Exercise 8 Page 351

Let's begin by plotting the triangle in a coordinate plane using the given coordinates.

To find the location of the centroid, we need to recall two definitions.

1. The centroid describes the point of concurrency for the lines containing the medians of a triangle.
2. A median of a triangle is the segment from a vertex to the midpoint of the opposite side of a triangle.

Let's first determine the midpoint of each side. We can do that using the Midpoint Formula.

To find the coordinates of the centroid, we should determine the equations for two of the medians and solve the system of these equations. Let's use the medians EI and FG.

Equation of the Median EI

Since we are given endpoints of EI, we can find its slope using the Slope Formula.

m_(EI)=y_2-y_1/x_2-x_1

SubstitutePoints

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

m_(EI)=-5-( -2)/5- 2

▼

Simplify right-hand side

SubNeg

a-(- b)=a+b

m_(EI)=-5+2/5-2

AddSubTerms

Add and subtract terms

m_(EI)=-3/3

CalcQuot

Calculate quotient

m_(EI)=-1

We found that the slope of EI is -1. y=-1x+b ⇔ y=- x+b To complete the equation of the line, we need its y-intercept. Let's substitute the vertex E(2,-2).

y=- x+b

SubstituteII

x= 2, y= -2

-2=-2+b

AddEqn

LHS+2=RHS+2

0=b

RearrangeEqn

Rearrange equation

b=0

Since b=0, the equation of the line containing the median EI is y=- x.

Equation of the Median FG

Now, let's find the equation of FG in the same manner.

m_(FG)=y_2-y_1/x_2-x_1

SubstitutePoints

Substitute ( 8,-2) & ( 2,-5)

m_(FG)=-5-( -2))/2- 8

▼

Simplify right-hand side

SubNeg

a-(- b)=a+b

m_(FG)=-5+2/2-8

AddSubTerms

Add and subtract terms

m_(FG)=-3/-6

ReduceFrac

a/b=.a /-2./.b /-2.

m_(FG)=1/2

The slope of FG is 12. y=1/2x+b Let's substitute the vertex F(8,-2) to find the y-intercept.

y=1/2x+b

SubstituteII

x= 8, y= -2

-2=1/2( 8)+b

▼

Solve for b

MoveRightFacToNumOne

1/b* a = a/b

-2=8/2+b

CalcQuot

Calculate quotient

-2=4+b

SubEqn

LHS-4=RHS-4

-6=b

RearrangeEqn

Rearrange equation

b=-6

Therefore the equation of the median FG is y= 12x-6.

Finding the Centroid

We want to find the intersection of the two medians that we found. To do it, we need to solve the system of their equations. y=- x y=1/2x-6 We can solve it by substituting y=- x in the second equation.

y=- x & (I) y=1/2x-6 & (II)

Substitute

(II): y= - x

y=- x - x=1/2x-6

▼

(II): Solve for x

SubEqn

(II): LHS-1/2x=RHS-1/2x

y=- x - x-1/2x=-6

OneToFrac

(II): Rewrite 1 as 2/2

y=- x -2/2x-1/2x=-6

SubFrac

(II): Subtract fractions

y=- x -3/2x=-6

MultEqn

(II): LHS * 2=RHS* 2

y=- x -3 x=-12

DivEqn

(II): .LHS /(-3).=.RHS /(-3).

y=- x x=4

Substitute

(I): x= 4

y=- 4 x=4

Therefore, the coordinates of the centroid are (4,-4).