Exercise 60 Page 887

Let's identify the coordinates of the given parallelogram.

First we will determine the most precise name for our parallelogram. Then we can find its area.

Name

To determine the most precise name for our parallelogram, let's review the definitions of the different types.

Parallelogram	Definition
Rhombus	Parallelogram with four congruent sides
Rectangle	Parallelogram with four right angles
Square	Parallelogram with four congruent sides and four right angles

Now, let's find the slopes of the sides using the Slope Formula.

Side	Slope Formula	Simplified
Slope of $\overline{WX}\text{:}$ $(0,0),$ $(4,0)$	$\frac{0-0}{4-0}$	$0$
Slope of $\overline{XY}\text{:}$ $(4,0),$ $(5,5)$	$\frac{5-0}{5-4}$	$5$
Slope of $\overline{YZ}\text{:}$ $(5,5),$ $(1,5)$	$\frac{5-(5)}{1-5}$	$0$
Slope of $\overline{ZW}\text{:}$ $(1,5),$ $(0,0)$	$\frac{0-5}{0-1}$	$5$

We can also tell that the consecutive side are not perpendicular, as their slopes are not opposite reciprocals. Therefore, our parallelogram is either a typical parallelogram or a rhombus. To check, we can find the lengths of its sides using the Distance Formula.

Side	Distance Formula	Simplified
Length of $WX\text{:}$ $(0,0),$ $(4,0)$	$\sqrt{(4-0{)}^2+(0-0{)}^2}$	$4$
Length of $XY\text{:}$ $(4,0),$ $(5,5)$	$\sqrt{(5-4{)}^2+(5-0{)}^2}$	$\sqrt{26}$
Length of $YZ\text{:}$ $(5,5),$ $(1,5)$	$\sqrt{(1-5{)}^2+(5-5{)}^2}$	$4$
Length of $ZW\text{:}$ $(1,5),$ $(0,0)$	$\sqrt{(0-1{)}^2+(0-5{)}^2}$	$\sqrt{26}$

Our parallelogram has two pairs of congruent sides. Therefore, the most precise name for this quadrilateral is a parallelogram.

Area

Since the given figure is typical parallelogram, we can use the formula for area of a parallelogram. If we choose the horizontal $\overline{WX}$ as the base of the parallelogram, then the vertical segment through $Z$ gives the height. By counting squares we can find the height of the parallelogram, which is $h=5.$ From previous calculations we already know that $b=WX=4.$ Let's substitute the values into the formula.