Question

Using Slopes Verify the given geometric property.

Use slopes to show that$A\left(1,1\right),B\left(7,4\right),C\left(5,10\right)$ and $D\left(-1,7\right)$are vertices of a parallelogram.

Short answer

It is proved that the given vertices are of a parallelogram.

Step-by-step solution

Step 1. Concept of Parallelogram.

A parallelogram is a quadrilateral with two pairs of parallel side hence, the slope of one pairs of side are equal.

Step 2. Evaluate the slope of each side.

The given points are $A\left(1,1\right),B\left(7,4\right),C\left(5,10\right)$and $D\left(-1,7\right)$.

The slope of$AB=\frac{7-1}{4-1}=\frac{6}{3}=2=m_1$ (say).

The slope of$BC=\frac{5-7}{10-4}=\frac{-2}{6}=\frac{-1}{3}=m_2$(say).

The slope of$CD=\frac{-1-5}{7-10}=\frac{-6}{-3}=2=m_2$(say).

The slope of $AD=\frac{-1-1}{7-1}=\frac{-2}{6}=\frac{-1}{3}=m_3$(say).

Step 3. Conclusion the answer.

Since, the slope of $AB$are equal to the slope of $CD$and slope of $BC$are equal to the slope of $AD$ hence,$AB$ is parallel to$CD$ and$BC$ are parallel to $AD$.Therefore$ABCD$ are the vertices of a parallelogram.