Question

Using Slopes Verify the given geometric property.

Use slopes to show that$A\left(1,1\right),B\left(11,3\right),C\left(10,8\right)$ and $D\left(0,6\right)$are vertices of a rectangle.

Short answer

It is proved that$A,B,C,D$ are the vertices are of a rectangle.

Step-by-step solution

Step 1. Concept of rectangle.

A rectangle is a shape which adjacent sides are parallel to each other that means the slope of adjacent lines are equal and each sides are perpendicular.

Step 2. Evaluate the slope of each side.

The given points are $A\left(1,1\right),B\left(11,3\right),C\left(10,8\right)$and $D\left(0,6\right)$.

The slope of$AB=\frac{11-1}{3-1}=\frac{10}{2}=5$.

The slope of$BC=\frac{10-11}{8-3}=\frac{-1}{5}$.

The slope of$CD=\frac{0-10}{6-8}=\frac{-10}{-2}=5$.

The slope of $AD=\frac{0-1}{6-1}=\frac{-1}{5}$.

Step 3. Conclusion the answer.

Since, the slope of $AB$and $CD$are same, that is why they are parallel to each other and slope of $AB$is the negative reciprocal of $BC$and $CD$ slope of are negative reciprocal of$AD$ and$BC$ hence, the lines are perpendicular to each other hence the points are vertices of a rectangle.