CorrectionKey=NL-D;CA-D 4 . 5 DO NOT EDIT--Changes must be ...

[Pages:8]4.5 L E S S O N

Equations of Parallel and Perpendicular Lines

Common Core Math Standards

The student is expected to:

COMMON CORE

G-GPE.B.5

... find the equation of a line parallel or perpendicular to a given line that passes through a given point ... .

Mathematical Practices

COMMON

CORE MP.2 Reasoning

Language Objective

Explain to a partner how to use the slope of a line to find the equation of a parallel or perpendicular line.

ENGAGE

Essential Question: How can you find the equation of a line that is parallel or perpendicular to a given line?

Possible answer: The slopes of parallel lines are equal. Substitute the known slope and the coordinates of a point on the other line into the point-slope form to find the equation of the parallel line. The product of the slopes of perpendicular lines is -1. Substitute the opposite reciprocal of the known slope and the coordinates of a point on the other line into the point-slope form to find the equation of the perpendicular line.

PREVIEW: LESSON PERFORMANCE TASK

View the Engage section online. Discuss the photo. Explain that GPS stands for Global Positioning System, a system of 24 orbiting satellites that enables a person to pinpoint his or her precise location on Earth's surface. Then preview the Lesson Performance Task.

205 Lesson 4.5

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Name

Class

Date

4.5 Equations of Parallel and Perpendicular Lines

Essential Question: How can you find the equation of a line that is parallel or perpendicular to a given line?

Resource Locker

Explore Exploring Slopes of Lines

Recall that the slope of a straight line in a coordinate plane is

t_ he AB

ratio of the

is

_ri_s_e run

=

_4_ 8

rise to = _12_.

the

run.

In

the

figure,

the

slope

of

y 3

1 -5 -3 -1 0 A(-4, -3)

B(4, 1)

x

3

Rise = 1-(-3)

= 4

-5 Run = 4-(-4) = 8

Graph the equations y = 2(x + 1) and y = 2x - 3.

What do you notice about the graphs of the two lines?

About the slopes of the lines? The lines are parallel. The slopes are equal.

y 4 2

-4 -2 0

2

-2

-4

x 4

The graphs of x + 3y = 22 and y = 3x - 14 are shown.

y

Use a protractor. What is the measure of the angle formed

8

by the intersection of the lines. What does that tell you

about the lines?

6

90?; the lines are perpendicular. 4

What are the slopes of the two lines? How are they related?

__-

1 3

and

3;

the

slopes

are

opposite

reciprocals.

2

x

0

2468

Complete the statements: If two nonvertical lines

are parallel , then they have equal slopes. If two nonvertical lines are perpendicular,

then the product of their slopes is ?1 .

Module 4

DCOorNreOctTioEnDKITe-y-=ChNaLn-Dg;eCsAm-Dust be made through "File info"

205

Date Class

4.5 Equations of Parallel Name

and Perpendicular Lines Exploring Slopes of Lines Essential Question: Hoorwpecrapnenyoduicufilnadr

COMMON CORE

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line

that

passes y

Explore Rt_ AheBecarilasltt_rirhoi_usa_enotf=ththe_4e8_sl=roispe_1e2_.toof

tahsetrrauing.hItnlitnheeifnigaucreo,otrhdeinsalotepeploafne

is

3

1 -5 -3 -10

ReLsoocukrecre

B(4, 1)

x

3

Rise = 1-(-3) = 4

A(-4, -3)

-5 Run = 4-(-4) = 8

? Houghton Mifflin Harcourt Publishing Company

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y 4 2

-4 -2 0 -2 -4

x 24

y

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246 are perpendicular,

8

x

205

Module 4

Lesson 5

HARDCOVER PAGES 179184

Turn to these pages to find this lesson in the hardcover student edition.

Lesson 5 2/25/16 11:37 PM

Reflect

1. Your friend says that if two lines have opposite slopes, they are perpendicular. He uses the slopes 1 and ?1 as examples. Do you agree with your friend? Explain. No; although lines with slopes of 1 and -1 are perpendicular, it's because the product of the slopes is -1. Slopes of 2 and -2 are opposites, but the corresponding lines are not

perpendicular.

