Chapter 13, Problem 26RP

(a)

To determine

To find: The quadrilateral’s area by using the encasement principle.

Answer to Problem 26RP

The area of the trapezoid is $60$ .

Explanation of Solution

Given:

The given points are $\left(3,1\right)\left(15,-3\right),\left(9,7\right)\left(5,7\right)$ .

Calculation:

Consider the given pointsof the trapezoid is $\left(3,1\right)\left(15,-3\right),\left(9,7\right)\left(5,7\right)$ .

From the given points the area of the trapezoid is of the form,

  $\begin{array}{c}A=A_{RECT}-A_I-A_{II}-A_{III}\\ =12\left(10\right)-\frac{6\left(2\right)}{2}-\frac{6\left(10\right)}{2}-\frac{4\left(12\right)}{2}\\ =120-6-30-24\\ =60\end{array}$

(b)

To determine

To find: The fact that is true about the diagonals.

Answer to Problem 26RP

The slope of both the diagonals shows that they are perpendicular to each other.

Explanation of Solution

Given:

The given points are $\left(3,1\right)\left(15,-3\right),\left(9,7\right)\left(5,7\right)$ .

Calculation:

The slope of the first diagonal is calculated as,

  $\begin{array}{c}\frac{7-1}{9-3}=\frac{6}{6}\\ =1\end{array}$

The slope of the second diagonal is calculated as,

  $\begin{array}{c}\frac{7-\left(-3\right)}{9-15}=\frac{10}{-10}\\ =-1\end{array}$

Thus, the slope of both the diagonals shows that they are perpendicular to each other.