Chapter 12 (Interactive), Exercises of Geometry

Download Chapter 12 (Interactive) and more Exercises Geometry in PDF only on Docsity!

820 Chapter 12

Extending

Transformational

Geometry

12A Congruence

Transformations

12-1 Reflections 12-2 Translations 12-3 Rotations Lab Explore Transformations with Matrices 12-4 Compositions of Transformations

12B Patterns

12-5 Symmetry 12-6 Tessellations Lab Use Transformations to Extend Tessellations 12-7 Dilations Ext Using Patterns to Generate Fractals

KEYWORD: MG7 ChProj

Let it Snow!

A blanket of snow is formed by trillions of symmetric crystals. You can use transformations and symmetry to explore snow crystals.

Extending Transformational Geometry 821

Vocabulary

Match each term on the left with a definition on the right.

1. image 2. preimage 3. transformation 4. vector

A. a mapping of a figure from its original position to a new position B. a ray that divides an angle into two congruent angles C. a shape that undergoes a transformation D. a quantity that has both a size and a direction E. the shape that results from a transformation of a figure

Ordered Pairs

Graph each ordered pair.

Congruent Figures

Can you conclude that the given triangles are congruent? If so, explain why.

11.
PQS and
PRS 12.
DEG and
FGE

*

+ ,

-

 

Identify Similar Figures

Can you conclude that the given figures are similar? If so, explain why.

13.
JKL and
JMN 14. rectangle PQRS and rectangle UVWX





  È

{

{°x

Î *

-

+ 8 1

, 7 6

n

{

™

£n

Angles in Polygons

15. Find the measure of each interior angle of a regular octagon.

 Ý Â

Ý Â

Ý Â

Ý Â

16. Find the sum of the interior angle measures of a convex pentagon. Ý Â

17. Find the measure of each exterior angle of a regular hexagon. 18. Find the value of x in hexagon ABCDEF.

Extending Transformational Geometry 823

Study Strategy: Prepare for Your Final Exam

Math is a cumulative subject, so your final exam will probably cover all of the material you have learned since the beginning of the course. Preparation is essential for you to be successful on your final exam. It may help you to make a study timeline like the one below.

Try This

1. Create a timeline that you will use to study for your final exam.

1 week before the final:1 week before the final:

• • Take the practice exam and check it.Take the practice exam and check it. For each problem I miss, find two orFor each problem I miss, find two or three similar ones and work those.three similar ones and work those.
• • Work with a friend in the class to quizWork with a friend in the class to quiz each other on formulas, postulates,each other on formulas, postulates, and theorems from my list.and theorems from my list.

2 weeks before the final:2 weeks before the final:

• • Look at previous exams and homework toLook at previous exams and homework to determine areas I need to focus on; reworkdetermine areas I need to focus on; rework problems that were incorrect or incomplete.problems that were incorrect or incomplete.
• • Make a list of all formulas, postulates, andMake a list of all formulas, postulates, and theorems I need to know for the final.theorems I need to know for the final.
• • Create a practice exam using problems fromCreate a practice exam using problems from the book that are similar to problems fromthe book that are similar to problems from each exam.each exam.

1 day before the final:1 day before the final:

• • Make sure I have pencils, calculatorMake sure I have pencils, calculator (check batteries!), ruler, compass,(check batteries!), ruler, compass, and protractor.and protractor.

824 Chapter 12 Extending Transformational Geometry

12-1 Reflections

Who uses this? Trail designers use reflections to find shortest paths. (See Example 3.)

An isometry is a transformation that does not change the shape or size of a figure. Reflections, translations, and rotations are all isometries. Isometries are also called congruence transformations or rigid motions.

Recall that a reflection is a transformation that moves a figure (the preimage) by flipping it across a line. The reflected figure is called the image. A reflection is an isometry, so the image is always congruent to the preimage.

E X A M P L E 1 Identifying Reflections Tell whether each transformation appears to be a reflection. Explain.

A B

Yes; the image appears No; the figure does not appear to be flipped across a line. to be flipped.

