parallel and perpendicular lines · functions and the translations of linear functions. sliding...

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527

10.1 SlidingLinesTranslations of Linear Functions ..................................529

10.2ParallelorPerpendicular?Slopes of Parallel and Perpendicular Lines .................. 541

10.3Up,Down,andAllAroundLine Transformations ..................................................555

ParallelandPerpendicularLines

The playing surfaces of most

sports rely on perpendicular and parallel lines. A football field has parallel yard lines,

baseball has perpendicular base lines, and a tennis net is

perpendicular to the court. Hockey is one of the few

sports with a curved playing surface.

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528 • Chapter 10 Parallel and Perpendicular Lines

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10.1 Translations of Linear Functions • 529

Learning GoalsIn this lesson, you will:

Translate linear functions horizontally and vertically.

Use multiple representations such as tables, graphs, and equations to represent linear

functions and the translations of linear functions.

SlidingLinesTranslations of Linear Functions

Look at the lines below each row of black and white squares. Are these lines

straight? Grab a ruler or other straightedge to test.

This very famous optical illusion is called the Zöllner illusion, named after its

discoverer, Johann Karl Friedrich Zöllner, who first wrote about it in 1860.

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Problem 1 Translating Linear Functions Up or Down

In a previous lesson in this course, geometric figures were translated vertically (up or

down) and horizontally (left or right). In this lesson, you will use that knowledge to translate

linear functions both vertically and horizontally.

1. Consider the equation y 5 x. Complete the table of values.

x y

23

22

21

0

1

2

3

2. Use the table of values and the coordinate plane provided

to graph the equation y 5 x.

x86

2

0

4

6

8

10–2–2

42–4

–4

–6

–6

–8

–8

–10

y

10

–10

How do you know that y = x is a function?

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3. Did you connect the points on the graph of the equation? Why or why not?

4. Suppose that a geometric figure is translated down 4 units.

a. How does this affect the value of the x-coordinate of each vertex?

b. How does this affect the value of the y-coordinate of each vertex?

5. Use your experience of translating a geometric figure to translate the graph of y 5 x

down 4 units. Draw the new line on the coordinate plane in Question 2 and then

complete the table of values.

x y

23

22

21

0

1

2

3

How will this table of

values compare to the table in Question 1?

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6. Compare the graph of y 5 x to the graph of y 5 x translated down 4 units.

a. What do you notice?

b. Write an equation in the form y 5 to represent

the translation.

c. Write an equation in the form x 5 to represent

the translation.

7. Translate the graph of y = x up 4 units. Draw the new line on the coordinate plane in

Question 2 and then complete the table of values.

x y

23

22

21

0

1

2

3

How does this table of values compare to the

other two?

Are the two equations

the same?

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8. Compare the graph of y 5 x to the graph of y 5 x translated up 4 units.


b. Write an equation in the form y 5 to represent the translation.

c. Write an equation in the form x 5 to represent the translation.

9. Label each equation on the coordinate plane in slope-intercept form. What do you

notice? What is similar about each line? What is different?

Problem 2 Translating Linear Functions Left or Right

1. Suppose that a geometric figure is translated to the left 4 units.

a. How does this affect the value of the

x-coordinate of each vertex?

b. How does this affect the value of the y-coordinate of

each vertex?

I think there is going to be a connection!

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2. Graph the equation y 5 x on the coordinate plane.

x86

2

0

4

6

8

10–2–2

42–4

–4

–6

–6

–8

–8

–10

y

10

–10

0

3. Use your experience of translating a geometric figure to translate the graph of y 5 x

to the left 4 units. Draw the new line on the coordinate plane and then complete the

table of values.

x y

23

22

21

0

1

2

3

4. Compare the graph of y 5 x to the graph of y 5 x translated to the left 4 units.


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5. Translate the graph of y 5 x to the right 4 units. Draw the new line on the coordinate

plane in Question 2 and then complete the table of values.

x y

23

22

21

0

1

2

3

6. Compare the graph of y 5 x to the graph of y 5 x translated to the right 4 units.




7. Label each equation on the coordinate plane in slope-intercept form. What do you

notice? What is similar about each line? What is different?

