3.4 Find and Use Slopes of Lines
Objectives:
1. To find the slopes of
lines
2. To find the slopes of
parallel and
perpendicular lines
Assignment:
• P. 175-7: 1-14 all, 16,
19, 23, 26-28, 33, 42,
43
• Challenge Problems
Warm-Up
How would you
describe the roof at
the right?
Rate of Change
A rate of change is how much one
quantity changes (on average) relative
to another.
Slope can be used to represent an
average rate of change.
For slope, we measure how 𝑦
changes relative to 𝑥.
Warmer-Upper
The slope or pitch of
a roof is quite a
useful
measurement.
How do you think a
contractor would
measure the slope
or pitch of a roof?
Warmer-Upper
The slope or pitch of
a roof is defined as
the number of
vertical inches of
rise for every 12
inches of horizontal
run.
Warmer-Upper
The steeper the roof,
the better it looks,
and the longer it
lasts. But the cost
is higher because
of the increase in
the amount of
building materials.
Investigation 1
Use the Slope Game,
a Geometer’s
Sketchpad activity,
to discover
something about
the actual value of
the slope of a line.
Then complete the
table on the next
slide.
Slope Summary
Summarize your findings about slope in the
table below:
m>0
m<0
m=0
m = undef
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Picture
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As the absolute value of the slope of a line
the line gets steeper.
increases, --?--.
Slope of a Line
The slope m of a
nonvertical line is
the ratio of vertical
change (the ryse)
to the horizontal
change (the run).
ryse
ryse
Exercise 1
Find the slope of the line containing the
given points. Then describe the line as
rising, falling, horizontal, or vertical.
1. (6, −9) and (−3, −9)
2. (8, 2) and (8, −5)
3. (−1, 5) and (3, 3)
4. (−2, −2) and (−1, 5)
Exercise 2
A line through points (5, -3) and (−4, y) has a
slope of −1. Find the value of y.
Objective 2
Investigation 2
Use the Geometer’s
Sketchpad activity
to complete the two
following postulates,
and then add them
to your Because I
Said So… Postulate
page.
Parallel and Perpendicular
Two lines are
parallel lines iff
they have the
same slope.
Two lines are
perpendicular lines
iff their slopes are
negative reciprocals.
Exercise 3
Tell whether the pair of lines are parallel,
perpendicular, or neither
1. Line 1: through (−2, 1) and (0, −5)
Line 2: through (0, 1) and (−3, 10)
2. Line 1: through (−2, 2) and (0, −1)
Line 2: through (−4, −1) and (2, 3)
Exercise 4
Line k passes through
(0, 3) and (5, 2).
Graph the line
perpendicular to k
that passes through
point (1, 2).
Exercise 5
Find the value of y so that the line passing
through the points (3, y) and (−5, −6) is
perpendicular to the line that passes
through the points (−2, −7) and (10, 1).
Exercise 6
Find the value of k so that the line through
the points (k – 3, k + 2) and (2, 1) is parallel
to the line through the points (−1, 1) and
(3, 9).
Tangent
Tangent
A line is a tangent if
and only if it
intersects a circle
in one point.
Investigation 3
Use the Geometer’s
Sketchpad activity
to discover the
relationship
between a radius
and a line tangent
to a circle.
Tangent Line Theorem
In a plane, a line is
tangent to a circle if
and only if the line
is perpendicular to a
radius of the circle
at its endpoint on
the circle.
Exercise 7
The center of a circle
has coordinates
(1, 2). The point
(3, -1) lies on this
circle. Find the
slope of the tangent
line at (3, -1).
6
4
2
5
-2
3.4 Find and Use Slopes of Lines
Objectives:
1. To find the slopes of
lines
2. To find the slopes of
parallel and
perpendicular lines
Assignment:
• P. 175-7: 1-14 all, 16,
19, 23, 26-28, 33, 42,
43
• Challenge Problems