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McGraw Hill Glencoe Geometry, 2012

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McGraw Hill Glencoe Geometry, 2012 View details

Extend: Geometry Lab, Equations of Perpendicular Bisectors

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Exercise 1 Page 206

The perpendicular bisector of a segment is the perpendicular through its midpoint.

y=- x+9

Practice makes perfect

We have to find the equation of the perpendicular bisector of the segment whose endpoints are P(5,2) and Q(7,4). We will do this in three steps.

Find the midpoint of PQ.
Find the slope of the perpendicular bisector.
Use the point-slope form to write the equation of the line.

Let's go for it!

Midpoint of PQ

A segment bisector contains the midpoint of the segment. We will use the Midpoint Formula to find the midpoint M of PQ. M( x_1+x_2/2,y_1+y_2/2 )To find the coordinates of the midpoint M, we will substitute (5,2) and (7,4) 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 ( 5,2) & ( 7,4)

M( 5+ 7/2,2+ 4/2 )

Add terms

M( 12/2, 6/2 )

Calculate quotient

M(6,3)

Therefore, M(6,3) is the midpoint of PQ.

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 PQ. To do so, we will use the Slope Formula. m = y_2-y_1/x_2-x_1 Let's substitute (5,2) and (7,4) for (x_1,y_1) and (x_2,y_2) in this formula.

m = y_2-y_1/x_2-x_1

SubstitutePoints

Substitute ( 5,2) & ( 7,4)

m=4- 2/7- 5

Subtract terms

m=2/2

Calculate quotient

m=1

The slope of PQ is 1. Let m_p be the slope of the perpendicular bisector. The product of the slopes of two perpendicular lines is - 1. 1 * m_p = - 1 ⇔ m_p = - 1 The slope of the perpendicular bisector of PQ is - 1.

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 - 1 for m and ( 6, 3) for (x_1,y_1) in the formula.

y-y_1=m(x-x_1)

SubstituteValues

Substitute values

y- 3= - 1(x- 6)

Distribute - 1

y-3=- x+6

LHS+3=RHS+3

y=- x+9

The equation of the perpendicular bisector of the segment whose endpoints are P(5,2) and Q(7,4) is y=- x+9.

Subchapter links

Exercises p.206