Transcript 3.8 Slopes of Parallel and Perpendicular Lines

3.8 Slopes of Parallel
and Perpendicular
Lines
SOL G3b
Objectives: 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?
y  y 5  4
9


 3
l1 m 
x x
1  2  3
2
1
2
1
3  4
7
y y


l2 m 
x  x  3  1  2
2
1
2
1
Since the two slopes are not equal
7
Then the lines are not parallel
3
2
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?
 4 1
y y
26



l3 m 
3
x  x  1  13 12
2
1
2
1
76 1
y y


l4 m 
63 3
x x
2
1
2
1
Since the two slopes are not equal
1 1
 Then the lines are not parallel
3 3
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)
Point-slope form
y – 8 = -3(x – (-1)) Substitute -3 for m, 8
for y1 and -1 for x1
y – 8 = -3(x + 1)
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?
y  y  4 2 6  3


l1 m 
2
0  4
x x
4
2
1
2
1
y  y 3  3 6  2


l2 m 
x  x 4  5 9 3
2
1
2
1
Since the product of the two slopes
3 2
*  1
2 3
Equal -1 then the lines are perpendicular
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?
4
73
y y


l3 m 
2  1 3
x x
2
1
2
1
76
1
y y


l4 m 
x  x 8  2 10
2
1
2
1
Since the product of the two slopes do not equal -1
4 1 4
* 
3 10 30
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
1
y = x – 5 that contains point (15, -4)?
5
m = -5
Negative reciprocal
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
Example 8:
Writing Equations of Perpendicular Lines
What is an equation of the line perpendicular to y = -3x – 5
that contains point (-3, 7)?
1
m=
Negative reciprocal
3
y – y1 = m(x – x1)
Point-slope form
1
y – 7 = (x – (-3))
3
1
y - 7 = (x + 3)
3
1
Substitute
for m,
3
7 for y1 and -3 for x1

HW: pg 201 – 203
# 7 – 21 odd, 27, 29, 41