Tell whether each transformation appears to be a reflection. 1a. 1b.

Draw a triangle and a line of reflection on a piece of patty paper.

Fold the patty paper back along the line of reflection.

Trace the triangle. Then unfold the paper.

 

Construction Reflect a Figure Using Patty Paper

Draw a segment from each vertex of the preimage to the corresponding vertex of the image. Your construction should show that the line of reflection is the perpendicular bisector of every segment connecting a point and its image.

Objective Identify and draw reflections.

Vocabulary isometry

To review basic transformations, see Lesson 1-7, pages 50−55.

826 Chapter 12 Extending Transformational Geometry

ACROSS THE x -AXIS ACROSS THE y -AXIS ACROSS THE LINE y = x

* Ý ]Ê Þ ®

* Ī Ý ]Ê  Þ ®

Ý

Þ

ä

Ý ]Ê Þ ®ÊÊƐ Ý ]Ê  Þ ®

* Ī  Ý ]Ê Þ ® * Ý ]Ê Þ ® Ý

Þ

ä

Ý ]Ê Þ ®ÊÊƐ  Ý ]Ê Þ ®

* Ý ]Ê Þ ®

* Ī Þ ]Ê Ý ® Ý

Þ

Ý ]Ê Þ ®ÊÊ Þ ]Ê Ý ®

ÞÊ  ÊÝ

Reflections in the Coordinate Plane

E X A M P L E 4 Drawing Reflections in the Coordinate Plane

Reflect the figure with the given vertices across the given line. A M (1, 2), N (1, 4), P (3, 3); y -axis

The reflection of ( x , y ) is ( - x , y ). Þ

Ý  {  Ó ä Ó {

Ó

{ * Ī *  Ī

Ī



M ( 1 , 2 ) → M '(- 1 , 2 )

N ( 1 , 4 ) → N '(- 1 , 4 )

P ( 3 , 3 ) → P '(- 3 , 3 )

Graph the preimage and image.

B D (2, 0), E (2, 2), F (5, 2), G (5, 1); y = x

The reflection of ( x , y ) is ( y , x ).

Þ

Ý  Ó Ó { È

{ Ī Ī

 Ī





 Ī

D ( 2 , 0 ) → D '( 0 , 2 )

E ( 2 , 2 ) → E '( 2 , 2 )

F ( 5 , 2 ) → F '( 2 , 5 )

G ( 5 , 1 ) → G '( 1 , 5 )

Graph the preimage and image.

4. Reflect the rectangle with vertices S (3, 4), T (3, 1), U (-2, 1),

and V (-2, 4) across the x -axis.

THINK AND DISCUSS

1. Acute scalene  ABC is reflected across

BC. Classify quadrilateral ABA ' C. Explain your reasoning.

2. Point A ' is a reflection of point A across line
   . What is the relationship of
   to

AA '?

3. GET ORGANIZED Copy and complete the graphic organizer.

ˆ˜iʜvÊ,iviV̈œ˜ “>}iʜvÊ >]ÊL ®Ê Ý>“«i Ý ‡>ÝˆÃ Þ ‡>ÝˆÃ Þ Ê 
Ý

12-1 Reflections 827

Exercises Exercises

12- KEYWORD: MG7 12-

KEYWORD: MG7 Parent

GUIDED PRACTICE

1. Vocabulary If a transformation is an isometry , how would you describe the relationship between the preimage and the image?

S E E E X A M P L E 1 p. 824

Tell whether each transformation appears to be a reflection.

2. 3.

S E E E X A M P L E 2

p. 825

Multi-Step Copy each figure and the line of reflection. Draw the reflection of the figure across the line.

6. 7.

S E E E X A M P L E 3

p. 825

8. City Planning The towns of San Pablo and Tanner are ->˜ *>Lœ

/>˜˜iÀ

ˆ} Ü>ÞÊ£äx

located on the same side of Highway 105. Two access roads are planned that connect the towns to a point P on the highway. Draw a diagram that shows where point P should be located in order to make the total length of the access roads as short as possible.

