Question

Using Slopes Verify the given geometric property.

Use slopes to verify $A\left(-3,-1\right),B\left(3,-3\right)$and $C\left(-9,8\right)$vertices of a right triangle or not.

Short answer

It is proved that $ABC$are not the vertices of a right triangle.

Step-by-step solution

Step 1. Concept of right triangle.

The two sides of a right triangle are perpendicular to each other that is their slopes are negative reciprocal,$\begin{array}{c}{m}_2=\frac{-1}{m_1}\\ m_1{m}_2=-1\end{array}$

Step 2. Evaluate the slope of each side.

The given points are$A\left(-3,-1\right),B\left(3,-3\right),C\left(-9,8\right)$.

The slope of localid="1646051777887" $AB=\frac{-3-(-1)}{3-(-3)}=\frac{-2}{6}=-\frac{1}{3}=m_1$.

The slope oflocalid="1646052327231" $BC=\frac{8-(-3)}{-9-3}=\frac{11}{-12}=-\frac{11}{12}.$

The slope of localid="1646051930005" $AC=\frac{8-(-1)}{-9-(-3)}=\frac{9}{-6}=-\frac{3}{2}=m_2$(say).

Step 3. Conclusion the answer.

Since, the slope of $AB=-\frac{1}{3}$and $AC=\frac{-3}{2}$and

$\begin{array}{c}{m}_1{m}_2=\frac{-1}{3}\times \frac{-3}{2}\\ \ne -1\end{array}$

Hence, $AB$and $AC$are not perpendicular to each other. Hence the given points are not the vertices of a right angled triangle.