2. The frets on a guitar are all perpendicular to one of the strings. Explain why the frets must be parallel to each other. The frets are lines that are perpendicular to the same line (the string), so the frets must

be parallel to each other.

Explain 1 Writing Equations of Parallel Lines

You can use slope relationships to write an equation of a line parallel to a given line. Example 1 Write the equation of each line in slope-intercept form.

The line parallel to y = 5x + 1 that passes through (-1, 2)

Parallel lines have equal slopes. So the slope of the required line is 5.

Use point-slope form. Substitute for m, x1, y1. Simplify.

y - y1 = m(x - x1)

y - 2 = 5(x - (-1))

y - 2 = 5x + 5

Solve for y.

y = 5x + 7

The equation of the line is y = 5x + 7.

The line parallel to y = -3x + 4 that passes through (9, -6)

Parallel lines have equal slopes. So the slope of the required line is -3 .

Use point-slope form. Substitute for m, x1, y1. Simplify.

y - y1 = m(x - x1)

( ) y - -6 = -3 x - 9

y + 6 = -3 x + 27

Solve for y.

y = -3 x + 21

The equation of the line is y = -3x + 21 .

? Houghton Mifflin Harcourt Publishing Company

EXPLORE

Exploring Slopes of Lines

INTEGRATE TECHNOLOGY

Students have used geometry software to construct perpendicular lines and calculate their slopes. They can use the calculation feature to find the product of slopes of perpendicular lines is always -1.

QUESTIONING STRATEGIES

What appears to be true about the slopes of non-vertical parallel lines? They are equal. What appears to be true about the slopes of two non-vertical perpendicular lines? The slopes are opposite reciprocals.

INTEGRATE MATHEMATICAL PRACTICES Focus on Critical Thinking MP.3 You may want to discuss the biconditional

nature of the slope criteria. Because they are if and only if statements, the criteria can be used in either direction. That is, if you know that two lines are parallel (perpendicular), you can conclude that they have the same (opposite reciprocal) slope. Conversely, if you know that two lines have the same (opposite reciprocal) slope, you can conclude that they are parallel (perpendicular).

Module 4

206

Lesson 5

PROFESSIONAL DEVELOPMENT

Math Background

In this lesson, students use the slope criterion for parallel lines and the slope criterion for perpendicular lines to solve problems. Note that the slope criteria given here assume that the lines are neither vertical nor horizontal. If the lines are vertical, the criteria for parallel and perpendicular lines do not apply, since slope is not defined for vertical lines. If the lines are horizontal, both lines have a slope of zero, and the criterion for parallel lines is trivial.

EXPLAIN 1

Writing Equations of Parallel Lines

QUESTIONING STRATEGIES

How do you know if an equation is written in slope-intercept form? It is of the form y = mx + b, with m the slope and b the y-intercept. How can you use graphing to check your answer? Graph the given line and your answer line. They should be parallel.

Equations of Parallel and Perpendicular Lines 206

AVOID COMMON ERRORS

Remind students that the x-coefficient gives the slope of a line only when the equation of the line is written in slope-intercept form. For example, some students might say that the slope of the line represented by the equation y - 2x = 4 is -2. However, the equation is not in slope-intercept form. Rewriting the equation in this form gives y = 2x + 4, which shows that the correct slope is 2.

EXPLAIN 2

Writing Equations of Perpendicular Lines

INTEGRATE MATHEMATICAL PRACTICES Focus on Technology MP.5 Students can use their graphing calculators to

check that two equations represent perpendicular lines. However, students should be aware that perpendicular lines may or may not appear to be perpendicular on a graphing calculator, depending upon the viewing window that is used. To ensure that perpendicular lines appear to be perpendicular, students should go to the ZOOM menu and choose 5:ZSquare.

QUESTIONING STRATEGIES

The given line has a positive slope. What does this tell you about the required perpendicular line? Why? It must have a negative slope because the product of the slopes is -1. How can you check your answers? Check that the product of the slopes is -1.

