If a border pattern consists of repeating a motif of symmetry type Z2, what is the

In Fig. 35, indicate which point is the image of P under (a) the reflection with axis l1. (b) the reflection with axis l2. (c) the reflection with axis l3. (d) the reflection with axis l4.

In Fig. 36, indicate which point is the image of P under (a) the reflection with axis l1. (b) the reflection with axis l2. (c) the reflection with axis l3. (d) the reflection with axis l4. figurE 36 C A B D E F G l2 l3 l4

In Fig. 37, A is the image of A under a reflection. (a) Find the axis of the reflection. (b) Find the image of A under the reflection.

In Fig. 38, P is the image of P under a reflection. (a) Find the axis of the reflection. (b) Find the image of P under the reflection. figurE 38

In Fig. 39, l is the axis of reflection. (a) Find the image of S under the reflection. (b) Find the image of quadrilateral PQRS under the reflection.

In Fig. 40, l is the axis of reflection. (a) Find the image of P under the reflection. (b) Find the image of the parallelogram PQRS under the reflection.

In Fig. 41, P is the image of P under a reflection. (a) Find the axis of the reflection. (b) Find the image of S under the reflection. (c) Find the image of the quadrilateral PQRS under the reflection. (d) Find a point on the quadrilateral PQRS that is a fixed point of the reflection.

In Fig. 42, P is the image of P under a reflection. (a) Find the axis of the reflection. (b) Find the image of S under the reflection. (c) Find the image of the quadrilateral PQRS under the reflection. (d) Find a point on the quadrilateral PQRS that is a fixed point of the reflection.

In Fig. 43, P is the image of P under a reflection. (a) Find the axis of the reflection. (b) Find the image of the shaded arrow under the reflection. figurE 43

In Fig. 44, is the image of R under a reflection. (a) Find the axis of reflection. (b) Find the image of the shaded arrow under the reflection.

In Fig. 45, A and B are fixed points of a reflection. Find the image of the shaded region under the reflection. figurE 45

In Fig. 46, A and B are fixed points of a reflection. Find the image of the shaded region under the reflection. figurE 46

In Fig. 47, indicate which point is (a) the image of B under a 90 clockwise rotation with rotocenter A. (b) the image of A under a 90 clockwise rotation with rotocenter B. (c) the image of D under a 60 clockwise rotation with rotocenter B. (d) the image of D under a 120 clockwise rotation with rotocenter B. (e) the image of I under a 3690 clockwise rotation with rotocenter A. figurE 47 H G C B A E D F

In Fig. 47, indicate which point is (a) the image of C under a 90 counterclockwise rotation with rotocenter B. (b) the image of F under a 60 clockwise rotation with rotocenter A. (c) the image of F under a 120 clockwise rotation with rotocenter B. (d) the image of I under a 90 clockwise rotation with rotocenter H. (e) the image of G under a 3870 counterclockwise rotation with rotocenter B.

In each case, give an answer between 0 and 360. (a) A clockwise rotation by an angle of 710 is equivalent to a counterclockwise rotation by an angle of .

In each case, give an answer between 0 and 360. (a) A clockwise rotation by an angle of 500 is equivalent to a clockwise rotation by an angle of . (b) A clockwise rotation by an angle of 500 is equivalent to a counterclockwise rotation by an angle of . (c) A clockwise rotation by an angle of 5000 is equivalent to a clockwise rotation by an angle of . (d) A clockwise rotation by an angle of 50,000 is equivalent to a counterclockwise rotation by an angle of .

In Fig. 48, a rotation moves B to B and C to C. (a) Find the rotocenter. (b) Find the image of triangle ABC under the rotation. figurE 48

In Fig. 49, a rotation moves A to A and B to B. (a) Find the rotocenter O. (b) Find the image of the shaded arrow under the rotation. figurE 49

In Fig. 50, a rotation moves A to A and B to B. (a) Find the rotocenter O. (b) Find the image of the shaded arrow under the rotation.

