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Lines, Lines, Lines, Lines, Lines!!! Lines!!! ~ ~ Parallel & Parallel & Perpendicular Lines Perpendicular Lines

Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines

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Page 1: Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines

Lines, Lines, Lines!!!Lines, Lines, Lines!!!~~

Parallel & Perpendicular Parallel & Perpendicular LinesLines

Page 2: Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines

What we Have Learned What we Have Learned so far….so far….

m = Slope

Where lower case m represents the slope for all

linear equations

Page 3: Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines

What we Have Learned What we Have Learned so far….so far….

Slope Intercept Form

bmxy

Where m is the slope and b is the intercept

Page 4: Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines

What we Have Learned What we Have Learned so far….so far….

Point-Slope Form

11 xxmyy

Where m is the slope and are a point on the line. 11,yx

Page 5: Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines

We will use our prior knowledge of We will use our prior knowledge of

SlopesSlopes& &

Slope-Intercept FormSlope-Intercept Form

To learn about To learn about ParallelParallel and and PerpendicularPerpendicular Lines Lines

Page 6: Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines

Parallel LinesParallel Lines

What are Parallel Lines?What are Parallel Lines?

Two lines with the Two lines with the Same SlopeSame Slope are said to be Parallel lines. are said to be Parallel lines.

When two Parallel Lines are graph they will When two Parallel Lines are graph they will NeverNever intersect. intersect.

We can decide algebraically if two lines are Parallel by finding We can decide algebraically if two lines are Parallel by finding the slope of each line and seeing if the the slope of each line and seeing if the Slopes are EqualSlopes are Equal to to each other.each other.

Page 7: Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines

Testing if Lines are ParallelTesting if Lines are Parallel

Are the lines y = 3x + 2 and 9x – 3y = -6 parallel?

First find the slope of y = 3x + 2

The slope m = 3

Second find the slope of 9x – 3y = - 6

The slope m = 3

Since the slopes are equal the lines are parallel.

9x – 9x – 3y = -9x – 6

-3y = -9x – 6

y = 3x + 2

Page 8: Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines

Example:Example:Bellow is the graph of two Parallel Bellow is the graph of two Parallel

Lines.Lines.

The red line is the graph of y = – 4x – 3 and the blue line is the graph ofy = – 4x – 7

Because the red line and the blue line have the same slope, they will NEVER intercept. Therefore they are PARALLEL LINES.

Page 9: Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines

Slope and Parallel LinesSlope and Parallel LinesIf two non-vertical lines are If two non-vertical lines are parallelparallel, then they have the , then they have the same same

slopeslope..

Write an equation of the line passing through (-3, 2) and parallel to the line whose equation is y = 2x + 1. Express the equation in point-slope form and y-intercept form.

y = 2x + 1

-5 -4 -3 -2 -1 1 2 3 4 5

5

4

3

2

1

-1-2

-3

-4-5

(-3, 2)

Rise = 2

Run = 1y – y1 = m(x – x1)

y1 = 2 m = 2 x1 = -3

Because the slope of the given line is 2, m = 2 for the new equation.

y – 2 = 2[x – (-3)]

y – 2 = 2(x + 3)

y – 2 = 2x + 6 Apply the distributive property.

y = 2x + 8

Add 2 to both sides. + 2 + 2

This is the slope-intercept form of the equation.

Page 10: Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines

Your turn!Your turn!Practice Testing if Lines are Practice Testing if Lines are

ParallelParallelAre the lines 6 3 5 and 2 4 4x y y x parallel?

6 3 5

3 6 5

52 32

x y

y x

y x

m

2 4 4

2 2

2

y x

y x

m

Since the slopes are differentthe lines are not parallel.

Are the lines 2 4 and 2 4 12x y x y parallel?

2 4

2 4

1 221

2

x y

y x

y x

m

2 4 12

4 2 12

1 321

2

x y

y x

y x

m

Since the slopes are equalthe lines are parallel.

Page 11: Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines

Your Turn!Your Turn!Practice Constructing Parallel LinesPractice Constructing Parallel Lines

Find the equation of the line going through the point (4,1) and parallel to 3 7y x

1 3 4

1 3 12

3 13

y x

y x

y x

Find the equation of the line going through the point (-2,7) and parallel to 2 8x y

7 2 2

7 2 2

7 2 4

2 3

y x

y x

y x

y x

Page 12: Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines

Perpendicular LinesPerpendicular LinesWhat are Perpendicular Lines?What are Perpendicular Lines?

Perpendicular lines are lines that intersect Perpendicular lines are lines that intersect forming a right angle (90forming a right angle (90˚̊))..