207 Lesson 4.5

? Houghton Mifflin Harcourt Publishing Company

Reflect

3. What is the equation of the line through a given point and parallel to the x-axis? Why? The equation is y = y1, where y1 is the y-coordinate of the given point. This is because the

x-axis is a horizontal line with equation y = 0.

Your Turn

Write the equation of each line in slope-intercept form.

4. The line parallel to y = -x that passes

through (5, 2.5)

y - 2.5 = -1(x - 5) y - 2.5 = -x + 5

y = -x + 7.5

5.

The line through

parallel

(-4, 0)

to

y

=

_32_x

+

4

tha

t

passes

_ y

-

(0)

=

3 2

(x

-

(-4))

_ y

=

3 2

x

+

6

Explain 2 Writing Equations of Perpendicular Lines

You can use slope relationships to write an equation of a line perpendicular to a given line.

Example 2 Write the equation of each line in slope-intercept form.

The line perpendicular to y = 4x - 2 that passes through (3, -1)

Perpendicular lines have slopes that are opposite reciprocals, which means that the product of the slopes will be -1. So the slope of the required line is -_41_.

y - y1 = m(x - x1)

Use point-slope form.

y - (-1) = -_41 (x - 3)

Substitute for m, x1, y1.

y

+

1

=

-_41 x

+

_ 3 4

Simplify.

y

=

-_41 x

-

_ 1 4

Solve for y.

The equation of the line is y = -_41_x - _14_.

The line perpendicular to y = -_52_x + 12 that passes through (-6, -8)

The product of the slopes of perpendicular lines is -1 . So the slope of the required line is

_5_ 2

.

y - y1 = m(x - x1)

Use point-slope form.

( ) y - -8 =

_5_ 2

x - -6

Substitute for m, x1, y1.

y + 8 =

_5_ 2

x + 15

Simplify.

y =

_5_ 2

x +

7

Solve for y.

The equation of the line is y y = _25_x + 7 .

Module 4

207

Lesson 5

COLLABORATIVE LEARNING

Whole Class Activity

Have groups of students create posters to describe the slope criteria. Then remind students about the biconditional nature of the criteria and ask them if the criteria are true in two directions. Ask them to display the criteria as graphic organizers. Sample organizers:

Lines are parallel.

Lines have the same slope.

Lines are perpendicular.

The product of slopes is ?1.

Reflect

6. A carpenter's square forms a right angle. A carpenter places the square so that one side is parallel to an edge of a board, and then draws a line along the other side of the square. Then he slides the square to the right and draws a second line. Why must the two lines be parallel? Both lines are perpendicular to the edge of the

board. If two coplanar lines are perpendicular to

the same line, then the two lines are parallel to each

other, so the lines must be parallel to each other.

Your Turn

Write the equation of each line in slope-intercept form.

__ 7.

The

line

perpendicular

to

y

=

3 2

x

+

2

that

8.

passes through (3, ?1)

y - (-1) = -_32_ (x - 3) y = -_32_x + 1

The line perpendicular to y = -4x that

passes through (0, 0)

y

-

0

=

_1_ 4

(x

-

0)

y = _41_x

Elaborate

9. Discussion Would it make sense to find the equation of a line parallel to a given line, and through a point on the given line? Explain. No; if the point is on the line, the line can't be parallel to the line, because it either

intersects it or it is the same line.

10. Would it make sense to find the equation of a line perpendicular to a given line, and through a point on the given line? Explain. Yes; the line will be perpendicular to the given line at the point.

11. Essential Question Check-In How are the slopes of parallel lines and perpendicular lines related? Assume the lines are not vertical. Parallel lines have the same slope; perpendicular lines have slopes whose product is -1.

ELABORATE

QUESTIONING STRATEGIES

What is the equation of the line through a given point and parallel to the x-axis? Why? The equation is y = b, where b is the y-coordinate of the point.

Can either of the lines referred to in the slope criterion for perpendicular lines be vertical? Why or why not? No; the slope criterion specifies that neither line is vertical. However, since the lines are perpendicular, if one line were horizontal, the other would be vertical.