In Fig. 51, a rotation moves Q to Q and R to R. (a) Find the rotocenter. (b) Find the angle of rotation. (c) Find the image of quadrilateral PQRS under the rotation

In Fig. 52, find the image of triangle ABC under a 60 clockwise rotation with rotocenter O. figurE 52

In Fig. 53, find the image of ABCD under a 60 counterclockwise rotation with rotocenter O. figurE 53

In Fig. 54, indicate which point is the image of P under (a) the translation with vector v1. (b) the translation with vector v2. (c) the translation with vector v3. (d) the translation with vector v4. figurE 54

In Fig. 55, indicate which point is the image of P under (a) the translation with vector v1. (b) the translation with vector v2. (c) the translation with vector v3. (d) the translation with vector v4. figurE 55

In Fig. 56, E is the image of E under a translation. (a) Find the image of A under the translation. (b) Find the image of the shaded figure under the translation. (c) Draw a vector for the translation. figurE 56

In Fig. 57, Q’ is the image of Q under a translation. (a) Find the image of P under the translation. (b) Find the image of the shaded quadrilateral under the translation. (c) Draw a vector for the translation.

In Fig. 58, D is the image of D under a translation. Find the image of the shaded trapezoid under the translation. figurE 58

In Fig. 59, Qis the image of Q under a translation. Find the image of the shaded region under the translation. figurE 59

Given a glide reflection with vector v and axis l as shown in Fig. 60, find the image of the triangle ABC under the glide reflection.

Given a glide reflection with vector v and axis l as shown in Fig. 61, find the image of the quadrilateral ABCD under the glide reflection. figurE 61

In Fig. 62, D is the image of D under a glide reflection having axis l. Find the image of the polygon ABCDE under the glide reflection. figurE 62

In Fig. 63, P is the image of P under a glide reflection having axis l. Find the image of the quadrilateral PQRS under the glide reflection. figurE 63

In Fig. 64, B is the image of B and D is the image of D under a glide reflection. (a) Find the axis of reflection. (b) Find the image of A under the glide reflection. (c) Find the image of the shaded figure under the glide reflection. figurE 64

In Fig. 65, A is the image of A and C is the image of C under a glide reflection. (a) Find the axis of reflection. (b) Find the image of B under the glide reflection. (c) Find the image of the shaded figure under the glide reflection. figurE 65 B C D

In Fig. 66, P is the image of P and Q is the image of Q under a glide reflection. Find the image of the shaded figure under the glide reflection. figurE 66

In Fig. 67, P is the image of P and Q is the image of Q under a glide reflection. Find the image of the shaded figure under the glide reflection.

In Fig. 68, D is the image of D and C is the image of C under a glide reflection. Find the image of the shaded figure under the glide reflection.

In Fig. 69, A is the image of A and D is the image of D under a glide reflection. Find the image of the shaded figure under the glide reflection.

List the symmetries of each object shown in Fig. 70. (Describe each symmetry by giving specificsthe axes of reflection, the centers and angles of rotation, etc.) figurE 70 A B D C (a) A B D C (b) A B D C (c)

List the symmetries of each object shown in Fig. 71. (Describe each symmetry by giving specificsthe axes of reflection, the centers and angles of rotation, etc.) A C B 45 45 (a) A C B 60 60 60 (b) A BC 60 30 (c) figurE 71

List the symmetries of each object shown in Fig. 72. (Describe each symmetry by giving specificsthe axes of reflection, the centers and angles of rotation, etc.)

List the symmetries of each object shown in Fig. 73. (Describe each symmetry by giving specificsthe axes of reflection, the centers and angles of rotation, etc.) (a) (b)

For each of the objects in Exercise 39, give its symmetry type.

For each of the objects in Exercise 40, give its symmetry type.

For each of the objects in Exercise 41, give its symmetry type

For each of the objects in Exercise 42, give its symmetry type.

Find the symmetry type for each of the following letters. (a) A (b) D (c) L (d) Z (e) H (f) N

Find the symmetry type for each of the following symbols. (a) $ (b) @ (c) % (d) : (e) &

Give an example of a capital letter of the alphabet that has symmetry type (a) Z1. (b) D1. (c) Z2. (d) D2.

Give an example of a one- or two-digit number that has symmetry type (a) Z1. (b) D1. (c) Z2. (d) D2.

