If , , and Is it possible for a polygon to have the following sum of Is it possible for a regular polygon to have the following

EXAMPLE 3 a Which figures have at least one line of symmetry? b Which figures have more than one line of symmetry? POINT SYMMETRY In Figure 2.41, rectangle ABCD is also said to have symmetry with respect to a point. As shown, point P is determined by the intersection of the diagonals of rectangle ABCD. Exs. 1–4 Solution a The isosceles triangle, square, and the regular pentagon all have a line of symmetry. b The square and regular pentagon have more than one line of symmetry, so these figures are shown with two lines of symmetry. There are actually more than two lines of symmetry. 쮿 Figure 2.40a Isosceles Triangle Square Quadrilateral Regular Pentagon Isosceles Triangle Figure 2.40b Square Regular Pentagon Figure 2.41 A figure has symmetry with respect to point P if for every point M on the figure, there is a second point N on the figure for which point P is the midpoint of MN. DEFINITION a b On the basis of this definition, each point on rectangle ABCD in Figure 2.41a has a corresponding point that is the same distance from P but lies in the opposite direction from P. In Figure 2.41b, M and N are a pair of corresponding points. Even though a figure may have multiple lines of symmetry, a figure can have only one point of sym- metry. Thus, the point of symmetry when one exists is unique. EXAMPLE 4 Which letters shown below have point symmetry? M N P S X Solution N, S, and X as shown all have point symmetry. 쮿 Geometry in the Real World Taking a good look at the hexagonal shape used in the Hampton Inn logo reveals both point symmetry and line symmetry. EXAMPLE 5 Which figures in Figure 2.42a have point symmetry? Solution Only the square, the rhombus, and the regular hexagon have point symmetry. In the regular pentagon, consider the “centrally” located point P and note that . 쮿 AP Z PM Figure 2.42a Isosceles Triangle Square Rhombus Regular Pentagon Regular Hexagon P A M Square YES Rhombus YES Regular Hexagon YES Regular Pentagon NO Figure 2.42b Exs. 5–8 Discover The uppercase block form of the letter O is shown below. Does it have symmetry with respect to a point? ANSWER Yes, point P centered is the point of symmetry. This point P is the only point of symmetry for the uppercase O. TRANSFORMATIONS In the following material, we will generate new figures from old figures by association of points. In particular, the transformations included in this textbook will preserve the shape and size of the given figure; in other words, these transformations lead to a © Hilton Hospitality, INC. second figure that is congruent to the given figure. The types of transformations included are 1 the slide or translation, 2 the reflection, and 3 the rotation. 왘 Slides Translations With this type of transformation, every point of the original figure is associated with a second point by locating it through a movement of a fixed length and direc- tion. In Figure 2.43, 䉭ABC is translated to the second triangle its image 䉭DEF by sliding each point through the distance and in the direction that takes point A to point D. The background grid is not necessary to demonstrate the slide, but it lends credibility to our claim that the same length and direction have been used to locate each point. 