Exercise 3 Page 64

Let's find the perimeter and the area of the figure one at a time.

Perimeter

To determine the perimeter of the polygon, we must find the sum of its side lengths. This polygon has four vertices, so it is a $\text{quadrilateral}.$ Let's denote the vertexes of this polygon.

Before we can find the sum of the side lengths, we must find the length of each side. We can use the Distance Formula to do this. Let's start with $\overline{AD}.$ We continue by calculating the length of the other three sides $\overline{AB},$ $\overline{BC},$ and $\overline{CD}.$

Side	Coordinates	$\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$	Length
$\overline{AB}$	$(\text{-}3,1)$ $(3,1)$	$\sqrt{(3-(\text{-}3){)}^2+(1-1{)}^2}$	$6$
$\overline{BC}$	$(3,1)$ $(3,\text{-}1)$	$\sqrt{(3-3{)}^2+(\text{-}1-1{)}^2}$	$2$
$\overline{CD}$	$(3,\text{-}1)$ $(\text{-}1,\text{-}1)$	$\sqrt{(\text{-}1-3{)}^2+(\text{-}1-(\text{-}1){)}^2}$	$4$

Now, let's calculate the quadrilateral's perimeter. We do so by adding the four sides. The quadrilateral's perimeter is equal to $12+\sqrt{2}$ units.

Area

Now, we can find the area of our quadrilateral. To do it, let's divide it into a triangle and a rectangle.

The area of our $\text{quadrilateral}$ is the sum of areas of the $\text{triangle}$ and the $\text{rectangle}.$ Let's begin by calculating the area of a triangle. To do so, we will use the formula for area of a triangle. Here, $AE$ is the base $b$ of our triangle and $ED$ is the height $h.$ We can use the Distance Formula again to find these lengths.

Side	Coordinates	$\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}$	Length
$\overline{AE}$	$(\text{-}3,1)$ $(\text{-}1,1)$	$\sqrt{(\text{-}1-(\text{-}3){)}^2+(1-1{)}^2}$	$2$
$\overline{ED}$	$(\text{-}1,1)$ $(\text{-}1,\text{-}1)$	$\sqrt{(\text{-}1-(\text{-}1){)}^2+(\text{-}1-1{)}^2}$	$2$

By substituting these lengths into our formula, we can calculate the area. The area of the triangle is $2\text{ square units}.$ Next, we will find the area of the rectangle. We can do it by multiplying its base by its height. Here, $CD$ is the base $b$ and $ED$ is the height $h.$ We already know these lengths, so we can substitute them into the formula to find the area. The area of the triangle is $8\text{ square units}.$ Now, we can add the areas of the $\text{triangle}$ and the $\text{rectangle},$ to find the area of the $\text{quadrilateral}.$ The quadrilateral's area is $10\text{ square units}.$