Classify each border pattern by its symmetry type. Use the standard crystallographic notation (mm, mg, m1, 1m, 1g, 12, or 11). (a) . . . A A A A A . . . . . . D D D D D . . . . . . Z Z Z Z Z . . . . . . L L L L L

Classify each border pattern by its symmetry type. Use the standard crystallographic notation (mm, mg, m1, 1m, 1g, 12, or 11). (a) N N N N N WWWWW H H H H H J J J J J (b) (c) (d

Classify each border pattern by its symmetry type. Use the standard crystallographic notation (mm, mg, m1, 1m, 1g, 12, or 11). (a) qpqpqpqp pdpdpdpd pbpbpbpb pqbdpqbd (b) (c) (d

Classify each border pattern by its symmetry type. Use the standard crystallographic notation (mm, mg, m1, 1m, 1g, 12, or 11). (a) qbqbqbqb qdqdqdqd dbdbdbdb qpdbqpdb (b) (c) (d)

If a border pattern consists of repeating a motif of symmetry type Z2, what is the symmetry type of the border pattern?

If a border pattern consists of repeating a motif of symmetry type Z1, what is the symmetry type of the border pattern?

Imagine a border pattern of type m1 placed between parallel lines l1 and l2 (Fig. 74). Create a new border pattern twice the height by reflecting a copy of the original pattern about l2. What is the symmetry type of the new border pattern?

Imagine a border pattern of type 12 placed between parallel lines l1 and l2 (Fig. 74). Create a new border pattern twice the height by rotating a copy of the original pattern 180 degrees and gluing both copies along line l2. What is the symmetry type of this new border pattern?

The minute hand of a clock is pointing at the number 9, and it is then wound clockwise 7080 degrees. (a) How many full hours has the hour hand moved? (b) At what number on the clock does the minute hand point at the end?

Name the rigid motion (translation, reflection, glide reflection, or rotation) that moves footprint 1 onto footprint (a) 2 (b) 3 (c) 4 (d) 5 1 2 4 3 5 figurE 75

In each case, determine whether the rigid motion is a reflection, rotation, translation, or glide reflection or the identity motion. (a) The rigid motion is proper and has exactly one fixed point. (b) The rigid motion is proper and has infinitely many fixed points. (c) The rigid motion is improper and has infinitely many fixed points. (d) The rigid motion is improper and has no fixed points.

Exercises 62 through 69 deal with combining rigid motions. Given two rigid motions  and , we can combine the two rigid motions by first applying  and then applying  to the result. The rigid motion defined by combining  and  ( goes first,  goes second) is called the product of  and .In Fig. 76, l1, l2, l3, and l4 are axes of reflection. In each case, indicate which point is the image of P under (a) the product of the reflection with axis l1 and the reflection with axis l2. (b) the product of the reflection with axis l2 and the reflection with axis l1. (c) the product of the reflection with axis l2 and the reflection with axis l3. (d) the product of the reflection with axis l3 and the reflection with axis l2. (e) the product of the reflection with axis l1 and the reflection with axis l4

Exercises 62 through 69 deal with combining rigid motions. Given two rigid motions  and , we can combine the two rigid motions by first applying  and then applying  to the result. The rigid motion defined by combining  and  ( goes first,  goes second) is called the product of  and .In Fig. 77, indicate which point is the image of P under (a) the product of the reflection with axis l and the 90 clockwise rotation with rotocenter A. (b) the product of the 90 clockwise rotation with rotocenter A and the reflection with axis l. (c) the product of the reflection with axis l and the 180 rotation with rotocenter A. (d) the product of the 180 rotation with rotocenter A and the reflection with axis l.

Exercises 62 through 69 deal with combining rigid motions. Given two rigid motions  and , we can combine the two rigid motions by first applying  and then applying  to the result. The rigid motion defined by combining  and  ( goes first,  goes second) is called the product of  and .In each case, state whether the rigid motion  is proper or improper. (a)  is the product of a proper rigid motion and an improper rigid motion. (b)  is the product of an improper rigid motion and an improper rigid motion. (c)  is the product of a reflection and a rotation. (d)  is the product of two reflections.