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Problem 3 Making Connections

1. Organize the equations you determined for the graph of each translation performed

on the linear equation y 5 x in the previous problem by completing the last two

columns of the table shown.

Original Equation

Translation Performed

Equation of Translation in the

Form of y 5

Equation of Translation in the

Form of x 5

y 5 x Down 4 Units y 5 x 5

y 5 x Up 4 Units y 5 x 5

y 5 x Left 4 Units y 5 x 5

y 5 x Right 4 Units y 5 x 5

2. Which translations of the linear equation y 5 x resulted in the same graph?

3. Kieran says that whenever a linear equation written in slope-intercept form shows a

plus sign, it is a translation right or up, and when it shows a minus sign it is a

translation left or down, because positive always means up and right on the

coordinate grid, and negative always means left and down. Is Kieran correct? Justify

your answer.

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4. Each graph shown is a result of a translation performed on the equation y 5 x.

Describe the translation. Then write an equation in slope-intercept form.

a.

x86

2

4

6

8

10–2–2

42–4

–4

–6

–6

–8

–8

–10

y

10

–10

0

b.

x86

2

4

6

8

10–2–2

42–4

–4

–6

–6

–8

–8

–10

y

10

–10

0

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5. Each graph shown is a result of a translation performed on the equation y 5 2x.

Describe the translation.

a.

x86

2

4

6

8

10–2–2

42–4

–4

–6

–6

–8

–8

–10

y

10

–10

0

b.

x86

2

4

6

8

10–2–2

42–4

–4

–6

–6

–8

–8

–10

y

10

–10

0

6. Each equation shown is a result of a translation performed on the equation y 5 x.


a. y 5 x 1 12.5

b. y 5 x 2 15.25

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7. Each equation shown is a result of a translation performed on the equation y 5 2x.


a. y 5 2x 2 1.2

b. y 5 2x 1 3.8

Talk the Talk

1. The equation shown is a result of a translation performed on the equation y 5 x.

For any real number h, describe the possible translations.

y 5 x 1 h

2. The equation shown is a result of a translation performed on the equation y 5 2x.

For any real number h, describe the possible translations.

y 5 2x 1 h

3. If a function is translated horizontally or vertically, is the resulting line still a function?

Be prepared to share your solutions and methods.

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10.2 Slopes of Parallel and Perpendicular Lines • 541

ParallelorPerpendicular?Slopes of Parallel and Perpendicular Lines

Key Terms reciprocal

negative reciprocal


Determine the slopes of parallel lines.

Determine the slopes of perpendicular lines.

Identify parallel lines.

Identify perpendicular lines.

Everything you see around you is made up of atoms—tiny particles (or waves?)

that are constantly moving. And most of an atom is actually empty space. So, why

is it that you can’t walk through walls?

The answer—or at least part of the answer—is the normal force. This force, which

is always perpendicular to the surface, is the one that pushes up on you. It’s the

force that keeps you from sinking into the floor—and unfortunately, the force that

makes it impossible for you to walk through walls.

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Problem 1 Graphing Equations, Part 1

1. Graph each equation on the coordinate plane.

● y 5 2x

● y 5 2x 1 3

● y 5 2x 2 5

● y 5 2x 1 5

x

2

4

6

8

–2

–4

–6

86–2 42–4–6–8

–8

y

0

a. Describe the relationship between the lines.

b. Describe a strategy for verifying the relationship between the lines.

Do you recall studying

transformations before?

Notice that all the equations

are in slope-intercept form, y = mx + b.

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c. Use measuring tools to verify the relationship between the lines.

d. What do all of the equations have in common?

2. Graph and label each equation on the coordinate plane.

● y 5 23x

● y 5 23x 2 2

● y 5 5 2 3x

● y 5 23x 2 8

x86–2 42–4–6–8

2

4

6

8

–2

–4

–6

–8

y

0


b. What do all of the equations have in common?

You might want to write

all the equations in slope-intercept form to make comparing

easier.

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3. Consider these equations.