S E E E X A M P L E 4 p. 826

Reflect the figure with the given vertices across the given line.

9. A (-2, 1), B (2, 3), C (5, 2); x -axis

10. R (0, - 1 ), S (2, 2), T (3, 0); y -axis

11. M (2, 1), N (3, 1), P (2, - 1 ), Q (1, - 1 ); y = x

12. A (-2, 2), B (-1, 3), C (1, 2), D (-2, - 2 ); y = x

PRACTICE AND PROBLEM SOLVING

For See Exercises Example 13–16 1 17–18 2 19 3 20–23 4

Independent Practice Tell whether each transformation appears to be a reflection.

13. 14.

Skills Practice p. S Application Practice p. S

Extra Practice

12-1 Reflections 829

38. Critical Thinking Sketch the next figure in the sequence below.

       

39. Critical Thinking Under a reflection in the coordinate plane, the point (3, 5)

is mapped to the point (5, 3). What is the line of reflection? Is this the only possible

line of reflection? Explain.

Draw the reflection of the graph of each function across the given line.

40. x -axis 41. y -axis

Ý

Þ

ä

Þ Ê  ÊÊ Ý ÊÓ

{

 {

 { {

Ý

Þ

Þ Ê  ÊÊÓÊ Ý

ä

{

 {

 { {

42. Write About It Imagine reflecting all the points in a plane across line
. Which points remain fixed under Ű this transformation? That is, for which points is the image the same as the preimage? Explain.

Construction Use the construction of a line perpendicular to a given line through a given point (see page 179) and the construction of a segment congruent to a given segment (see page 14) to construct the reflection of each figure across a line.

43. a point 44. a segment 45. a triangle 46. Daryl is using a coordinate plane to plan a garden. He draws a flower

bed with vertices (3, 1), (3, 4), (-2, 4), and (-2, 1). Then he creates

a second flower bed by reflecting the first one across the x -axis. Which of these is a vertex of the second flower bed?

37. This problem will prepare you for the Multi-Step Test Prep on page 854.

The figure shows one hole of a miniature golf course. a. Is it possible to hit the ball in a straight line from the tee T to the hole H? b. Find the coordinates of H ', the reflection of H across

BC.

c. The point at which a player should aim in order to make a hole in one is the intersection of

TH ' and

BC. What are the coordinates of this point?

Þ

Ý Ó { È

{

È

Ó

ä

/

 

830 Chapter 12 Extending Transformational Geometry

47. In the reflection shown, the shaded figure is the 

*  ^7

preimage. Which of these represents the mapping? - MJNP → DSWG JMPN → GWSD DGWS → MJNP PMJN → SDGW

48. What is the image of the point (-3, 4) when it is

reflected across the y -axis?

(4,^ -^3 ) (3, 4)

(-3,^ -^4 ) (-4,^ -^3 )

CHALLENGE AND EXTEND Find the coordinates of the image when each point is reflected across the given line.

49. (4, 2); y = 3 50. (-3, 2); x = 1 51. (3, 1); y = x + 2

52. Prove that the reflection image of a segment is congruent to the preimage. Given:

A'B' is the reflection image of

AB across line
.

Ī

Ī

Ű

Prove:

AB 

A'B'

Plan: Draw auxiliary lines

AA' and

BB' as shown. First prove that  ACD   A'CD. Then use CPCTC to conclude that ∠ CDA  ∠ CDA'. Therefore ∠ ADB  ∠ A'DB' , which makes it possible to prove that  ADB   A'DB'. Finally use CPCTC to conclude that

AB 

A'B'.

Once you have proved that the reflection image of a segment is congruent to the preimage, how could you prove the following? Write a plan for each proof.

53. If

A ' B ' is the reflection of

AB , then AB = A ' B '.