SUMMARIZE THE LESSON

Given the equation of a line and a point not on the line, how do you find the equation of a line parallel and perpendicular to the given line? Sample answer: Parallel: Use the slope-intercept form of a line, y = mx + b, and replace m with the slope from the given line. Use the given point and the slope to solve for b and then rewrite the equation using the same slope m and the new y-intercept b.

Perpendicular: Do the same steps as for parallel except replace m with the opposite reciprocal of m.

? Houghton Mifflin Harcourt Publishing Company ? Image Credits: ?Zoran Zeremski/Shutterstock

Module 4

208

Lesson 5

DIFFERENTIATE INSTRUCTION

Communicating Math

Group students in pairs and give each pair a sheet of graph paper with a non-vertical, non-horizontal line drawn on it. Have students draw axes and find the equation of the line. Then have each student plot a point that is not on the line, and find the equation of the line that is parallel, and the equation of the line that is perpendicular to the original line and that passes through the partner's point. When they are done, they should compare the slopes of their lines to show that the two new lines are parallel (or perpendicular) to each other.

Equations of Parallel and Perpendicular Lines 208

EVALUATE

ASSIGNMENT GUIDE

Concept & Skills

Explore Exploring Slopes of Lines

Example 1 Writing Equations of Parallel Lines

Example 2 Writing Equations of Perpendicular Lines

Practice

Exercises 1?4

Exercises 5?7, 11?12, 14?15 Exercises 8?10, 13, 16?19

AVOID COMMON ERRORS

A common error students make when finding slopes of perpendicular lines is using the same sign for both slopes. One slope must be the opposite reciprocal of the other, not just the reciprocal of the other, so that the product is -1, not 1.

209 Lesson 4.5

? Houghton Mifflin Harcourt Publishing Company

Evaluate: Homework and Practice

Use the graph for Exercises 1?4.

1. A line with a positive slope is parallel to one of the lines shown.

What is its slope?

_ 6-2

5-0

=

_ 4 5

2. A line with a negative slope is perpendicular to one of the lines shown.

What is its slope?

-_5 4

3. A line with a positive slope is perpendicular to one of the lines shown.

What is its slope?

The line will be perpendicular to the line with

_ _ slope

5-1 1-2

=

-4. So the slope

is

1 4

.

4. A line with a negative slope is parallel to one of the lines shown.

What is its slope?

The line will be parallel to the line with

_ slope

5-1 1-2

=

-4. So the slope

is -4.

? Online Homework ? Hints and Help ? Extra Practice

y 6

4

2 x

0 246

Find the equation of the line that is parallel to the given line and passes through the given point.

5. y = ?3x + 1; (9, 0)

y - 0 = -3 (x - 9) y = -3x + 27

6. y = 0.6x ? 3; (?2, 2)

y - 2 = 0.6 (x - (-2))

y = 0.6x + 3.2

( ) 7. y = 5(x + 1); _12, -_21

y = 5x + 5

( ) ( ) y -

- _21_

= 5

x

-

_1_ 2

y = 5x - 3

Find the equation of the line that is perpendicular to the given line

and passes through the given point.

8. y = 10x; (1, -3)

9. y = -_31x - 5; (12, 0)

10. y = _ 5x 3+ 1; (1, 1)

y - (-3) = -0.10(x - 1)

y = 3(x - 12)

3y = 5x + 1

y = -0.1x - 2.9

y = 3x - 36

y

=

_35_x

+

_1_ 3

y - 1 = -_53_(x - 1)

y =

_-_3_x_+__8_ 5

Module 4

209

Lesson 5

COMMON

Exercise Depth of Knowledge (D.O.K.) CORE Mathematical Practices

1?4 5?10 11 12?15 16 17?19

1 Recall of Information 1 Recall of Information 2 Skills/Concepts 2 Skills/Concepts 2 Skills/Concepts 2 Skills/Concepts

MP.6 Precision MP.2 Reasoning MP.2 Reasoning MP.4 Modeling MP.4 Modeling MP.2 Reasoning

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