Exercises 62 through 69 deal with combining rigid motions. Given two rigid motions  and , we can combine the two rigid motions by first applying  and then applying  to the result. The rigid motion defined by combining  and  ( goes first,  goes second) is called the product of  and .Suppose that a rigid motion  is the product of a reflection with axis l1 and a reflection with axis l2, where l1 and l2 intersect at a point C. Explain why  must be a rotation with center C.

Exercises 62 through 69 deal with combining rigid motions. Given two rigid motions  and , we can combine the two rigid motions by first applying  and then applying  to the result. The rigid motion defined by combining  and  ( goes first,  goes second) is called the product of  and .Suppose that the rigid motion  is the product of the reflection with axis l1 and the reflection with axis l3, where l1 and l3 are parallel. Explain why  must be a translation.

Exercises 62 through 69 deal with combining rigid motions. Given two rigid motions  and , we can combine the two rigid motions by first applying  and then applying  to the result. The rigid motion defined by combining  and  ( goes first,  goes second) is called the product of  and .In Fig. 78, l1 and l2 intersect at C, and the angle between them is a. (a) Give the rotocenter, angle, and direction of rotation of the product of the reflection with axis l1 and the reflection with axis l2.

Exercises 62 through 69 deal with combining rigid motions. Given two rigid motions  and , we can combine the two rigid motions by first applying  and then applying  to the result. The rigid motion defined by combining  and  ( goes first,  goes second) is called the product of  and .In Fig. 79, l1 and l3 are parallel and the distance between them is d. (a) Give the length and direction of the vector of the translation that is the product of the reflection with axis l1 and the reflection with axis l3. (b) Give the length and direction of the vector of the translation that is the product of the reflection with axis l3 and the reflection with axis l1. figurE 79

Exercises 62 through 69 deal with combining rigid motions. Given two rigid motions  and , we can combine the two rigid motions by first applying  and then applying  to the result. The rigid motion defined by combining  and  ( goes first,  goes second) is called the product of  and .In Fig. 80, P is the image of P under a translation  and Q is the image of Q under a translation . (a) Find the images of P and Q under the product of  and . (b) Show that the product of  and  is a translation. Give a geometric description of the vector of the translation.

(a) Explain why a border pattern cannot have a reflection symmetry along an axis forming a 45 angle with the direction of the pattern. (b) Explain why a border pattern can have only horizontal and/or vertical reflection symmetry

A rigid motion  moves the triangle PQR into the triangle PQR as shown in Fig. 81. Explain why the rigid motion  must be a glide reflection.

Construct border patterns for each of the seven symmetry types using copies of the symbol 1 (and rotated versions of it).

Let the six symmetries of the equilateral triangle ABC shown in Fig. 82 be denoted as follows: r1 denotes the reflection with axis l1; r2 denotes the reflection with axis l2; r3 denotes the reflection with axis l3; R1 denotes the 120 clockwise rotation with rotocenter O; R2 denotes the 240 clockwise rotation with rotocenter O; I denotes the identity symmetry. Complete the symmetry multiplication table by entering, in each row and column of the table, the product of the row and the column (i.e., the result of applying first the symmetry in the row and then the symmetry in the column). For example, the entry in row r1 and column r2 is R1 because the product of the reflection r1 and the reflection r2 is the rotation R1

Let the eight symmetries of the square ABCD shown in Fig. 83 be denoted as follows: r1 denotes the reflection with axis l1; r2 denotes the reflection with axis l2; r3 denotes the reflection with axis l3; r4 denotes the reflection with axis l4; R1 denotes the 90 clockwise rotation with rotocenter O; R2 denotes the 180 clockwise rotation with rotocenter O; R3 denotes the 270 clockwise rotation with rotocenter O; I denotes the identity symmetry. Complete the symmetry multiplication table below (Hint: Try Exercise 73 first.) A l1 l2 l3 l4 B D C O

List all the symmetries of the wallpaper pattern shown in Fig. 84.

List all the symmetries of the wallpaper pattern shown in Fig. 85.

List all the symmetries of the wallpaper pattern shown in Fig. 86

List all the symmetries of the wallpaper pattern shown in Fig. 87.

List all the symmetries of the wallpaper pattern shown in Fig. 88. figurE 88

List all the symmetries of the wallpaper pattern shown in Fig. 89. figurE 89