54. If ∠ A ' B ' C ' is the reflection of ∠ ABC , then m∠ ABC = m∠ A ' B ' C '. 55. The reflection  A ' B ' C ' is congruent to the preimage  ABC. 56. If point C is between points A and B , then the reflection C ' is between A ' and B '. 57. If points A , B , and C are collinear, then the reflections A ', B ', and C ' are collinear.

SPIRAL REVIEW A jar contains 2 red marbles, 6 yellow marbles, and 4 green marbles. One marble is drawn and replaced, and then a second marble is drawn. Find the probability of each outcome. (Previous course)

58. Both marbles are green. 59. Neither marble is red. 60. The first marble is yellow, and the second is green.

The width of a rectangular field is 60 m, and the length is 105 m. Use each of the following scales to find the perimeter of a scale drawing of the field. (Lesson 7-5)

61. 1 cm : 30 m 62. 1.5 cm : 15 m 63. 1 cm : 25 m

Find each unknown measure. Round side lengths to the nearest

Ó

ÊÊ Ȗ
е ÇÊ^
Ê

hundredth and angle measures to the nearest degree. (Lesson 8-3)

64. BC 65. m∠ A 66. m∠ C

832 Chapter 12 Extending Transformational Geometry

A translation is a transformation along a vector
Ī

/À>˜Ã>̈œ˜ ÛiV̜ÀÊÊÊ

such that each segment joining a point and its image has the same length as the vector and is parallel to the vector.

Translations

E X A M P L E 2 Drawing Translations Copy the triangle and the translation vector. Draw the translation of the triangle along v
. Ь ÛÊ Step 1 Draw a line parallel to the vector

Ь Û Ê

through each vertex of the triangle.

Step 2 Measure the length of the vector.

Û Ь Ê

Then, from each vertex mark off this distance in the same direction as the vector, on each of the parallel lines.

Step 3 Connect the images of the vertices.

Ь Û^ Ê

2. Copy the quadrilateral and the

ÜÊ Ь

translation vector. Draw the translation of the quadrilateral along w
.

Recall that a vector in the coordinate plane can be written as 〈 a , b 〉, where a is the horizontal change and b is the vertical change from the initial point to the terminal point.

HORIZONTAL TRANSLATION ALONG VECTOR 〈 a , 0 〉

VERTICAL TRANSLATION ALONG VECTOR 〈 0, b 〉

GENERAL TRANSLATION ALONG VECTOR 〈 a , b 〉

* Ý ]Ê Þ ® (^) * Ī Ý Ê Ê > ]Ê Þ ® Ý

Þ

ä

Ý ]Ê Þ ®ÊÊƐ Ý Ê Ê > ]Ê Þ ®

* Ý ]Ê Þ ®

* Ī Ý ]Ê Þ Ê Ê L ® Ý

Þ

ä

Ý ]Ê Þ ®ÊÊƐ Ý ]Ê Þ Ê Ê L ®

* Ý ]Ê Þ ® * Ī ÝÊ
> ]Ê ÞÊ Ê L ® Ý

Þ

ä

Ý ]Ê Þ ®ÊÊƐ ÝÊ
> ]Ê ÞÊ Ê L ®

Translations in the Coordinate Plane

12-2 Translations 833

E X A M P L E 3 Drawing Translations in the Coordinate Plane

Translate the triangle with vertices A (-2, - 4 ),

 { Ó {

 {

ä

Þ

Ý
Ī

Ī

Ī

B (-1, - 2 ), and C (-3, 0) along the vector 〈2, 4〉. The image of (^) ( x , y ) is (^) ( x + 2, y + (^4) ).

A ( - 2 , - 4 ) → A '( - 2 + 2 , - 4 + 4 ) = A '( 0 , 0 )

B ( - 1 , - 2 ) → B '( - 1 + 2 , - 2 + 4 ) = B '( 1 , 2 )

C ( - 3 , 0 ) → C '( - 3 + 2 , 0 + 4 ) = C '( - 1 , 4 )

Graph the preimage and image.

3. Translate the quadrilateral with vertices R (2, 5), S (0, 2),

T (1, - 1 ), and U (3, 1) along the vector 〈-3, - 3 〉.

