Exercise 30 Page 161

To find the midpoint, we can use the Midpoint Formula. M(x_1+x_2/2,y_1+y_2/2) Then we can use the midpoint to write the equation of the perpendicular bisector.

Finding the Midpoint

The coordinates of the given endpoints are P(- 7,0) and Q(1,8). Let's use these to find the coordinates of the midpoint.

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

SubstitutePoints

Substitute ( - 7,0) & ( 1,8)

M(- 7+ 1/2,0+ 8/2)

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Simplify

AddTerms

Add terms

M(- 6/2,8/2)

MoveNegNumToFrac

Put minus sign in front of fraction

M(-6/2,8/2)

CalcQuot

Calculate quotient

M(- 3,4)

The midpoint of PQ is at M(- 3,4). Let's see what this looks like on a coordinate plane.

Finding the Perpendicular Line

To find the line that runs perpendicular to PQ, we first need to find the slope of the segment. We can do this by substituting P(- 7,0) and Q(1,8) into the Slope Formula.

m=y_2-y_1/x_2-x_1

SubstitutePoints

Substitute ( - 7,0) & ( 1,8)

m=8- 0/1-( - 7)

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Simplify right-hand side

SubNeg

a-(- b)=a+b

m=8-0/1+7

AddSubTerms

Add and subtract terms

m=8/8

QuotOne

a/a=1

m=1

Perpendicular lines have opposite reciprocal slopes. With this information, we know that all lines that are perpendicular to our given line will have a slope of - 1. y=- 1x+b ⇔ y=- x+b By substituting the midpoint M( - 3, 4) into this equation for x and y, we can solve for the y-intercept b of the perpendicular line.

y=- x+b

SubstituteII

x= - 3, y= 4

4=-( - 3)+b

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Solve for b

NegNeg

- (- a)=a

4=3+b

SubEqn

LHS-3=RHS-3

1=b