equilateral triangle.

Using the three vertices of the triangle $ABC$ .

  $A\left(-2,3\right),B\left(0,-3\right),C\left(3,-2\right)$ .

Using the distance formula:

From distance formula its cleared that the triangle $ABC$ is a scalene triangle.

A triangle is a right triangle if two of its sides are perpendicular, forming a right angle.

Using the slope formula, calculate the slope of $\overline{AB}$ .

For two line to be perpendicular to each other the product od slope of both the line should be $-1.$

  $\begin{array}{l}{m}_1\times m_2=-1\\ \overline{AB}\perp \overline{AC}\end{array}$

Perpendicular liens form a right angle.

If a triangle contains a right angle then its right triangle.

Hence, it is right scalene triangle.