Introduction to Coordinate Algebrachambleems.dekalb.k12.ga.us/Downloads/TB-Chapter...

© 2

012

Car

negi

e Le

arni

ng

879 879

Introduction to Coordinate Algebra

18.1 Sliding LinesTranslations of Linear Functions .................................. 881

18.2 Parallel or Perpendicular?Slopes of Parallel and Perpendicular Lines ..................893

18.3 Up, Down, and All AroundLine Transformations ..................................................907

18.4 Back on the GridSimilar Triangles on the Coordinate Plane ................... 917

The London

Eye–situated on the banks of the River Thames–is one of the

most famous Ferris wheels in the world. It rotates at about 10 inches per second. One trip all the way around

takes about 30 minutes. It moves so slowly that

passengers can just walk on and off.

880 • Chapter 18 Introduction to Coordinate Algebra

© 2

012

Car

negi

e Le

arni

ng

© 2

012

Car

negi

e Le

arni

ng

18.1 Translations of Linear Functions • 881

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.

Sliding LinesTranslations 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.


© 2

012

Car

negi

e Le

arni

ng

Problem 1 Translating Linear Functions Up or Down

In the previous lesson, 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?

© 2

012

Car

negi

e Le

arni

ng


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

4. In the previous lesson, a geometric figure was translated down 4 units.

a. How did that affect the value of the x-coordinate of each vertex?

b. How did that 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?


© 2

012

Car

negi

e Le

arni

ng

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?

© 2

012

Car

negi

e Le

arni

ng


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. In the previous lesson, a geometric figure was translated to the left 4 units.

a. How did that affect the value of the

x-coordinate of each vertex?

b. How did that affect the value of the y-coordinate of

each vertex?

I think there is going to be a connection!


© 2

012

Car

negi

e Le

arni

ng

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.


© 2

012

Car

negi

e Le

arni

ng




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?


© 2

012

Car

negi

e Le

arni

ng

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.

© 2

012

Car

negi

e Le

arni

ng


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


© 2

012

Car

negi

e Le

arni

ng

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

© 2

012

Car

negi

e Le

arni

ng


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.


© 2

012

Car

negi

e Le

arni

ng

© 2

012

Car

negi

e Le

arni

ng

18.2 Slopes of Parallel and Perpendicular Lines • 893

Parallel or Perpendicular?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.


© 2

012

Car

negi

e Le

arni

ng

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.

© 2

012

Car

negi

e Le

arni

ng


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.


© 2

012

Car

negi

e Le

arni

ng

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.

© 2

012

Car

negi

e Le

arni

ng


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?


© 2

012

Car

negi

e Le

arni

ng

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

© 2

012

Car

negi

e Le

arni

ng


Problem 2 Graphing Equations, Part 2

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

● 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.


© 2

012

Car

negi

e Le

arni

ng

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

● y 5 4 __ 5 x 1 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.

© 2

012

Car

negi

e Le

arni

ng


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.


© 2

012

Car

negi

e Le

arni

ng

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?

© 2

012

Car

negi

e Le

arni

ng


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?


© 2

012

Car

negi

e Le

arni

ng

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 negative reciprocals of one another.

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

© 2

012

Car

negi

e Le

arni

ng


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


© 2

012

Car

negi

e Le

arni

ng

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


© 2

012

Car

negi

e Le

arni

ng

18.3 Line Transformations • 907

Up, Down, and All AroundLine Transformations


Explore transformations related to parallel lines.

Explore transformations related to perpendicular lines.

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?


© 2

012

Car

negi

e Le

arni

ng

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?

© 2

012

Car

negi

e Le

arni

ng


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.


© 2

012

Car

negi

e Le

arni

ng

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?

© 2

012

Car

negi

e Le

arni

ng


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?


© 2

012

Car

negi

e Le

arni

ng


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?

© 2

012

Car

negi

e Le

arni

ng


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 the last chapter. Can you use

any of those three methods with the information given?


© 2

012

Car

negi

e Le

arni

ng

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.

© 2

012

Car

negi

e Le

arni

ng


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 Triangle Sum Theorem states 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.


© 2

012

Car

negi

e Le

arni

ng

18.4 Similar Triangles on the Coordinate Plane • 917

Back on the GridSimilar Triangles on the Coordinate Plane


Graph similar triangles and determine the dilation factor.

Dilate triangles to form similar triangles.

Verify that triangles are similar using Similarity Theorems.

Determine the coordinates of a point needed to form similar triangles.

You know that the Earth is farther away from the Sun than the planet Venus.

But is there a way to tell just by looking? Amazingly, the ancient Greeks used

similar triangles to figure this out.

They observed that Venus was never more than 47° high in the sky at sunset.

Venus

Sun Earth

47o

And they knew something that you will learn later in mathematics—that for any

angle in a right triangle, the ratio of the opposite side length to the hypotenuse is

always the same for all similar triangles, no matter what these lengths are.

The ancient Greeks knew that for a 47° angle, the ratio of the opposite side to the

hypotenuse was about 0.72. This meant that the Venus–Sun distance was about 7 ___ 10

of the Earth–Sun distance. Venus is closer!

Can you see how they did it?


© 2

012

Car

negi

e Le

arni

ng

Problem 1 Similar Triangles Resulting from Dilations

Let’s explore methods for showing two triangles are similar through dilations.