Perpendicular lines are lines that have Perpendicular lines are lines that have exact opposite slopesexact opposite slopes. .

We can decide algebraically if two lines are perpendicular by finding the slope We can decide algebraically if two lines are perpendicular by finding the slope of each line and seeing if the slopes are negative reciprocals of each other. of each line and seeing if the slopes are negative reciprocals of each other. This is equivalent to multiplying the two slopes together and seeing if their This is equivalent to multiplying the two slopes together and seeing if their product is –1. product is –1.

If one line has the slope ‘m’ If one line has the slope ‘m’

then a perpendicular line to that line will have a slope = then a perpendicular line to that line will have a slope = ..

Page 13: Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines

Slope and Perpendicular Slope and Perpendicular LinesLines

Slope and Perpendicular Lines• If two non-vertical lines are perpendicular,

then the product of their slopes is –1.(5/2) • (-2/5) = -1

• Slopes are negative reciprocals of each other

Slope and Perpendicular Lines• If two non-vertical lines are perpendicular,

then the product of their slopes is –1.(5/2) • (-2/5) = -1

• Slopes are negative reciprocals of each other

90°

Find the slope of any line that is perpendicular to the line whose equation is 2x + 4y – 4 = 0.

-2x + 4 -2x + 4 To isolate the y-term, subtract 2x and add 4 on both sides.

Slope is –1/2.

y = -1/2x + 1The given line has slope –1/2.

4 4

Any line perpendicular to this line has a slope that is the negative reciprocal, 2.

4y = -2x + 4Divide both sides by 4.

Page 14: Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines

The red line is the graph of y = – 2x + 5

and the blue line is the graph ofy = – 1/2 x +4

Example:Example:Bellow is the graph of two Bellow is the graph of two

Perpendicular Lines.Perpendicular Lines.

Because the red line and the blue line have the exact opposite slope, they will always intercept and form a 90˚ angle. Therefore the lines are PERPENDICULAR.

Page 15: Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines

Testing if Lines Are Testing if Lines Are PerpendicularPerpendicular

Find the slope of 2 5 2

2 5

x y m

y x

1 1Find the slope of 4

2 2y x m

Since the slopes are negative reciprocals of each other the lines are perpendicular. 1

2 12

Are the lines 2x + y = 5 and y = 1/2x + 4 perpendicular?

Page 16: Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines

A Point? A Line?Write an equation of the line passing through (-3,6) and perpendicular to the line whose equation is y=1/3 x +4 Express in point-slope form and slope-intercept form.

perpendicular slope: 3

1 3

y y1 m(x x1)

y 6 3(x ( 3))y 6 3x 9

6 6

y 3x 3

Page 17: Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines

Your Turn!Your Turn!Practice Testing if Lines Are Practice Testing if Lines Are

PerpendicularPerpendicularAre the lines 6 3 5 and 2 4 4 perpendicular?x y y x

6 3 5

3 6 5

52 32

x y

y x

y x

m

2 4 4

2 2

2

y x

y x

m

Since the slopes are not negative reciprocals of each other (their product is not -1) the lines are not perpendicular

Are the lines 2 4 and 4 2 6 perpendicular?x y x y 2 4

2 4

1 221

2

x y

y x

y x

m

4 2 6

2 4 6

2 3

2

x y

y x

y x

m

Since the slopes are negative reciprocals of each other (their product is -1) the lines are perpendicular.

Page 18: Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines

Your Turn!Your Turn!Constructing Perpendicular LinesConstructing Perpendicular Lines

Find the equation of a line going through the point (3, -5) and perpendicular to 2 83y x

The slope of the perpendicular line will be m = 3/2 Using

the point-slope equation where the slope m = 3/2 and

the point is (3, -5) we get 35 323 95 2 2

3 192 2

y x

y x

y x

Page 19: Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines

Your Turn!Your Turn!Practice Constructing Practice Constructing Perpendicular LinesPerpendicular Lines

Find the equation of the line going through the point (4,1) and perpendicular to 3 7y x

11 431 41 3 31 1

3 3

y x

y x

y x

Find the equation of the line going through the point (-2,7) and perpendicular to 2 8x y

17 2217 2217 121 82

y x

y x

y x

y x

Page 20: Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines

Student ActivityStudent ActivityYou will now receive a You will now receive a worksheet. Turn the worksheet. Turn the worksheet in when worksheet in when

completed.completed.

Page 21: Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines Lines, Lines, Lines!!! ~ Parallel & Perpendicular Lines

Do Not DisturbDo Not DisturbWork In ProgressWork In Progress