3.8 Slopes of Parallel and Perpendicular Lines

SOL G3bObjectives: TSW …• Relate slope to perpendicular and parallel lines.• Applying slope to verify and determine whether lines

are parallel or perpendicular• Write equations of lines that are perpendicular or

parallel to each other.

Key Concepts: Slopes of Parallel Lines

If two nonvertical lines are parallel, then their slopes are equal.

If the slope of two distinct nonvertical lines are equal, then the lines are parallel.

Any two vertical lines or horizontal lines are parallel.

Example 1: Check for parallel lines.

Are lines l1 and l2 parallel? Explain?

12

12

xx

yym

2145

39

3l1

12

12

xx

yym

13

43

27

l2

Since the two slopes are not equal

27

3

Then the lines are not parallel

Example 2: Check for parallel lines.

Are lines l3 contains A(-13, 6) and B(-1, 2). Line l4 contains C(3, 6) and D(6, 7). Are l3 and l4 parallel? Explain?

12

12

xx

yym

131

62

124l3

12

12

xx

yym

36

67

31l4

Since the two slopes are not equal

31

31

Then the lines are not parallel

31

Writing Equations of Parallel Lines

Identify the slope of the given line. Since the lines are parallel; the

slopes are the same (equal) You know a point and the slope for the

new line. Use point-slope form to write the equation.

Example 3: Writing Equations of Parallel Lines

What is an equation of the line parallel to

y = -3x – 5 that contains point (-1, 8)?

m = -3

y – y1 = m(x – x1)

y – 8 = -3(x – (-1))

y – 8 = -3(x + 1)

Point-slope form

Substitute -3 for m, 8 for y1 and -1 for x1

Example 4: Writing Equations of Parallel Lines

What is an equation of the line parallel to

y = -x – 7 that contains point (-5, 3)?

m = -1

y – y1 = m(x – x1)

y – 3 = - (x – (-5))

y – 3 = -(x + 5)

Point-slope form

Substitute -1 for m, 3 for y1 and -5 for x1

Key Concepts: Slopes of Perpendicular Lines

If two nonvertical lines are perpendicular, then the product of their slopes is -1. (negative reciprocals)

If the slopes of two lines have a product of -1, then the lines are perpendicular.

Any horizontal line and vertical line are perpendicular.

Example 5: Check for parallel lines.

Lines l1 and l2 are neither horizontal nor vertical? Are they perpendicular? Explain?

12

12

xx

yym

40

24

46l1

12

12

xx

yym

54

33

96l2

Since the product of the two slopes

132

*23

Equal -1 then the lines are perpendicular

23

3

2

Example 6: Check for perpendicular lines.

Are lines l3 contains A(2, 7) and B(3, -1). Line l4 contains C(-2, 6) and D(8, 7). Are l3 and l4 parallel? Explain?

12

12

xx

yym

1237

34l3

12

12

xx

yym

28

67

101l4

Since the product of the two slopes do not equal -1

304

101

*34 Then the lines are not perpendicular

Writing Equations of Perpendicular Lines

Identify the slope of the given line. Recall perpendicular have a product of -1. Negative Reciprocals.

You know a point and the slope for the new line. Use point-slope form to write the equation.

Example 7: Writing Equations of Perpendicular Lines

What is an equation of the line perpendicular to

y = x – 5 that contains point (15, -4)?

m = -5

y – y1 = m(x – x1)

y – (-4) = -5(x – 15)

y + 4 = -5(x - 15)

Point-slope form

Substitute -5 for m, -4 for y1 and 15 for x1

51

Negative reciprocal

Example 8: Writing Equations of Perpendicular Lines

What is an equation of the line perpendicular to y = -3x – 5 that contains point (-3, 7)?

31

m =

y – y1 = m(x – x1) Point-slope form

Substitute for m,

7 for y1 and -3 for x1

Negative reciprocal

y – 7 = (x – (-3))31

y - 7 = (x + 3)31

31

HW: pg 201 – 203

# 7 – 21 odd, 27, 29, 41