6-6:PARALLELANDPERPENDICULARLINESLessonObjectives:

• Determinewhetherlinesareparallelorperpendicular• Relateslopetoperpendicularityandparallelism.

PROPERTY:SlopesofParallelLinesNon-verticallinesareparalleliftheyhavethesameslopeanddifferenty-intercepts.Anytwoverticallinesareparallel.EXAMPLE1:DeterminingWhetherLinesareParallel1.Arethegraphsof𝑦 = −!

!𝑥 + 5and2𝑥 + 6𝑦 = 12parallel?Explain.2.Arethegraphsof−6𝑥 + 8𝑦 = −24and𝑦 = !

!𝑥 − 7parallel?Explain.

Inthegraphattheright,thetwolinesareparallel.PARALLELLINESarelinesinthesameplanethatneverintersect.Theequationofthetoplineisy=½x+1.Theequationofthebottomlineisy=½x–3.

Youcanusethefactthattheslopesofparallellinesarethesametowritetheequationofalineparalleltoagivenline.Towritetheequation,youusetheslopeofthegivenlineandthepoint-slopeformofalinearequation.EXAMPLE2:WritingEquationsofParallelLines3.Writeanequationforthelinethatcontains(5,1)andisparallelto𝑦 = !

!𝑥 − 4.4.Writeanequationforthelinethatisparalleltothegivenlineandthatpassesthroughthegivenpoint.a) 3,4 ; 𝑦 = 2𝑥 − 7 b) 1,3 ; 𝑦 = 4𝑥 + 5 c) −8,−4 ; 𝑦 = −!

!𝑥 + 55.Writeanequationforthelinethatisparalleltothegivenlineandthatpassesthroughthegivenpoint.

a) b)

PROPERTY:SlopesofPerpendicularLinesTwolinesareperpendiculariftheproductoftheirslopesis-1.ThatistheyareOPPOSITERECIPROCALSofeachother.Averticalandahorizontallinearealsoperpendiculartoeachother.EXAMPLE3:WritingEquationsforPerpendicularLines6.Writetheequationforthelinethatcontains(0,-2)andisperpendicularto𝑦 = 5𝑥 + 3.7.Writeanequationforthelinethatisperpendiculartothegivenlineandthatpassesthroughthegivenpoint.a) 6,4 ; 𝑦 = 3𝑥 − 2 b) −5,5 ; 𝑦 = −5𝑥 + 9 c) −1,−4 ; 𝑦 = !

!𝑥 + 1

Thelinesattherightareperpendicular.PERPENDICULARLINESarelinesthatintersecttoformrightangles.Theequationsofthesetwolinesare𝑦 = !!!𝑥 − 1and𝑦 = 4𝑥 + 2.

8.Writeanequationforthelinethatisperpendiculartothegivenlineandthatpassesthroughthegivenpoint.

a) b)

9.Tellwhetherthelinesforeachpairofequationsareparallel,perpendicular,orneither.

a) b) c)

10.AbikepathforanewcityparkwillconnecttheparkentrancetoParkRoad.ThepathwillbeperpendiculartoParkRoad.Writeanequationforthelinerepresentingthebikepath.

Lesson 6-6 Parallel and Perpendicular Lines 345

You can use the negative reciprocal of the slope of a given line to write anequation of a line perpendicular to that line.

Writing Equations for Perpendicular Lines

Multiple Choice Which equation describes a line that contains (0,-2) and isperpendicular to y = 5x + 3?

y = 5x - 2 y = y = y =

Step 1 Identify the slope of Step 2 Find the negative reciprocal the given line. of the slope.

y = 5x + 3 The negative reciprocal of 5 is .c

slope

Step 3 Use the slope-intercept form to write an equation.

y = mx + b

y = x + (-2) Substitute for m, and –2 for b.

y = x - 2 Simplify.

The equation is y = - 2. So D is the correct answer.

Write an equation of the line that contains (1, 8) and is perpendicular to y = + 1.

You can use equations of parallel and perpendicular lines to solve some real-world problems.

Urban Planning A bike path for a new city park will connect the park entrance toPark Road. The path will be perpendicular to Park Road. Write an equation for theline representing the bike path.

Step 1 Find the slope m of Park Road.

m = = = = 2 Points (2, 1) and (4, 5) are on Park Road.

Step 2 Find the negative reciprocal of the slope.

The negative reciprocal of 2 is So the slope of the bike path is The y-intercept is 4.

The equation for the bike path is y = + 4.

A second bike path is planned. It will be parallel to Park Road and will alsocontain the park entrance. Write an equation for the line representing this bike path. y ≠ 2x ± 4

44Quick Check2

12 x

2 12 .2

12 .

42

5 2 14 2 2

y2 2 y1x2 2 x1

!2 2 4

4

2

O

y

x

new pathperpendicularto Park Road

y-intercept

(2, 1)

(4, 5)

!2 2 4

4

2

O

y

xPark

Roa

d

parkentrance

Park

EXAMPLEEXAMPLE Real-World Problem Solving44

34 x33Quick Check

2 15 x

2 15

2 152

15

2 15

2 15x 2 21

5x 1 215x 2 2

EXAMPLEEXAMPLE33

ConnectionReal-World

Careers Urban planners usemathematics to makedecisions on communityproblems such as trafficcongestion and air pollution.

y ≠ – x ± 913

43

Test-Taking Tip

1 A B C D E

2 A B C D E

3 A B C D E

4 A B C D E

5 A B C D E

B C D E

After finding theslopes of theperpendicular lines,multiply the twoslopes as a check.

5(215) 5 21

345

Error Prevention

Students may think that anegative reciprocal is alwaysnegative. Tell students that the phrase negative reciprocalactually means the negative ofthe reciprocal. So they are reallyjust finding the opposite of thereciprocal.

Additional Examples

Find an equation of the line that contains (6, 2) and isperpendicular to y = -2x + 7. y ≠ x – 1

The line in the graph representsthe street in front of a new house.The point is the front door. Thesidewalk from the front door willbe perpendicular to the street.Write an equation representingthe sidewalk. y ≠ x – 3

Resources• Daily Notetaking Guide 6-6• Daily Notetaking Guide 6-6—

Adapted Instruction

Closure

Ask: Compare the equations of non-vertical parallel lines.Parallel lines have the sameslope, but different y-intercepts.Compare the equations ofperpendicular lines.Perpendicular lines have slopesthat are negative reciprocals ofeach other. They will have thesame y-intercept only if that iswhere the two lines intersect.

L1

L3

O

4

2

!4

!2 2 6x

y

frontdoor

street

32

44

12

33

EXAMPLEEXAMPLE33

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