3-5
3-5 Slopes
SlopesofofLines
Lines
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Geometry
Holt
Geometry
3-5 Slopes of Lines
Warm Up
Find the value of m.
1.
2.
3.
4.
undefined
Holt Geometry
0
3-5 Slopes of Lines
Objectives
Find the slope of a line.
Use slopes to identify parallel and
perpendicular lines.
Holt Geometry
3-5 Slopes of Lines
Vocabulary
rise
run
slope
Holt Geometry
3-5 Slopes of Lines
The slope of a line in a coordinate plane
is a number that describes the
steepness of the line. Any two points on
a line can be used to determine the
slope.
Holt Geometry
3-5 Slopes of Lines
Holt Geometry
3-5 Slopes of Lines
Example 1A: Finding the Slope of a Line
Use the slope formula to determine the slope of
each line.
AB
Substitute (–2, 7) for (x1, y1)
and (3, 7) for (x2, y2) in the
slope formula and then simplify.
Holt Geometry
3-5 Slopes of Lines
Example 1B: Finding the Slope of a Line
Use the slope formula to determine the slope of
each line.
AC
Substitute (–2, 7) for (x1, y1)
and (4, 2) for (x2, y2) in the
slope formula and then simplify.
Holt Geometry
3-5 Slopes of Lines
Example 1C: Finding the Slope of a Line
Use the slope formula to determine the slope of
each line.
AD
Substitute (–2, 7) for (x1, y1)
and (–2, 1) for (x2, y2) in the
slope formula and then simplify.
The slope is undefined.
Holt Geometry
3-5 Slopes of Lines
Remember!
A fraction with zero in the
denominator is undefined because
it is impossible to divide by zero.
Holt Geometry
3-5 Slopes of Lines
Example 1D: Finding the Slope of a Line
Use the slope formula to determine the slope of
each line.
CD
Substitute (4, 2) for (x1, y1) and
(–2, 1) for (x2, y2) in the slope
formula and then simplify.
Holt Geometry
3-5 Slopes of Lines
Check It Out! Example 1
Use the slope formula to determine the slope
of JK through J(3, 1) and K(2, –1).
Substitute (3, 1) for (x1, y1) and (2, –1) for
(x2, y2) in the slope formula and then simplify.
Holt Geometry
3-5 Slopes of Lines
One interpretation of slope is a rate of change. If
y represents miles traveled and x represents time
in hours, the slope gives the rate of change in
miles per hour.
Holt Geometry
3-5 Slopes of Lines
Example 2: Transportation Application
Justin is driving from home to his college
dormitory. At 4:00 p.m., he is 260 miles from
home. At 7:00 p.m., he is 455 miles from
home. Graph the line that represents Justin’s
distance from home at a given time. Find and
interpret the slope of the line.
Use the points (4, 260) and
(7, 455) to graph the line
and find the slope.
Holt Geometry
3-5 Slopes of Lines
Example 2 Continued
The slope is 65, which
means Justin is
traveling at an average
of 65 miles per hour.
Holt Geometry
3-5 Slopes of Lines
Check It Out! Example 2
What if…? Use the graph below to estimate
how far Tony will have traveled by 6:30 P.M.
if his average speed stays the same.
Since Tony is traveling at an
average speed of 60 miles
per hour, by 6:30 P.M. Tony
would have traveled 390
miles.
Holt Geometry
3-5 Slopes of Lines
Holt Geometry
3-5 Slopes of Lines
If a line has a slope of
perpendicular line is
The ratios
Holt Geometry
and
, then the slope of a
.
are called opposite reciprocals.
3-5 Slopes of Lines
Caution!
Four given points do not always
determine two lines.
Graph the lines to make sure the
points are not collinear.
Holt Geometry
3-5 Slopes of Lines
Example 3A: Determining Whether Lines Are Parallel,
Perpendicular, or Neither
Graph each pair of lines. Use their slopes to
determine whether they are parallel,
perpendicular, or neither.
UV and XY for U(0, 2),
V(–1, –1), X(3, 1),
and Y(–3, 3)
The products of the slopes is –1, so the lines are
perpendicular.
Holt Geometry
3-5 Slopes of Lines
Example 3B: Determining Whether Lines Are Parallel,
Perpendicular, or Neither
Graph each pair of lines. Use their slopes to
determine whether they are parallel,
perpendicular, or neither.
GH and IJ for G(–3, –2),
H(1, 2), I(–2, 4), and J(2, –4)
The slopes are not the same, so the lines are not
parallel. The product of the slopes is not –1, so the
lines are not perpendicular.
Holt Geometry
3-5 Slopes of Lines
Example 3C: Determining Whether Lines Are Parallel,
Perpendicular, or Neither
Graph each pair of lines. Use their slopes to
determine whether they are parallel,
perpendicular, or neither.
CD and EF for C(–1, –3),
D(1, 1), E(–1, 1), and F(0, 3)
The lines have the same slope, so they are parallel.
Holt Geometry
3-5 Slopes of Lines
Check It Out! Example 3a
Graph each pair of lines. Use slopes to
determine whether the lines are parallel,
perpendicular, or neither.
WX and YZ for W(3, 1),
X(3, –2), Y(–2, 3), and
Z(4, 3)
Vertical and horizontal lines are perpendicular.
Holt Geometry
3-5 Slopes of Lines
Check It Out! Example 3b
Graph each pair of lines. Use slopes to
determine whether the lines are parallel,
perpendicular, or neither.
KL and MN for K(–4, 4),
L(–2, –3), M(3, 1), and
N(–5, –1)
The slopes are not the same, so the lines are not
parallel. The product of the slopes is not –1, so the
lines are not perpendicular.
Holt Geometry
3-5 Slopes of Lines
Check It Out! Example 3c
Graph each pair of lines. Use slopes to
determine whether the lines are parallel,
perpendicular, or neither.
BC and DE for B(1, 1),
C(3, 5), D(–2, –6), and
E(3, 4)
The lines have the same slope, so they are parallel.
Holt Geometry
3-5 Slopes of Lines
Lesson Quiz
1. Use the slope formula to determine the slope of the
line that passes through M(3, 7) and N(–3, 1).
m=1
Graph each pair of lines. Use slopes to determine
whether they are parallel, perpendicular, or
neither.
2. AB and XY for A(–2, 5),
B(–3, 1), X(0, –2), and Y(1, 2)
4, 4; parallel
3. MN and ST for M(0, –2),
N(4, –4), S(4, 1), and T(1, –5)
Holt Geometry