● y 5 4x

● y 5 6 1 4x

● y 5 4x 2 3

● y 5 22 1 4x

a. Without graphing these equations, describe the relationship between the lines.

b. Explain how you determined the relationship between the lines.

4. Create four linear equations that represent lines with the same slope.

a. b.

c. d.

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e. Graph and label your equations.

x86–2 42–4–6–8

2

4

6

8

–2

–4

–6

–8

y

0

f. Describe the relationship between the lines.

g. Compare the graphs of your equations with those of your classmates. What can

you conclude about the slopes of parallel lines?

5. What is the slope of a line that is parallel to the line represented by the equation

y 5 200x 1 93?

6. What is the slope of a line that is parallel to the line represented by the equation

y 5 20 2 7x?

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7. Write equations for four different lines that are parallel to the line y 5 x.

a. b.

c. d.

e. Verify that your lines are parallel by graphing them on the coordinate plane.

x86–2 42–4–6–8

2

4

6

8

–2

–4

–6

–8

y

0

8. Identify the slope value in each of the equations shown to determine if the lines

represented by the equations are parallel to each other.

y 5 5x 1 4

y 5 7 1 5x

y 1 2x 5 3x 1 10

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Problem 2 Graphing Equations, Part 2


● y 5 2 __ 3 x

● y 5 2 3 __ 2

x

x86–2 42–4–6–8

2

4

6

8

–2

–4

–6

–8

y

0


b. Describe a strategy for verifying the relationship between the lines.

c. Use measuring tools to verify the relationship between the lines.

d. Calculate the product of the slopes.

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● y 5 4 __ 5 x1 1

● y 5 2 2 5 __ 4 x

x86–2 42–4–6–8

2

4

6

8

–2

–4

–6

–8

y

0


b. Calculate the product of the slopes.

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3. Consider these equations.

● y 5 6x

● y 5 2 2 1 __ 6 x

a. Without graphing these equations, describe the relationship between the lines.

b. Explain how you determined the relationship between the lines.

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4. Write two equations where the product of the slope values is 21.

a. Graph and label your equations.

x86–2 42–4–6–8

2

4

6

8

–2

–4

–6

–8

y

0

b. Describe the relationship between the lines.

c. Compare the graphs of your equations with those of your classmates.

What do you notice?

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5. What can you conclude about the slope values in equations that represent

perpendicular lines?

6. What is the slope of a line that is perpendicular to the line y 5 200x 1 93?

7. What is the slope of a line that is perpendicular to the line y 5 20 2 7x?

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8. Write four different equations for lines that are perpendicular to the line y 5 x.

a. b.

c. d.

e. Verify that your lines are perpendicular by graphing them on the grid.

x86–2 42–4–6–8

2

4

6

8

–2

–4

–6

–8

y

0

9. Identify the slope value in each of the equations shown to determine if the lines they

represent are perpendicular.

y 5 3 1 10x

y 2 7 5 1 ___ 10

x

When the product of two numbers is 1, the numbers are reciprocals of one another. When

the product of two numbers is 21, the numbers are negativereciprocalsof one another.

So, the slopes of perpendicular lines are negative reciprocals of each other.

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Talk the Talk

1. Determine if the two equations are parallel, perpendicular, or neither.

a. y 5 x 1 8

y 5 10 1 x

b. 4y 5 12 2 x

y 5 24x 2 5

c. 3y 5 12 2 x

y 5 3x 1 4

d. 2y 5 x 1 8

y 5 10 2 x

e. 4y 5 12 1 x

y 5 24x 2 5

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2. Graph points A(3, 1), B(8, 1), C(10, 5), and D(5, 5).

a. Use slopes to determine if opposite sides of the figure are parallel.

b. Use slopes to determine if the diagonals of the figure are perpendicular.

x8 94 62 73 51

6

8

4

2

y

9

5

7

3

1

00


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10.3 Line Transformations • 555

Up,Down,andAllAroundLine Transformations


Explore transformations related to parallel lines.

Explore transformations related to perpendicular lines.

Use angles formed by parallel lines and transversals to

identify similar triangles.

