Exercise 15 Page 57

To find the perimeter and area of the polygon, let's do each step separately.

Calculating the Perimeter

To determine the perimeter of the polygon, we must find the sum of its side lengths. This polygon has three vertices, so it is a triangle. 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 GE.

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

SubstitutePoints

Substitute ( -1,5) & ( 6,-2)

GE = sqrt(( 6-( -1))^2+( -2- 5)^2)

SubNeg

a-(- b)=a+b

GE=sqrt((6+1)^2+(-2-5)^2)

AddSubTerms

Add and subtract terms

GE=sqrt(7^2+(-7)^2)

CalcPow

Calculate power

GE=sqrt(49+49)

AddTerms

Add terms

GE=sqrt(98)

▼

Simplify right-hand side

SplitIntoFactors

Split into factors

GE=sqrt(49*2)

SqrtProd

sqrt(a* b)=sqrt(a)*sqrt(b)

GE=sqrt(49)*sqrt(2)

CalcRoot

Calculate root

GE=7sqrt(2)

We continue by calculating the length of the other two sides EF and FG.

Side	Coordinates	sqrt((x_2-x_1)^2+(y_2-y_1)^2)	Length
EF	( 6,-2) ( 6,5)	sqrt(( 6- 6)^2+( 5-( -2))^2)	7
FG	( 6,5) ( -1,5)	sqrt(( -1- 6)^2+( 5- 5)^2)	7

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

P=GE+EF+FG

SubstituteIII

Substitute GE= 7sqrt(2), EF= 7, FG= 7

P= 7sqrt(2)+ 7+ 7

UseCalc

Use a calculator

P= 23.899494...

RoundDec

Round to 2 decimal place(s)

P≈ 23.90

The triangle's perimeter is approximately 23.90 units.

Calculating the Area

Now, let's calculate the area of the triangle. Note that, because EF is vertical and FG is horizontal, they are perpendicular. Therefore, the given triangle is a right triangle.

To calculate the area of a triangle, we can use the following formula. A=1/2bh In this formula, b is the length of the base of the triangle and h is the height. Since we are given a right triangle, we can arbitrarily choose the base and the height to be either of the perpendicular sides. Let's use EF as the height and FG as the base. They are both 7 units long, so now we can calculate the area.

A=1/2bh

SubstituteII

b= 7, h= 7

A=1/2( 7)( 7)

Multiply

Multiply

A=1/2*49

MoveRightFacToNum

a/c* b = a* b/c

A=49/2

CalcQuot

Calculate quotient

A=24.5

The triangle's area is 24.5square units.