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Holt McDougal Geometry
3-5 Slopes of Lines 3-5 Slopes of Lines
Holt Geometry
Warm Up
Lesson Presentation
Lesson Quiz
Holt McDougal Geometry
Holt McDougal Geometry
3-5 Slopes of Lines
Warm Up Find the value of m.
1. 2.
3. 4.
undefined 0
Holt McDougal Geometry
3-5 Slopes of Lines
Find the slope of a line.
Use slopes to identify parallel and perpendicular lines.
Objectives
Holt McDougal Geometry
3-5 Slopes of Lines
rise
run
slope
Vocabulary
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.
Substitute (–2, 7) for (x1, y1) and (3, 7) for (x2, y2) in the slope formula and then simplify.
AB
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
The slope is undefined.
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.
Holt McDougal Geometry
3-5 Slopes of Lines
A fraction with zero in the denominator is undefined because it is impossible to divide by zero.
Remember!
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
Holt McDougal Geometry
3-5 Slopes of Lines
If a line has a slope of , then the slope of a
perpendicular line is .
The ratios and are called opposite reciprocals.
Holt McDougal Geometry
3-5 Slopes of Lines
Four given points do not always determine two lines. Graph the lines to make sure the points are not collinear.
Caution!
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.
4, 4; parallel
2. AB and XY for A(–2, 5), B(–3, 1), X(0, –2), and Y(1, 2)
3. MN and ST for M(0, –2), N(4, –4), S(4, 1), and T(1, –5)