Chapter 2.4, Problem 56PPSE

Solution

Prove that the quadrilateral with given vertices is a rectangle.

Given:

Vertices are $\left(2,5\right),\left(4,8\right),\left(7,6\right),\left(5,3\right)$ .

Formula Used:

Slope of a line is given by $m=\frac{y_2-y_1}{x_2-x_1}$ .

Slopes of parallel lines are equal.

Slopes of perpendicular lines are negative reciprocal of each other.

The opposite sides of a rectangle are parallel and each interior angle is 90 degrees.

Calculation:

The given vertices of the triangle are $\left(2,5\right),\left(4,8\right),\left(7,6\right),\left(5,3\right)$ .

The slope of the line joining the points $\left(2,5\right),\left(4,8\right)$ is

  $\begin{array}{c}{m}_1=\frac{8-5}{4-2}\\ =\frac{3}{2}\end{array}$

Now, the slope of the line with points $\left(4,8\right),\left(7,6\right)$ is

  $\begin{array}{c}{m}_2=\frac{6-8}{7-4}\\ =-\frac{2}{3}\end{array}$

Since, these two slopes are negative reciprocal to each other. Hence, the line joining these points are perpendicular to each other.

Now, the slope of the line with points $\left(7,6\right),\left(5,3\right)$ is

  $\begin{array}{c}{m}_3=\frac{3-6}{5-7}\\ =\frac{3}{2}\end{array}$

Now, the slope of the line with points $\left(2,5\right),\left(5,3\right)$ is

  $\begin{array}{c}{m}_4=\frac{3-5}{5-2}\\ =-\frac{2}{3}\end{array}$

Slopes $m_1\text{and }{m}_3$ are equal. Moreover, slopes $m_4\text{and }{m}_2$ are equal

Hence, the lines joining these points are parallel to each other.

Therefore, we can see that these points satisfying the properties of a rectangle.

Hence, we can conclude that these vertices are of a rectangle.