Lesson 9-2 Re ections 553

9-2

Objective To fi nd refl ection images of fi gures

Re ections

Look at the shapes at the right. Visualize flipping each shape across its yellow line. What word do the images of the shapes form? Copy the shapes as they are shown and sketch the results of flipping them.

In the Solve It, you visualized shapes fl ipping across lines. Notice that when a fi gure fl ips across a line, the preimage and its image are congruent and have opposite orientations. You can tell that two fi gures have opposite orientations if the corresponding vertices of the preimage and image read in opposite directions.

Essential Understanding Th e size and shape of a geometric fi gure stay the same when you fl ip the fi gure across a line. Its orientation reverses.

The vertices ofNBUG readclockwise.

The vertices ofNBUG readcounterclockwise.G G

B B

U U

Key Concept Re ection Across a Line

Re ection across a line r, called the line of re ection, is a transformation with these two properties: If a point A is on line r, then the

image of A is itself (that is, Ar 5 A). If a point B is not on line r, then r is

the perpendicular bisector of BBr .

A refl ection across a line is an isometry.

The preimage B andits image B areequidistant fromthe line of reection.

B

C

A

r

C

B

A

sWtsaf

Flipping a figure across a line is another type of transformation.

Dynamic ActivityRe ections

AC T I V I T I

E S

D

AAAAAAAAC

ACC

I ESSSSSSSS

DYNAMIC

Lesson Vocabulary

re ection line of re ection

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Problem 1

Got It?

Problem 2

Got It?

554 Chapter 9 Transformations

Re ecting a Point Across a Line

Multiple Choice If point P(3, 4) is refl ected across the line y 5 1, what are the coordinates of its refl ection image?

(3, 24) (0, 4) (3, 22) (23, 22)

Graph point P and the line of refl ection y 5 1. P and its refl ection image across the line must be equidistant from the line.

P is 3 units above the line y 5 1, so Pr is 3 units below the line y 5 1. Th e line y 5 1 is the perpendicular bisector of PPr if Pr is (3, 22). Th e correct answer is C.

1. What is the image of P(3, 4) refl ected across the line x 5 21?

Graphing a Re ection Image

Coordinate Geometry Graph points A(23, 4), B(0, 1), and C(4, 2). What is the image of kABC refl ected across the y-axis?

Step 1 Graph nABC . Show they-axis as the dashedline of refl ection.

Step 2 Find Ar, Br, and Cr . Br is in the same position as B because B is on the line of refl ection. Locate Ar and Cr so that the y-axis is the perpendicular bisector of

AAr and CCr .

Step 3 Draw nArBrCr.

2. Graph nABC from Problem 2. What is the image of nABC refl ected across the x-axis?

Stop when the distances of P and Pto the line of reection are the same.

Move along the line through P thatis perpendicular to the line of reection.

xy 1

222

2

4

O 4

yP

P

2

y

x

5

O4 2 4

A

B

C

A

BB

A

C C

5

24 2 4Ox

y

A

BB

A

C C

5

24 2 4Ox

y

kABC intersects the line of re ection. How will the image relate to the line of re ection?The image will also intersect the line of re ection.

P

t

How does a graph help you visualize the problem?A graph shows that y 5 1 is a horizontal line, so the line through P that is perpendicular to the line of re ection is a vertical line.

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Lesson Check

Problem 3

Got It?

Lesson 9-2 Re ections 555

Minimizing a Distance

Hiking Beginning from a point on Summit Trail, a hiking club will build a trail to the Overlook and a trail to Balance Rock. Working under a tight budget, the club members want to minimize the total length of the two trails. How can you fi nd the point on Summit Trail where the two new trails should start? Assume the trails will be straight and that they will cover similar terrain.

You need to fi nd the point P on line t such that OP 1 PR is as small as possible. Locate Or, the refl ection image of O across t. Because t is the

perpendicular bisector of OOr, PO 5 POr and OP 1 PR 5 OrP 1 PR. By the Triangle Inequality Th eorem, the sum OrP 1 PR is smallest when R, P, and Or are collinear. So, the trails should start at the point P where ROr intersects line t.

3. Reasoning Your classmate began to solve the problem above by refl ecting point R in line t. Will her method work? Explain.

Balance RockSummit Trail

OverlookO

R

O

R

P

tO

The ending points, O and R, of the two trails

The point P on Summit Trail for which OP 1 PR is as small as possible

Re ect O across Summit Trail to Or. Then nd the intersection of Summit Trail and ROr .

YouOPrefl

perOPTh eand

poi

Thof

To see how the Triangle Inequality Theorem applies, look at another possible starting point, S. You can use nSROr to explain why OrS 1 SR . OrP 1 PR.

O Ot

P

RS

Do you know HOW?Use the graph of kFGH.