E X A M P L E 4 Entertainment Application

In a marching drill, it takes 8 steps to march 5 yards. A drummer starts 8 steps to the left and 8 steps up from the center of the field. She marches 16 steps to the right to her second position. Then she marches 24 steps down the field to her final position. What is the drummer’s final position? What single translation vector moves her from the starting position to her final position? The drummer’s starting coordinates

n]Ên®

Ͱ £È]Êä ͱ

  

 n]Ên®

n]Ê  £È®

Ͱ £È]Ê  Ó{ ͱ Ͱ ä]Ê  Ó{ ͱ

are (-8, 8).

Her second position is

Her final position is

The vector that moves her directly from her starting position to her final position is 〈16, 0〉 + 〈0, - 24 〉 = 〈16, - 24 〉.

4. What if…? Suppose another drummer started at the center of the field and marched along the same vectors as above. What would this drummer’s final position be?

THINK AND DISCUSS

1. Point A ' is a translation of point A along v
   . What is the relationship of v
   to

AA '?

AB is translated to form

A ' B '.

Classify quadrilateral AA ' B ' B. Explain your reasoning.

3. GET ORGANIZED Copy and

/À>˜Ã>̈œ˜Ã

ivˆ˜ˆÌˆœ˜

complete the graphic organizer. Ý>“«i œ˜iÝ>“«i

12-2 Translations 835

Each frame of a computer-animated feature represents __ 241 of a second of film. Source: www.pixar.com

Animation

Multi-Step Copy each figure and the translation vector. Draw the translation of the figure along the given vector. 15.

ÛÊ Ь

Ü Ь

Translate the figure with the given vertices along the given vector.

17. P (-1, 2), Q (1, - 1 ), R (3, 1), S (2, 3); 〈-3, 0〉

18. A (1, 3), B (-1, 2), C (2, 1), D (4, 2); 〈-3, - 3 〉

19. D (0, 15), E (-10, 5), F (10, - 5 ); 〈5, - 20 〉

20. Animation An animator draws the ladybug shown and then y

–5^0 x

translates it along the vector 〈1, 1〉, followed by a translation of the new image along the vector 〈2, 2〉, followed by a translation of the second image along the vector 〈3, 3〉. a. Sketch the ladybug’s final position. b. What single vector moves the ladybug from its starting position to its final position?

Draw the translation of the graph of each function along the given vector.

21. 〈3, 0〉 22. 〈-1, - 1 〉

 {  Ó {

 {

Ó

{

ä

Þ

Ý

Þ Ê  ÊÊ Ý ÊÎ

 {  Ó Ó {

 {

Ó

{

Þ

Ý

Þ Ê 
 Ê Ý ÊÓ

23. Probability The point P (3, 2) is translated along one of the following four vectors

chosen at random: 〈-3, 0〉, 〈-1, - 4 〉, 〈3, - 2 〉, and 〈2, 3〉. Find the probability of each of the following. a. The image of P is in the fourth quadrant. b. The image of P is on an axis. c. The image of P is at the origin.

24. This problem will prepare you for the Multi-Step Test Prep on page 854. The figure shows one hole of a miniature golf course

Ó { È

Ó

{

È

/ 

Ý

Þ

ä

and the path of a ball from the tee T to the hole H. a. What translation vector represents the path of the ball from T to

DC?

b. What translation vector represents the path of the ball from

DC to H? c. Show that the sum of these vectors is equal to the vector that represents the straight path from T to H.

836 Chapter 12 Extending Transformational Geometry

Each figure shows a preimage (blue) and its image (red) under a translation. Copy the figure and draw the vector along which the polygon is translated.

25. 26. 27. Critical Thinking The points of a plane are translated along the given vector AB  . Do any points
remain fixed under this transformation? That is, are there any points for which the image coincides with the preimage? Explain. 28. Carpentry Carpenters use a tool called adjustable parallels to set up level work areas and to draw parallel lines. Describe how a carpenter could use this tool to translate a given point along a given vector. What additional tools, if any, would be needed?

