Exercise 56 Page 433

We have to find the equation of the perpendicular bisector of the segment whose endpoints are D(- 2,- 4) and E(2,1). We will do this in three steps.

1. Find the midpoint of DE.
2. Find the slope of the perpendicular bisector.
3. Use the point-slope form to write the equation of the line.

Let's go for it!

Midpoint of DE

A segment bisector contains the midpoint of the segment. We will use the Midpoint Formula to find the midpoint M of DE. M( x_1+x_2/2,y_1+y_2/2 )To find the coordinates of the midpoint M, we will substitute (- 2,- 4) and (2,1) for (x_1,y_1) and (x_2,y_2) in the formula.

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

SubstitutePoints

Substitute ( - 2,- 4) & ( 2,1)

M( - 2+ 2/2,- 4+ 1/2 )

AddTerms

Add terms

M( 0/2, - 3/2 )

CalcQuot

Calculate quotient

M(0, - 3/2 )

MoveNegNumToFrac

Put minus sign in front of fraction

M(0, - 3/2 )

Therefore, M(0,- 32 ) is the midpoint of DE.

Slope of the Perpendicular Bisector

A perpendicular bisector is perpendicular to the segment through the midpoint. In order to find the slope of the bisector, we will first find the slope of DE. To do so, we will use the Slope Formula. m = y_2-y_1/x_2-x_1 Let's substitute (- 2,- 4) and (2,1) for (x_1,y_1) and (x_2,y_2) in this formula.

m = y_2-y_1/x_2-x_1

SubstitutePoints

Substitute ( - 2,- 4) & ( 2,1)

m=1-( - 4)/2-( - 2)

SubNeg

a-(- b)=a+b

m=5/4

The slope of DE is 54. Let m_p be the slope of the perpendicular bisector. The product of the slopes of two perpendicular lines is - 1. 5/4 * m_p = - 1 ⇔ m_p = - 4/5 The slope of the perpendicular bisector of DE is - 45.

Equation of the Perpendicular Bisector

Since we know a point and the slope of the bisector, we will use the point-slope form of a line to write its equation. y-y_1=m(x-x_1) Let's substitute - 45 for m and ( 0, - 32 ) for (x_1,y_1) in the formula.

y-y_1=m(x-x_1)

SubstituteValues

Substitute values

y-( - 3/2) = - 4/5(x- 0 )

▼

Solve for y

SubNeg

a-(- b)=a+b

y+3/2=- 4/5(x-0)

Distr

Distribute - 4/5

y+3/2=- 4/5x+0

IdPropAdd

Identity Property of Addition

y+3/2=- 4/5x

SubEqn

LHS-3/2=RHS-3/2

y=- 4/5x-3/2

The equation of the perpendicular bisector of the segment whose endpoints are D(- 2,- 4) and E(2,1) is y=- 45x- 32.