MATHPOWERTM 10, WESTERN EDITION

Chapter 6 Coordinate Geometry6.7

6.7.1

Parallel Lines

A(-3, 0)

B(0, 5)

C(0, -5)

D(3, 0)

mAB =5−00−−3

mCD =0−−53−0

mCD =53

mAB =53

If the slopes of two lines areequal, the lines are parallel.

If two lines are parallel, their slopes are equal.

AB is parallel to CD.6.7.2

Show that the line segment AB with endpoints A(2, 3) and B(6, 5) is parallel to the line segment CD with endpoints C(-1, 4) and D(3, 6).

m=y2 −y1

x2 −x1

mAB =5−36−2

mAB =12

mCD =6−43−−1

mCD =12

Since the slopes are equal, the line segments are parallel.6.7.3

Verifying Parallel Lines

The following are slopes of parallel lines. Find the value of k.

a) 23,

4k

b) -15

, 2k

c) -k5

, 32 d)

-k3

, -27

23

=4k

2k = 12 k = 6

−15

=2k

-1k = 10 k = -10

−k5

=32

-2k = 15 −k3

=−27

-7k = -6

6.7.4

Using Parallel Slopes to Find k

−152

k = 67

k =

Perpendicular Lines

A(-2, -2)

B(4, 2)

C(3, -2)

D(-1, 4)mAB =

2−−24−−2

mCD =4−− 2−1−3

mCD =−32

mAB =23

If the slopes of two linesare negative reciprocals, the lines are perpendicular.

If two lines are perpendicular, their slopes are negative reciprocals.

AB is perpendicular to CD.

6.7.5

Show that the line segment AB with endpoints A(0, 2) and B(-3, -4) is perpendicular to the line segment CDwith endpoints C(2, -4) and D(-8, 1).

m=y2 −y1

x2 −x1

mAB =−4−2−3−0

mAB =2

mCD =1−−4−8−2

mCD =−12

The slopes are equal so line segments are perpendicular.6.7.6

Perpendicular Line Segments

The following are slopes of perpendicular lines. Find the value of k.

a) 23

, 4k

b) -15

, 2k

c) -k5

, 32

d) -k3

, -27

23

=−k4

-3k = 8 −15

=−k2

-5k = -2

−k5

=−23

-3k = -10 −k3

=72

-2k = 21

6.7.7

Using Perpendicular Slopes to Find k

k = −83

k =25

k =103

k =−212

Given the following equations of lines, determine which are parallel and which are perpendicular.

A) 3x + 4y - 24 = 0 B) 3x - 4y + 10 = 0

C) 4x + 3y - 16 = 0 D) 6x + 8y + 15 = 0

4y = -3x + 24

y = x + 6

-4y = -3x - 10

y = x + 5/2

3y = -4x + 16 8y = -6x - 15

Lines A and D have the same slope, so they are parallel.Lines B and C have negative reciprocal slopes, so they areperpendicular. 6.7.8

−34

34

y=−43x+

163

y=−34x−

158

Slope = −34

Slope =34

Slope = −43 Slope = −

34

Parallel and Perpendicular Lines

Find the equation of the line through the point A(-1, 5) and parallel to 3x - 4y + 16 = 0.

Find the slope.

3x - 4y + 16 = 0 -4y = - 3x - 16

4y - 20 = 3(x + 1)4y - 20 = 3x + 30 = 3x - 4y + 23

3x - 4y + 23 = 0

6.7.9

Writing the Equation of a Line

y = x + 434

Slope =34

y - y1 = m(x - x1)

y - 5 = (x - -1)34

Find the equation of the line through the point A(-1, 5) and perpendicular to 3x - 4y + 16 = 0.

Find the slope.3x - 4y + 16 = 0 -4y = -3x - 16

3y - 15 = -4(x + 1)3y - 15 = -4x - 44x + 3y - 11 = 0

4x + 3y - 11 = 0Therefore, use the slope

6.7.10

Writing the Equation of a Line

Slope =34

y = x + 434

y - y1 = m(x - x1)

y - 5 = (x - -1)−43

−43

.

Determine the equation of the line parallel to 3x + 6y - 9 = 0and with the same y-intercept as 4x + 4y - 16 = 0.

3x + 6y - 9 = 06y = -3x + 9

4x + 4y - 16 = 0For the y-intercept, x = 0:4(0) + 4y - 16 = 0 4y = 16 y = 4

A point is (0, 4).

2y - 8 = -1xx + 2y - 8 = 0

6.7.11

Writing the Equation of a Line

y=−12x+

32

.The slope is−12

y - y1 = m(x - x1)

y - 4 = (x - 0)−12

.

Determine the equation of the line that is perpendicular to 3x + 6y - 9 = 0 and has the same x-intercept as 4x + 4y - 16 = 0.

3x + 6y - 9 = 06y = -3x + 9

The slope is 2.

4x + 4y - 16 = 0

For the x-intercept, y = 0:4x + 4(0)- 16 = 0 4x = 16 x = 4

A point is (4, 0).y - y1 = m(x - x1) y - 0 = 2(x - 4) y = 2x - 8 0 = 2x - y - 8 The equation of the

line is 2x - y - 8 = 0.6.7.12

y=−12x+

32

Writing the Equation of a Line

Determine the equation of each of the following lines.

A) perpendicular to 5x - y - 1 = 0 and passing through (4, -2)

B) perpendicular to 2x - y - 3 = 0 and intersects the y-axis at -2

C) parallel to 2x + 5y + 10 = 0 and same x-intercept as 4x + 8 = 0

D) passing through the point (3, 6) and parallel to the x-axis

x + 2y + 4 = 0

2x + 5y + 4 = 0

x + 5y + 6 = 0

y = 6 or y - 6 = 0 E) passing through the y-intercept of 6x + 5y + 25 = 0 and parallel to 4x - 3y + 9 = 0

4x - 3y - 15 = 0F) passing through the x-intercept of 6x + 5y + 30 = 0 and perpendicular to 4x - 3y + 9 = 0

3x + 4y + 15 = 0 6.7.13

Writing the Equation of a Line

Pages 294 and 2951 - 25 odd,27ace, 28 - 42 even,44 - 50 6.7.14

Suggested Questions: