Transcript Unit D Chapter 3.5 (Slopes of Lines Parallel/Perpendicular)

3-5
3-5 Slopes
SlopesofofLines
Lines
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Geometry
Holt
McDougal
Geometry
3-5 Slopes of Lines
Warm Up
Find the value of m.
1.
2.
3.
4.
undefined
Holt McDougal Geometry
0
3-5 Slopes of Lines
Objectives
Find the slope of a line.
Use slopes to identify parallel and
perpendicular lines.
Holt McDougal Geometry
3-5 Slopes of Lines
Vocabulary
rise
run
slope
Holt McDougal 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 McDougal Geometry
3-5 Slopes of Lines
Holt McDougal 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 McDougal 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 McDougal 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 McDougal 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 McDougal 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 McDougal 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 McDougal 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 McDougal 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 McDougal 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 McDougal 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 McDougal Geometry
3-5 Slopes of Lines
What does it mean when lines
have equal slopes?
Draw an example.
Holt McDougal Geometry
3-5 Slopes of Lines
How can you tell if lines are
perpendicular?
Draw an example.
Using their slopes?
Holt McDougal Geometry
3-5 Slopes of Lines
Holt McDougal Geometry
3-5 Slopes of Lines
If a line has a slope of
perpendicular line is
The ratios
Holt McDougal 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 McDougal 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 McDougal 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 McDougal 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 McDougal 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 McDougal 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 McDougal 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 McDougal 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 McDougal Geometry
3-5 Slopes of Lines
The equation of a line can be written in
many different forms. The point-slope and
slope-intercept forms of a line are
equivalent. Because the slope of a vertical
line is undefined, these forms cannot be
used to write the equation of a vertical line.
Holt McDougal Geometry
3-5 Slopes of Lines
Holt McDougal Geometry
3-5 Slopes of Lines
Remember!
A line with y-intercept b contains the point (0, b).
A line with x-intercept a contains the point (a, 0).
Holt McDougal Geometry
3-5 Slopes of Lines
Example 1A: Writing Equations In Lines
Write the equation of each line in the given
form.
the line with slope 6 through (3, –4) in pointslope form
y – y1 = m(x – x1)
y – (–4) = 6(x – 3)
Holt McDougal Geometry
Point-slope form
Substitute 6 for m, 3 for
x1, and -4 for y1.
3-5 Slopes of Lines
Example 1B: Writing Equations In Lines
Write the equation of each line in the given form.
the line through (–1, 0) and (1, 2) in slopeintercept form
Find the slope.
y = mx + b
Slope-intercept form
0 = 1(-1) + b
Substitute 1 for m, -1 for x,
and 0 for y.
1=b
y=x+1
Holt McDougal Geometry
Write in slope-intercept form
using m = 1 and b = 1.
3-5 Slopes of Lines
Example 1C: Writing Equations In Lines
Write the equation of each line in the given form.
the line with the x-intercept 3 and y-intercept
–5 in point slope form
Use the point (3,-5) to
find the slope.
y – y1 = m(x – x1)
y – 0 = 5 (x – 3)
3
5
y = 3 (x - 3)
Holt McDougal Geometry
Point-slope form
Substitute5
for m, 3 for
x , and 0 3
for y .
1
Simplif
y.
1
3-5 Slopes of Lines
Check It Out! Example 1a
Write the equation of each line in the given form.
the line with slope 0 through (4, 6) in slopeintercept form
y – y1 = m(x – x1)
Point-slope form
y – 6 = 0(x – 4)
Substitute 0 for m, 4 for
x1, and 6 for y1.
y=6
Holt McDougal Geometry
3-5 Slopes of Lines
Check It Out! Example 1b
Write the equation of each line in the given form.
the line through (–3, 2) and (1, 2) in pointslope form
Find the slope.
y – y1 = m(x – x1)
Point-slope form
y – 2 = 0(x – 1)
Substitute 0 for m, 1 for x1,
and 2 for y1.
y-2=0
Simplif
y.
