obj. 2 slopes of lines
TRANSCRIPT
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Obj. 2 Slopes of LinesObjectives
Find the slope of a line.
Use slopes to identify parallel and perpendicular lines.
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rise
run
The difference in theyyyy----valuesvaluesvaluesvalues of two points
on a line.
The difference in the xxxx----valuesvaluesvaluesvalues of two points
on a line.
2 4 6 8
-2
2
4
6
8
x
y
(xxxx1111,yyyy1111)
(xxxx2222,yyyy2222)
run = 4run = 4run = 4run = 4
rise = 6rise = 6rise = 6rise = 6
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slope The ratio of riseriseriserise to runrunrunrun. If (xxxx1111,yyyy1111) and
(xxxx2222,yyyy2222) are any two points on a line, theslope of the line is
So, for the previous example, substitute(1111, 1111) for (xxxx1111,yyyy1111) and (5555, 7777) for (xxxx2222,yyyy2222):
Note: Always reduce fractions to theirsimplest forms. Also, its usually better toleave improper fractions improper.
2 1
2 1x
ym
y
x
=
1
2 1
2
x x
y y 7 6 3m
41 2
1
5
= = = =
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Summary: Slope of a Line
positive slope negative slope
zero slope undefined slope
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horizontal line
vertical line
reciprocal
The slope of a horizontal linehorizontal linehorizontal linehorizontal line is 0000.
The slope of a vertical linevertical linevertical linevertical line is undefinedundefinedundefinedundefined.
The reciprocalreciprocalreciprocalreciprocal of is . The product of
a number and its reciprocal is 1.
a
b
b
a
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Parallel Lines TheoremIn a coordinate plane, two nonvertical lines are parallel if andonly if they have the same slope.
Any two vertical lines are parallel
x
y
ms = mt
s t
s t
s
1 3 4
m 22 0 2
= = =
t
3 1 4m 2
1 1 2
= = =
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Perpendicular Lines TheoremIn a coordinate plane, two nonvertical lines are perpendicularif and only if the product of their slopes is 1 (negative
reciprocals).
Vertical and horizontal lines are perpendicular.
x
y
p
q
p
2 4 6m 3
2 0 2
+= = =
q
0 1 1 1m
3 0 3 3
= = =
p qm m 1= i p q
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PracticeFor each slope, find the parallel and perpendicular slopes:
1. 2.
3. 4. m = 5
2m3
= 1m2
=
4m
5=
2 3m ,m
3 2
= =
1m ,m 2
2
= =
4 5m ,m5 4= = 1m 5,m 5
= =
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PracticeFind the slopes between the following points:
5. (2, 1) and (8, 4) 6. (3, 10) and (5, 6)
7. (1, 12) and (10, 10) 8. (22, 4) and (0, 28)
( ) = = =
4 1 3 1m
8 2 6 2( )
= = =
6 10 16m 2
5 3 8
= = =
10 12 22
m 210 1 11
= = =
28 4 24 12
m 0 22 22 11
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Practice9. Given A(3, 1), B(3, 3), C(4, 4), and D(0, 2), is AB
parallel or perpendicular to CD?
The two slopes are not equal, so the lines are not
parallel. The product of the slopes is 1, so the lines areperpendicularperpendicularperpendicularperpendicular.
3 ( 1) 4 2AB : m
3 ( 3) 6 3
= = =
2 4 6 3
CD: m 0 ( 4) 4 2
= = =