1. Triangle ABC has vertices A(3, 23), B(8, 3), and C(8, 23).

a. Graph triangle ABC.

x86

2

4

8

10 12–2–2

42–4

–4

–6

–8

–8

–10–12

y

10

12

–10

–12

6

–60

b. Triangle DEF is the image that resulted from the dilation of triangle ABC. The

coordinates of triangle DEF are D (1.5, 21.5), E (4, 1.5), and F (4, 21.5). Graph

triangle DEF on the coordinate plane in part (a).

c. What scale factor was used?

d. How did you determine the scale factor?

e. Is the dilation an enlargement or reduction? Explain your reasoning.

© 2

012

Car

negi

e Le

arni

ng


f. How can you verify that triangle ABC and triangle DEF are similar?

g. Use the SAS Similarity Theorem to verify triangle ABC is similar to triangle DEF.


© 2

012

Car

negi

e Le

arni

ng

2. Triangle MAP is the image that resulted from the dilation of triangle QRN.

x

86

2

4

8

10–2–2

42–4

–4

–6

–8

–8

–10–12–14

y

–10

–12

(7, 1)

(7, 7)(5, 7) R

(214, 22)P

(210, 214)M

(214, 214)A

Q

N

6

–6

10

–14

0

a. What scale factor was used?

b. How did you determine the scale factor?

c. Is the dilation an enlargement or reduction? Explain your reasoning.

d. How can you verify that triangle MAP and triangle QRN are similar?

© 2

012

Car

negi

e Le

arni

ng


e. Use the SSS Similarity Theorem to verify triangle MAP is similar to triangle QRN.


© 2

012

Car

negi

e Le

arni

ng

3. Analyze triangle PWN.

a. Dilate triangle PWN shown using a scale factor of 4 to form triangle GKA.

x86

2

4

8

10 12–2–2

42–4

–4

–6

–8

–8

–10–12

y

10

12

–10

–12

(3, 3)

(–1, 1)(3, 0)

P

W

N

6

–60

b. What are the coordinates of the dilated image?

c. How did you determine the coordinates of the dilated image?

d. Is the dilation an enlargement or reduction? Explain your reasoning.

e. How can you verify that triangle PWN and triangle GKA are similar?

f. Use the AA Similarity Theorem to verify triangle PWN is similar to triangle GKA.

© 2

012

Car

negi

e Le

arni

ng


4. Analyze triangle ZEN.

a. Dilate triangle ZEN shown using a scale factor of 1 __ 2

to form triangle FRB.

x86

2

4

8

10 12–2–2

42–4

–4

–6

–8

–8

–10–12

y

10

12

–10

–12

(25, 3)

(8, 8)

Z

E

(10, 1)N

6

–60

b. What are the coordinates of the dilated image?

c. How did you determine the coordinates of the dilated image?

d. Is the dilation an enlargement or reduction? Explain your reasoning.

e. Use the AA Similarity Theorem to verify triangle ZEN is similar to triangle FRB.


© 2

012

Car

negi

e Le

arni

ng


Talk the Talk

This chapter focused on three methods to show that two triangles are similar. Complete

the graphic organizer by listing the methods and provide an illustration of each method.

© 2

012

Car

negi

e Le

arni

ng

Chapter 18 Summary • 925

Chapter 18 Summary

Key Terms reciprocal (18.2)

negative reciprocal (18.2)

Triangle Sum Theorem (18.3)

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

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


© 2

012

Car

negi

e Le

arni

ng

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 x 1 5

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 reciprocals of 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

© 2

012

Car

negi

e Le

arni

ng


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.


© 2

012

Car

negi

e Le

arni

ng

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)

5 3 __ 7

line CD:

m 5y2 2 y1 _______ x2 2 x1

5 7 2 4 _________ 4 2 (23)

5 3 __ 7

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

© 2

012

Car

negi

e Le

arni

ng


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)

5 5 __ 2

line AC:

m 5y2 2 y1 _______ x2 2 x1

521 2 (23)

__________ 27 2 (22)

5 2 2 __ 5

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

perpendicular to line AB.


© 2

012

Car

negi

e Le

arni

ng

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

© 2

012

Car

negi

e Le

arni

ng


Similar Triangles on the Coordinate Plane

Properties of similar triangles can be used to graph similar triangles, determine the scale

factor that was used to create the triangles, and verify that the triangles are similar.

Example

Triangle ABC has vertices A(1,2), B(1, 5), C(5, 2). Graph triangle ABC.

A

B

C

D

E

F

2 4 6 8 10 12 14 16 18

18

16

14

12

10

8

6

4

2

00

Triangle DEF is the image that resulted from the dilation of triangle ABC. The coordinates

of triangle DEF are D(3,6), E(3, 15), F(15, 6). Graph triangle DEF.

The scale factor used was 3.

Use the SAS Similarity Theorem to verify triangle ABC is similar to triangle DEF.

/A is congruent to /D because they are both right angles and all right angles are

congruent.

AB ___ DE

5 AC ___ DF

3 __ 9 5 4 ___

12

1 __ 3 5 1 __

3

The lengths of the sides that include /A and /D are proportional.

Triangle ABC is similar to triangle DEF by the SAS Similarity Theorem.

Introduction to Coordinate Algebrachambleems.dekalb.k12.ga.us/Downloads/TB-Chapter...

Documents

Transcript of Introduction to Coordinate Algebrachambleems.dekalb.k12.ga.us/Downloads/TB-Chapter...