Find the vector associated with each translation.

 {  Ó Ó {

 {

 Ó

Ó

{

Þ

Ý

ä

Then use arrow notation to describe the mapping of the preimage to the image.

29. the translation that maps point A to point B 30. the translation that maps point B to point A 31. the translation that maps point C to point D 32. the translation that maps point E to point B 33. the translation that maps point C to the origin 34. Multi-Step The rectangle shown is translated

Îʈ˜°

nʈ˜°

two-thirds of the way along one of its diagonals. Find the area of the region where the rectangle and its image overlap.

35. Write About It Point P is translated along the vector 〈 a , b 〉. Explain how to find the distance between point P and its image.

Construction Use the construction of a line parallel to a given (^)
Ī

line through a given point (see page 163) and the construction of a segment congruent to a given segment (see page 14) to construct the translation of each figure along a vector.

36. a point 37. a segment 38. a triangle

39. What is the image of P (1, 3) when it is translated along the vector 〈-3, 5〉?

40. After a translation, the image of A (-6, - 2 ) is B (-4, - 4 ). What is the image of

the point (3, - 1 ) after this translation?

838 Chapter 12 Extending Transformational Geometry

Transformations

of Functions

Transformations can be used to graph complicated functions by using the graphs of simpler functions called parent functions. The following are examples of parent functions and their graphs.

Try This

For each parent function, write a function rule for the given transformation and graph the preimage and image.

1. parent function: y = x^2 transformation: a translation down 1 unit and right 4 units 2. parent function: y = √ x transformation: a reflection across the x -axis 3. parent function: y =  x  transformation: a translation up 2 units and left 1 unit

y =  x  (^) Þ

Ý

y = √ x (^) Þ

Ý

y = x^2 Þ

Ý

Transformation of Parent Function y = f ( x ) Reflection Vertical Translation (^) Horizontal Translation

Across x -axis: y = - f ( x ) Across y -axis: y = f (- x )

y = f ( x ) + k Up k units if k > 0 Down k units if k < 0

y = f ( x - h )

Right h units if h > 0 Left h units if h < 0

Example

For the parent function y = x^2 , write a function rule for the given transformation and graph the preimage and image. A a reflection across the x -axis function rule: y = - x^2 graph:

 {  Ó Ó {

 {

Ó

{

Þ Ê  Ý ÊÓ

Þ Ê   Ý ÊÓ

Þ

Ý ä

B a translation up 2 units and right 3 units

function rule: y = ( x - 3 )^2 + 2

graph:

 {  Ó Ó {

 {

 Ó

Ó

{ Þ Ê  Ý ÊÓ Þ Ê  ÝÊ 
ήÊÓ
ÓÊ

Þ

Ý ä

See Skills Bank page S

Algebra

12-3 Rotations 839

12-3 Rotations

Who uses this? Astronomers can use properties of rotations to analyze photos of star trails. (See Exercise 35.)

Remember that a rotation is a transformation that turns a figure around a fixed point, called the center of rotation. A rotation is an isometry, so the image of a rotated figure is congruent to the preimage.

E X A M P L E 1 Identifying Rotations Tell whether each transformation appears to be a rotation. Explain.

A B

Yes; the figure appears to No; the figure appears be turned around a point. to be flipped, not turned.

Tell whether each transformation appears to be a rotation. 1a. 1b.

On a sheet of paper, draw a triangle and a point. The point will be the center of rotation.

Place a sheet of patty paper on top of the diagram. Trace the triangle and the point.

Hold your pencil down on the point and rotate the bottom paper counterclockwise. Trace the triangle.

 

Construction Rotate a Figure Using Patty Paper

Draw a segment from each vertex to the center of rotation. Your construction should show that a point’s distance to the center of rotation is equal to its image’s distance to the center of rotation. The angle formed by a point, the center of rotation, and the point’s image is the angle by which the figure was rotated.

Objective Identify and draw rotations.