Holt McDougal Geometry
3-5 Slopes of Lines
A system of two linear equations in two variables
represents two lines. The lines can be parallel,
intersecting, or coinciding. Lines that coincide
are the same line, but the equations may be
written in different forms.
Holt McDougal Geometry
3-5 Slopes of Lines
Holt McDougal Geometry
3-5 Slopes of Lines
Example 3A: Classifying Pairs of Lines
Determine whether the lines are parallel,
intersect, or coincide.
y = 3x + 7, y = –3x – 4
The lines have different slopes, so they intersect.
Holt McDougal Geometry
3-5 Slopes of Lines
Example 3B: Classifying Pairs of Lines
Determine whether the lines are parallel,
intersect, or coincide.
Solve the second equation for y to find the slopeintercept form.
6y = –2x + 12
Both lines have a slope of
, and the y-intercepts
are different. So the lines are parallel.
Holt McDougal Geometry
3-5 Slopes of Lines
Example 3C: Classifying Pairs of Lines
Determine whether the lines are parallel,
intersect, or coincide.
2y – 4x = 16, y – 10 = 2(x - 1)
Solve both equations for y to find the slopeintercept form.
2y – 4x = 16
2y = 4x + 16
y = 2x + 8
y – 10 = 2(x – 1)
y – 10 = 2x - 2
y = 2x + 8
Both lines have a slope of 2 and a y-intercept of 8, so
they coincide.
Holt McDougal Geometry
3-5 Slopes of Lines
Check It Out! Example 3
Determine whether the lines 3x + 5y = 2 and
3x + 6 = -5y are parallel, intersect, or coincide.
Solve both equations for y to find the slopeintercept form.
3x + 5y = 2
3x + 6 = –5y
5y = –3x + 2
Both lines have the same slopes but different
y-intercepts, so the lines are parallel.
Holt McDougal Geometry
3-5 Slopes of Lines
Example 4: Problem-Solving Application
Erica is trying to decide between two
car rental plans. For how many miles
will the plans cost the same?
Holt McDougal Geometry
3-5 Slopes of Lines
1
Understand the Problem
The answer is the number of miles for which
the costs of the two plans would be the
same. Plan A costs $100.00 for the initial fee
and $0.35 per mile. Plan B costs $85.00 for
the initial fee and $0.50 per mile.
Holt McDougal Geometry
3-5 Slopes of Lines
2
Make a Plan
Write an equation for each plan, and then graph
the equations. The solution is the intersection of
the two lines. Find the intersection by solving
the system of equations.
Holt McDougal Geometry
3-5 Slopes of Lines
3
Solve
Plan A: y = 0.35x + 100
Plan B: y = 0.50x + 85
0 = –0.15x + 15
Subtract the second
equation from the first.
x = 100
Solve for x.
y = 0.50(100) + 85 = 135
Substitute 100 for x in
the first equation.
Holt McDougal Geometry
3-5 Slopes of Lines
4
Look Back
Check your answer for each plan in the
original problem.
For 100 miles, Plan A costs
$100.00 + $0.35(100) = $100 + $35 =
$135.00.
Plan B costs $85.00 + $0.50(100) = $85 +
$50 = $135, so the plans cost the same.
Holt McDougal Geometry
3-5 Slopes of Lines
Check It Out! Example 4
What if…? Suppose the rate for Plan B
was also $35 per month. What would be
true about the lines that represent the
cost of each plan?
The lines would be parallel.
Holt McDougal Geometry
3-5 Slopes of Lines
Lesson Quiz: Part I
Write the equation of each line in the given
form. Then graph each line.
1. the line through (-1, 3)
and (3, -5) in slopeintercept form.
y = –2x + 1
2. the line through (5, –1)
with slope in point-slope
form.
y + 1 = 2 (x – 5)
5
Holt McDougal Geometry
3-5 Slopes of Lines
Lesson Quiz: Part II
Determine whether the lines are parallel,
intersect, or coincide.
3. y – 3 = –
1
x, y – 5 = 2(x + 3)
2
intersect
4. 2y = 4x + 12, 4x – 2y = 8
parallel
Holt McDougal Geometry