Parallel and Perpendicular Lines

Parallel Lines //

All parallel lines have the same slope.Parallel lines will NEVER have the same

y-intercept.The slope of all vertical lines is

undefined. (No Slope)The slope of all horizontal lines is zero.

Perpendicular Lines Lines that form a 90° Angle.Perpendicular Lines CAN have the same

y-intercept IF that is where they cross.Perpendicular Lines have slopes that are

negative reciprocals.– This means to change the sign and flip the

slope.

Ex. If line “m” has a slope of 5, then it’s negative reciprocal is

1

5

You try it!!IF line “p” has a

slope of -2, then a line to it has a slope of ……

For line “n” the

slope is

the slope is...

1

2

REMEMBERChange the sign

AndFlip it over.

1

3

3

1

3

Let’s compare Vertical and Horizontal Lines.

Vertical lines are ┴ to horizontal lines.

AND

Horizontal lines are ┴ to vertical lines.

0m

m undefined

Examples

Name the slope of each line, thenGive the PARALLEL slope and thePERPENDICULAR slope.

Equation m // m m

y = 3x + 5

7x + y = 4

y = 2

x = -4

3 31

3

7 4y x 7 7 1

7

0 0 .Undef

.Undef .Undef 0

Why do we need to be able to identify the Parallel & Perpendicular Slopes?

So that we can write equations for new lines.– Either lines that are Parallel– OR lines that are Perpendicular

HOW?– 1. Name the slope of the line you are given.– 2. List the new slope.– 3. Use the new slope and the point you are

given in the slope-intercept formula to write a new equation.

Example 5

3 4y x 5. 3m

Write an equation that is PARALLEL to the given line passing through the given point.

// 3m; (1,5)

y mx b ( , )x y

5 ( 3) (1) b3 b533b8

y mx b y

New //Equation 3x 8

Write an equation that is PARALLEL to the given line passing through the given point.

6. 6 4x y ; ( 2,3)To get the

Slope, solveFor “y”

6x 6x6 4y x

6m // 6m•Find the PRGM key on your calculator.

•Select program ASLOPE

•Which option?

•#2 because you have a point and a slope.

•Enter NEW (parallel) slope

•Enter X and Y from your ordered pair

6 9y x

Parallel LinesHave SAME

Slope (m)

But…

DIFFERENTY-int.(b)

Example 6

7. x = 5; (3, 4)

Choose program ASLOPEOption #2– Name the slope

• Undefined – No number value – so…..

– Name the “x” coordinate in the ordered pair.

Parallel LinesHave SAME

Slope (m)

Both are Undefined

But…

DIFFERENTY-int.(b)

No y-int, but different “x”

8. y = 3x – 2; (6, -1)Choose program ASLOPEOption #2

– Name the slope of this line but do not type it in.• m = 3

• What is perpendicular to 3?

• - 1/3

– type this one in because you are looking for a perpendicular equation.

– Enter the X and Y from the ordered pair.

Write an equation that is PERPENDICULAR to the given line passing through the given point.

11

3y x

Perpendicular LinesHave

OPPOSITESlope (m)

AND….

DIFFERENTY-int.(b)

Write an equation that is PERPENDICULAR to the given line passing through the given point.

9. 5 2 6x y ; (4, 2)To get the

Slope, solveFor “y”

5x 5x

53

2y x 5

2m 2

5m

•Find the PRGM key on your calculator.

•Select program ASLOPE

•Which option?

•#2 because you have a point and a slope.

•Enter NEW (perpendicular) slope

•Enter X and Y from your ordered pair2 2

5 5y x

Perpendicular LinesHave

OPPOSITESlopes (m)

AND….

DIFFERENTY-int.(b)

2 2 22 5 6y x

Example 9

10. y = 8; (-2, 8)Choose program ASLOPEOption #2– Name the slope

• ZERO – but don’t enter it yet.– What is perpendicular to ZERO?

• Undefined – has no number value so…

– Name the “x” coordinate in the ordered pair.

PerpendicularLinesHave

OPPOSITESlopes (m)

AND…

DIFFERENTY-int.(b)

No y-int, but “x”-int.

Example 10