Exercise 16 Page 343

Let's start by labeling the coordinates of the vertices of the given triangle.

To find the circumcenter, we need equations for the perpendicular bisectors of at least two sides of the triangle. Recall that a bisector cuts something in half, so we want perpendicular lines that perfectly divide the side of the triangle in half.

Finding Perpendicular Bisectors

By the Slopes of Perpendicular Lines Theorem, we know that horizontal and vertical lines are perpendicular. Since DF is horizontal, any perpendicular line will be vertical. Examining the diagram, we can also identify the midpoints of this side.

Given the information, we know that the perpendicular bisector through DF has the equation x=4. To find the other perpendicular bisector, we need to find the equation of the line perpendicular to DE, passing through the midpoint between the D and E. This will be a four-step process.

1. Find the slope of the line DE,
2. Find the slope of the line perpendicular to DE,
3. Find the midpoint between D and E,
4. Find the equation of the perpendicular bisector.

Finding the Slope of DE

To find the slope of DE, let's use the Slope Formula. m = y_2 - y_1/x_2 - x_1 The (x_1, y_1) and (x_2, y_2) are the coordinates of two points lying on DE. Since we know the coordinates of D and E, we can use them to find the slope.

m = y_2 - y_1/x_2 - x_1

SubstitutePoints

Substitute ( 5,-1) & ( -1,3)

m=3-( -1)/-1- 5

SubNeg

a-(- b)=a+b

m=3+1/-1-5

AddSubTerms

Add and subtract terms

m=4/-6

MoveNegDenomToFrac

Put minus sign in front of fraction

m=-4/6

ReduceFrac

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

m=-2/3

The slope is - 23.

Finding the Slope of the Line Perpendicular to DE

By The Slopes of Perpendicular Lines Theorem, the slope of any line perpendicular to DE will be the negative reciprocal of the slope of DE, that is, - 23.

m_1 m_2 = -1

Substitute

m_1= -2/3

-2/3m_2 = -1

MoveRightFacToNum

a/c* b = a* b/c

-2m_2/3=-1

MultEqn

LHS * (-1)=RHS* (-1)

2m_2/3=1

MultEqn

LHS * 3=RHS* 3

2m_2=3

DivEqn

.LHS /2.=.RHS /2.

m_2 = 3/2

The slope of the perpendicular line is 32. This allows us to write the following partial formula for the perpendicular bisector of DE. y = 3/2x + b To find the value of b we need to find the coordinates of one point from the bisector. Since the midpoint between D and E must lie on the perpendicular bisector of DE, this will be the point we will find.

Finding The Midpoint Between D and E

To find the midpoint between D and E we will use the Midpoint Formula. M( x_1 + x_2/2, y_1 + y_2/2 ) Let's substitute the coordinates of D and E into this formula.

M ( x_1 + x_2/2, y_1 + y_2/2 )

SubstitutePoints

Substitute ( 5,-1) & ( -1,3)

M( 5+( -1)/2, -1+ 3/2)

AddNeg

a+(- b)=a-b

M(5-1/2,-1+3/2)

AddSubTerms

Add and subtract terms

M(4/2,2/2)

CalcQuot

Calculate quotient

M(2,1)

The midpoint of DE is (2,1).

Finding the Equation of the Perpendicular Bisector

The last step in finding the equation of the perpendicular bisector is to substitute the found coordinates for x and y into the partial formula for the bisector to find the value of b.

y=3/2x+b

SubstituteII

x= 2, y= 1

1=3/2( 2)+b

FracMultDenomToNumber

a/2* 2 = a

1=3+b

SubEqn

LHS-3=RHS-3

-2=b

RearrangeEqn

Rearrange equation

b=-2

Now we are able to write the full formula of the perpendicular bisector. y=3/2x + ( -2) ⇔ y=3/2x-2 Having found the equation of the perpendicular bisector DE, we can graph it on the same graph we used for the perpendicular bisector of EF. Note that this line should also pass through (0, -2), because b= -2 is the y-value of the y-intercept of our line.

Finding the Circumcenter

The triangle's circumcenter is the point at which the perpendicular bisectors intersect.

The circumcenter is located at (4,4).