2.6 A C B D F E Figure 2.43 EXAMPLE 6 Slide 䉭 horizontally in Figure 2.44 to form 䉭 . In this example, the distance length of the slide is XR. Solution 쮿 RST XYZ Y X Z T R S Y X Z R Figure 2.44 In Example 6, 䉭 䉭 and also 䉭 䉭 . In every slide, the given figure and the produced figure its image are necessarily congruent. In Example 6, the correspondence of vertices is given by , , and . Z 4 S Y 4 T X 4 R XYZ RTS ⬵ RTS XYZ ⬵ 왘 Reflections With the reflection, every point of the original figure is reflected across a line in such a way as to make the given line a line of symmetry. Each pair of corresponding points will lie on opposite sides of the line of reflection and at like distances. In Figure 2.46, obtuse triangle MNP is reflected across the vertical line to produce the image . The vertex N of the given obtuse angle corresponds to the vertex H of the obtuse angle in the image triangle. It is possible for the line of reflection to be horizontal or oblique slanted. With the vertical line as the axis of reflection, a drawing such as Figure 2.46 is some- times called a horizontal reflection, since the image lies to the right of the given figure. 䉭GHK Í AB EXAMPLE 7 Where , complete the slide of quadrilateral ABCD to form quadrilateral EFGH. Indicate the correspondence among the remaining vertices. Solution 쮿 A 4 E D C A B H G F E Figure 2.45 D C A B E N P K H M G A B Figure 2.46 EXAMPLE 8 Draw the reflection of right a across line 艎 to form . 䉭XYZ 䉭ABC A B C , , and in Figure 2.45. D 4 H C 4 G B 4 F b across line m to form . 䉭PQR A B C m Solution As shown in Figure 2.47 쮿 A X B Y C Z Figure 2.47 A B R C P Q m With the horizontal axis line of reflection, the reflection in Example 8a is often called a vertical reflection. In the vertical reflection of Figure 2.47a, the image lies be- low the given figure. In Example 9, we use a side of the given figure as the line line segment of reflection. This reflection is neither horizontal nor vertical. EXAMPLE 9 Draw the reflection of across side to form in Figure 2.48. How are and related? Solution The triangles are congruent; also, notice that , , and . 쮿 C 4 C B 4 B D 4 A 䉭DBC 䉭ABC 䉭DBC BC 䉭ABC A B C D Figure 2.48 A B C a b 왘 Rotations In this transformation, every point of the given figure leads to a point its image by ro- tation about a given point through a prescribed angle measure. In Figure 2.50, ray AB rotates about point A clockwise through an angle of 30° to produce the image ray AC. This has the same appearance as the second hand of a clock over a five-second period of time. In this figure, and . B 4 C A 4 A EXAMPLE 10 Complete the figure produced by a reflection across the given line in Figure 2.49. Solution 쮿 EXAMPLE 11 In Figure 2.51, square WXYZ has been rotated counterclockwise about its center intersection of diagonals through an angle of 45° to form congruent square QMNP. What is the name of the eight-pointed geometric figure that is formed by the two intersecting squares? Solution The eight-pointed figure formed is a regular octagram. 쮿 Figure 2.49 Geometry in the Real World The logo that identifies the Health Alliance Corporation begins with a figure that consists of a rectangle and an adjacent square. The logo is completed by rotating this basic unit through angles of 90°. 30° Figure 2.50 45° Z W Q M Y P X Figure 2.51 EXAMPLE 12 Shown in Figure 2.52 are the uppercase A, line 艎, and point O. Which of the pairs of transformations produce the original figure? a The letter A is reflected across 艎, and that image is reflected across 艎 again. b The letter A is reflected across 艎, and that image is rotated clockwise 60° about point O. c The letter A is rotated 180° about O, followed by another 180° rotation about O. Solution a and c 쮿 Figure 2.52 Exs. 9–14 Exercises 2.6 1. Which letters have symmetry with respect to a line? M N P T X 2. Which letters have symmetry with respect to a line? I K S V Z 3. Which letters have symmetry with respect to a point? M N P T X 4. Which letters have symmetry with respect to a point? I K S V Z