Use angles formed by parallel lines and transversals to

justify the Triangle Sum Theorem.

Key Term Triangle Sum Theorem

In an earlier lesson, you learned that when you rotate a point (x, y)

90 degrees counterclockwise about the origin, the location of the new point is

(2y, x). But what happens when you rotate an entire line 90 degrees?

If you rotate the line described by the equation y 5 x counterclockwise

90 degrees, what would be the equation for the rotated line?

Can you graph the two lines?

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Problem 1 Translating Lines

1. Points A(3, 1) and B(8, 4) are given.

● Connect points A and B to form line AB.

● Create points A and B by vertically translating points A and B 10 units.


12

16

8

4

x8 94 62 73 51

y

18

10

14

6

2A

B

00

2. Calculate the slope of line AB. 3. Calculate the slope of line AB.

4. Is line AB parallel to line AB? Why or why not?

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5. How could a transversal help to prove that lines AB and AB

are parallel?

6. Draw a transversal on the graph and use a protractor to verify that

line AB is parallel to line AB.

Problem 2 Rotating Lines

1. Points A(3, 1) and B(8, 4) are given.


● Use point A as the point of rotation and rotate line AB 90° counterclockwise.

● Sketch this image line and label its y-intercept C.

x16

12

16

188

8

124

4

146 10200

y

18

10

14

6

2

A

B

Think about all the angle relationships

when a transversal cuts parallel lines.

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2. Calculate the slope of line AB.

3. Calculate the slope of line AC.

4. Is line AB perpendicular to line AC? Why or why not?

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Problem 3 Reflecting Lines

1. Points A(3, 2), B(8, 1), C(3, 0), and D(8, 21) are given.

● Connect point A to point B to form line segment AB, and connect point C to point

D to form line segment CD.

● Graph the reflection line y 5 2x.

● Reflect point A over the reflection line y 5 2x to create point A.

● Reflect point B over the reflection line y 5 2x to create point B.

● Reflect point C over the reflection line y 5 2x to create point C.

● Reflect point D over the reflection line y 5 2x to create point D.

● Connect point A to point B to form line segment AB, and connect point C

to point D to form line segment CD.

x8 10642–2–4–6

2

4

6

–2

–4

–6

–8

–10

y

A

C

D

B

0

2. What are the coordinates of points A, B, C, and D?

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4. Calculate the slope of line CD.

5. Is line AB parallel to line CD? Why or why not?


7. Calculate the slope of line CD.

8. Is line AB parallel to line CD? Why or why not?

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Problem 4 Triangle Relationships

Use what you have learned about triangle similarity to answer the following questions.

Given: ‹

___ › BD i

‹

___ › HG , ‹

___ › AH i

‹

___ › DF ,

‹

___ › AH '

‹

___ › AG ,

‹

___ › DF '

‹

___ › AG

A C G

D

E

F

H

B

1. Identify all of the triangles in the diagram.

2. Is nABC , nAHG? Explain your reasoning.

You studied three ways to show

triangle similarity in a previous chapter. Can you

use any of those three methods with the information given?

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3. Is nABC , nEDC? Explain your reasoning.

4. Is nEDC , nEFG? Explain your reasoning.

5. Is nABC , nEFG? Explain your reasoning.

6. Is nAHG , nEFG? Explain your reasoning.

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Problem 5 Parallel Lines and the Triangle Sum Theorem

Given: ‹

___ › AB i

‹

___ › CE , ‹

___ › AC i

‹

___ › BD ,

‹

___ › AD i

‹

___ › BE

A B

C D E

1 3

2

1. Label all other angles in the diagram congruent to /1 by writing a 1 at the location

of each angle.


of each angle.


of each angle.

The TriangleSumTheoremstates that the sum of the measures of the three interior

angles of a triangle is equal to 180°.

4. Explain how this diagram can be used to justify the Triangle Sum Theorem.


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Chapter 10 Summary • 565

Chapter 10 Summary

Key Terms reciprocal (10.2)

negative reciprocal (10.2)

Triangle Sum Theorem (10.3)

Translating Linear Functions

You learned that a translation is a transformation that “slides” each point of a geometric

figure the same distance and direction. That knowledge can also be applied to linear

functions on a coordinate plane.

