Exercise 52 Page 66

We have to graph and find the perimeter and area of the rectangle. Let's begin by finding the perimeter.

Perimeter

To determine the perimeter of the polygon, we must find the sum of its side lengths. From the exercise, we know that our polygon will be rectangle. Let's draw it in a coordinate plane.

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 AB.

AB = sqrt((x_2-x_1)^2 + (y_2-y_1)^2)

SubstitutePoints

Substitute ( -2, -3) & ( -2, -6)

AB = sqrt(( -2-( -2))^2 + ( -3-( -6))^2)

SubNeg

a-(- b)=a+b

AB=sqrt((-2+2)^2+(-3+6)^2)

AddSubTerms

Add and subtract terms

AB=sqrt(0^2+3^2)

CalcPow

Calculate power

AB=sqrt(9)

CalcRoot

Calculate root

AB=3

We continue by calculating the length of the other three sides BC, CD, and DA.

Side	Coordinates	sqrt((x_2-x_1)^2+(y_2-y_1)^2)	Length
BC	( -2, -3) ( 3, -3)	sqrt(( 3-( -2))^2+( - 3-( -3))^2)	5
CD	( 3, -3) ( 3,-6)	sqrt(( 3- 3)^2+( -6-( - 3))^2)	3
DA	( 3, -6) ( - 2,-6)	sqrt(( - 2- 3)^2+( -6-( - 6))^2)	5

Now, let's calculate the rectangle's perimeter. We do so by adding the three sides.

P=AB+BC+CD+DA

SubstituteValues

Substitute values

P=3+5+3+5

AddTerms

Add terms

P=16

The rectangle's perimeter is 16 units.

Area

Next, we will find the area. The area of a rectangle can be found by multiplying its base by its height. A= b h Here DA is the base, b and AB is the height h of the rectangle. Since we know that DA= 5 and AB= 3, we can put these values for b and h into the formula to find the area.

A=bh

SubstituteII

b= 5, h= 3

A=( 5)( 3)

Multiply

Multiply

A=15

The rectangle's area is 15 square units.