5. Which geometric figures have symmetry with respect to at

least one line? a b c

6. Which geometric figures have symmetry with respect to at

least one line? a b c

7. Which geometric figures have symmetry with respect

to a point? a b c

8. Which geometric figures have symmetry with respect

to a point? a b c 9. Which words have a vertical line of symmetry? DAD MOM NUN EYE 10. Which words have a vertical line of symmetry? WOW BUB MAM EVE

11. Complete each figure so that it has symmetry with respect

to line 艎. a b

12. Complete each figure so that it has symmetry with respect

to line m. a b

13. Complete each figure so that it reflects across line 艎.

a b

14. Complete each figure so that it reflects across line m.

a b

15. Suppose that slides to the right to the position of

. a If , find . b Is ? c Is congruent to ?

16. Suppose that square RSTV slides point for point to form

quadrilateral WXYZ. a Is WXYZ a square? b Is RSTV WXYZ ? c If , find WX.

17. Given that the vertical line is a line of symmetry,

complete each letter to discover the hidden word.

18. Given that the horizontal line is a line of symmetry,

complete each letter to discover the hidden word.

19. Given that each letter has symmetry with respect to the

indicated point, complete each letter to discover the hidden word. RS = 1.8 cm ⬵ 䉭DEF 䉭ABC AC ⬵ DF m ∠D m ∠A = 63° 䉭DEF 䉭ABC

20. What word is produced by a 180° rotation about the

point?

21. What word is produced by a 180° rotation about the

point?

22. What word is produced by a 360° rotation about the

point?

23. In which direction clockwise or counterclockwise will

pulley 1 rotate if pulley 2 rotates in the clockwise direction? a b

24. In which direction clockwise or counterclockwise will

gear 1 rotate if gear 2 rotates in the clockwise direction? a b

25. Considering that the consecutive dials on the electric

meter rotate in opposite directions, what is the current reading in kilowatt hours of usage?

26. Considering that the consecutive dials on the natural gas

meter rotate in opposite directions, what is the current reading in cubic feet of usage? m A B C D F G E m m J L K H C A B F D E V R S T Z Y W X MOH MOH MOM MOM FRED FRED 1 2 1 2 1 2 1 3 2 1 2

3 4

6 5

7 8

9 1

2 3 4 6 5 7 8 9 1 2

3 4

6 5

7 8

9 1

2 3 4 6 5 7 8 9 1 2

3 4

6 5

7 8 9 KWH 1 2

3 4

6 5

7 8

9 1

2 3 4 6 5 7 8 9 1 2

3 4

6 5

7 8

9 1

2 3 4 6 5 7 8 9 1 2

3 4

6 5

7 8 9 Cu FT ©alslutsky Shutterstock m

31. A regular hexagon is rotated about a centrally located

point as shown. How many rotations are needed to repeat the given hexagon vertex for vertex if the angle of rotation is a 30°? b 60°? c 90°? d 240°?

32. A regular octagon is rotated about a centrally located

point as shown. How many rotations are needed to repeat the given octagon vertex for vertex if the angle of rotation is a 10°? b 45°? c 90°? d 120°?

33. is the image of

following the reflection of across line 艎 . If and , find x.

34. is the image of

following a 100° counterclockwise rotation of about point Y. If and , find x . m ∠X¿YZ¿ = 130° m ∠XYZ = 5x 6 ∠XYZ ∠XYZ ∠X¿YZ¿ m ∠ABC = x 2 + 5 m ∠A ¿ B ¿ C ¿ = x 5 + 20 ∠ABC ∠ABC ∠A¿B¿C¿

27. Describe the types of symmetry displayed by each of

these automobile logos. a Toyota b Mercury c Volkswagen

28. Describe the types of symmetry displayed by each of

these department store logos. a Kmart b Target c Bergner’s

29. Given a figure, which of the following pairs of

transformations leads to an image that repeats the original figure? a Figure slides 10 cm to the right twice. b Figure is reflected about a vertical line twice. c Figure is rotated clockwise about a point 180° twice. d Figure is rotated clockwise about a point 90° twice.

30. Given a figure, which of the following pairs of

transformations leads to an image that repeats the original figure? a Figure slides 10 cm to the right, followed by slide of 10 cm to the left. b Figure is reflected about the same horizontal line twice. c Figure is rotated clockwise about a point 120° twice. d Figure is rotated clockwise about a point 360° twice. The Toyota brand and logos as well as Toyota model names are trademarks of Toyota Motor The Kmart logo is a registered trademarks of Sears Brands, LLC. Target and the Bullseye Design are registered trademarks of Target Brands, Inc. All rights reserved. Bergners Courtesy of Ford Motor Company Used with permission of Volkswagen Group of America, Inc.