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MATHPOWERTM 10, WESTERN EDITION
Chapter 6 Coordinate Geometry6.7
6.7.1
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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
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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
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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 =
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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
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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
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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
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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
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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
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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
.
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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
.
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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
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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
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Pages 294 and 2951 - 25 odd,27ace, 28 - 42 even,44 - 50 6.7.14
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