Example

Complete the table of values using the function y 5 x.

x y

22 22

21 21

1 1

2 2

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Graph the function using the table of values.

2

4

6

8

-2

-4

-6

-8

2-2-4-6-8 4 6 8x

y

y = x

+ 5

y = x

0

Translate the graph of y 5 x up 5 units and complete the table of values.

x y

22 3

21 4

1 6

2 7

Write an equation in the form y 5 to represent the translation.

y 5 x1 5

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Determining Slopes of Parallel and Perpendicular Lines

The equations of parallel lines have equal slope values. The equations of perpendicular

lines have slope values that are negative reciprocalsof each other. The product of the

slope values of perpendicular lines is 21.

Example

Consider each linear equation.

a. 16x 1 4y 5 32

b. 220x 2 5y 5 15

c. 2x 2 8y 5 48

First, determine the slope of each line.

a. 16x 1 4y 5 32

4y 5 216x 1 32

y 5 24x 1 8

slope 5 24

b. 220x 2 5y 5 15

25y 5 20x 1 15

y 5 24x 2 3

slope 5 24

c. 2x 2 8y 5 48

28y 5 22x 1 48

y 5 2 __ 8 x 2 6

y 5 1 __ 4

x 2 6

slope 5 1 __ 4

Next, compare the slopes to determine if the lines are parallel or perpendicular.

The slopes of part (a) and part (b) are equal, so the lines are parallel.

The product of the slopes of part (a) and part (c) is 21, so the lines are perpendicular.

The product of the slopes of part (b) and part (c) is 21, so the lines are perpendicular.

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Translating Lines

Translating a line 908 will form a new line that is parallel to the original line.

Example

Line AB was translated vertically 5 units to create line CD. You can calculate the slope of

each line to determine if the lines are parallel.

x86–2 42–4–6–8

2

4

6

8

–2

–4

–6

–8

y

A

B

D

C

0

line AB:

m 5y2 2 y1 _______ x2 2 x1

52 2 (21)

________ 4 2 (23)

53 __ 7

line CD:

m 5y2 2 y1 _______ x2 2 x1

5 7 2 4 _________ 4 2 (23)

53 __ 7

The slope of line AB is equal to the slope of line CD, so line AB is parallel to line CD.

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Rotating Lines

Rotating a line will form a new line that is perpendicular to the original line.

Example

Line AB was rotated 90°counterclockwise around point A to form line AC. You can

calculate the slope of each line to determine if the lines are perpendicular.

x86

2

4

6

8

–2

–2

42–4

–4

–6

–6

–8

–8

y

A

C

B

0

line AB:

m 5y2 2 y1 _______ x2 2 x1

52 2 (23)

________ 0 2 (22)

55 __ 2

line AC:

m 5y2 2 y1 _______ x2 2 x1

521 2 (23)

___________ 27 2 (22)

5 22 __ 5

The slope of line AC is the negative reciprocal of the slope of line AB, so line ACis

perpendicular to line AB.

462351_CL_C3_CH10_pp527-570.indd 569 28/08/13 2:47 PM


© C

arne

gie

Lear

ning

Reflecting Lines

Reflecting two parallel lines will form two new parallel lines.

Example

Line segment AB has been reflected over the reflection line y 5 2x to form line

segment CD.

Line segment EF has been reflected over the reflection line y 5 2x to form line

segment GH.

x6

2

4

6

8

–2

–4

–6

8–2 42–4–6–8

–8

y

A

C

D

G

H

B

E

F

0

slope of ____

AB 5 2 4 __ 5

slope of ___

EF 5 2 4 __ 5

____

AB i ___

EF

slope of ____

CD 5 2 5 __ 4

slope of ____

GH 5 2 5 __ 4

___

CD i ___

GH

462351_CL_C3_CH10_pp527-570.indd 570 28/08/13 2:47 PM

parallel and perpendicular lines · functions and the translations of linear functions. sliding...

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