1. What are the coordinates of H refl ected across the y-axis?

2. What are the coordinates of G refl ected across the line x 5 3?

3. Graph nFGH and its refl ection image across the line y 5 4.

Do you UNDERSTAND? 4. Vocabulary What is the relationship between a line

of refl ection and a segment joining corresponding points of the preimage and image?

5. Error Analysis A classmate refl ected point A across line r as shown in the diagram. Explain your classmates error. Copy point A and line r and show the correct location of Ar.

6. What are the coordinates of a point P(x, y) refl ected across the y-axis? Across the x-axis?

y4

2

24

4

O 4x

F

H

G

A

r

A

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556 Chapter 9 Transformations

Practice and Problem-Solving Exercises

Each point is refl ected across the line indicated. Find the coordinates of each image.

7. Q across x 5 1 8. R across y 5 21

9. S across the y-axis 10. T across y 5 0.5

11. U across x 5 23 12. V across the x-axis

Coordinate Geometry Given points J(1, 4), A(3, 5), and R(2, 1), graph kJAR and its refl ection image across each line.

13. the x-axis 14. the y-axis 15. y 5 2 16. y 5 5 17. x 5 21 18. x 5 2

19. Civil Engineering Town offi cials in Waterville and Drighton plan to construct a water pumping station along the Franklin Canal. Th e station will provide both towns with water. Where along the canal should the offi cials build the pumping station to minimize the total length of pipe needed?

Copy each fi gure and line

Lesson 9-2 Re ections 557

25. Recreation When you play pool, you can use the fact that the ball bounces off the side of the pool table at the same angle at which it hits the side. Suppose you want to put the ball at point B into the pocket at point P by bouncing it

off side RS. Off what point on RS should the ball bounce? Draw a diagram and explain your reasoning.

26. Think About a Plan Th e coordinates of the vertices of nFGH are F(2, 21), G(22, 22), and H(24, 3). Graph nFGH and its image nFrGrHr for a refl ection across the line y 5 x 2 3.

What is the relationship between the line y 5 x 2 3 and FFr, GGr, and HHr? How can you use slope to fi nd the image of each vertex?

Copy each pair of fi gures. Th en draw the line of refl ection you can use to map one fi gure onto the other.

27. 28. 29.

30. History Th e work of artist and scientist Leonardo da Vinci (14521519) has an unusual characteristic. His handwriting is a mirror image of normal handwriting.

a. Write the mirror image of the sentence, Leonardo da Vinci was left-handed. Use a mirror to check how well you did.

b. Explain why the fact about da Vinci in part (a) might have made mirror writing seem natural to him.

31. Open-Ended Give three examples from everyday life of objects that come in a left-handed version and a right-handed version.

Find the image of O(0, 0) after two refl ections, fi rst across line

558 Chapter 9 Transformations

43. Use the diagram at the right. Find the coordinates of the given point.

a. Ar , the refl ection image of A across y 5 x b. Arr , the refl ection image of Ar across y 5 2x c. Arrr , the refl ection image of Arr across y 5 x d. Arrrr , the refl ection image of Arrr across y 5 2x e. How are A and Arrrr related?

Reasoning Can you form the given type of quadrilateral by drawing a triangle and then refl ecting one or more times? Explain.

44. parallelogram 45. isosceles trapezoid 46. kite

47. rhombus 48. rectangle 49. square

50. Coordinate Geometry Show that B(b, a) is the refl ection image of A(a, b) across the line y 5 x. (Hint: Show that y 5 x is the perpendicular bisector of AB.)

x

y A (1, 3)

y x

y x

2

2

22

ChallengeC

Mixed Review

For the given points, ArBr is a translation of AB. Write a rule to describe each translation.

55. A(21, 5), B(2, 0), Ar(3, 3), Br(6, 22) 56. A(29, 24), B(27, 1), Ar(24, 23), Br(22, 2)

57. Maps A map of Alberta, Canada, uses the scale 1 cm 5 25 km. On the map, the distance from Calgary to Edmonton is 11.1 cm. How far apart are the two cities?

Get Ready! To prepare for Lesson 9-3, do Exercises 5863.

Use a protractor to draw an angle with the given measure.

58. 1208 59. 908 60. 728 61. 608 62. 458 63. 368

See Lesson 9-1.

See Lesson 7-2.

See p. 824.

Standardized Test Prep

51. What is the refl ection image of (a, b) across the line y 5 26 ?

(a 2 6, b) (a, b 2 6) (212 2 a, b) (a, 212 2 b)

52. Th e diagonals of a quadrilateral are perpendicular and bisect each other. What is the most precise name for the quadrilateral?

rectangle parallelogram rhombus kite

53. AD) bisects /A of nCAB, with point D on CB. Which of the following is true?

CDDB 5CABA

CDDB 5

BACA

CBCD 5

CABA

CACB 5

ABDB

54. Write an indirect proof of the following statement: Th e hypotenuse of a right triangle is the longest side of the right triangle.

SAT/ACT

ExtendedResponse

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