Great Ideas in Mathematics

Math 110 - Fall 2007

Ch. 11: Symmetry

Practice questions

 Answers below.

1.         Which of the following rigid motions has exactly one fixed point?

A            Glide reflection

B            Reflection

C            Rotation

D            Translation

2.         Which of the following rigid motions has infinitely many fixed points?

A            Glide reflection

B            Reflection

C            Rotation

D            Translation

3.         Which TWO of the following are proper rigid motions?

A            Glide reflection

B            Reflection

C            Rotation

D            Translation

4.         How many point-image pairs are necessary to completely determine a rotation?

_________

5.         How many point-image pairs are necessary to completely determine a translation?

_________

6.         How many point-image pairs are necessary to completely determine a glide reflection?

_________

7.         How many point-image pairs are necessary to completely determine a reflection?

_________

Questions 8 to 15 involve the square shown below to the right.

8.         Which of the following reflections is not a symmetry of the square shown at the right:

A         a reflection with axis line AB

B          a reflection with axis line MN

C         a reflection with axis line PQ

D         a reflection with axis line AC

E          All of the above are symmetries of the square.

9.         Which of the following rotations is a symmetry of the square?

A         a 90° clockwise rotation with center O

B         a 90° clockwise rotation with center P

C         a 90° counterclockwise rotation with center N

D         a 180°  clockwise rotation with center M

E         None of the above

10.       Which of the following translations is a symmetry of the square?

A         a translation that sends A to C

B          a translation that sends P to Q

C         a translation that sends A to O

D         a translation that sends A to B

E          none of the above

11.       The image of A under the reflection with axis the line through M and N is

A         O

B          B

C         C

D         D

E          None of the above

12.       The image of A under a 90° clockwise rotation with center O is

A         M

B          B

C         C

D         D

E          none of the above

13        A translation sends the point A to the point Q.  The image of P under this translation is

A         O

B          B

C         C

D         N

E          none of the above

14.       A glide reflection sends the point A to the point Q and the point P to the point C.  The image of B under this glide reflection is

A         A

B         N

C         P

D         D

E          none of the above

15.       A glide reflection sends the point A to the point Q and the point P to the point C.  The axis of this glide reflection is a line passing through the points

A         M and N

B         A and Q

C         P and Q

D         A and B

E         none of the above

16.       What are the symmetries of the letter D?

A         90° rotation

B         The letter D has no symmetries

C         a reflection with a vertical axis

D         a reflection with a horizontal axis

E         None of the above

17        What is the symmetry type of an isosceles triangle that is not equilateral?

A         D₁

B         D₂

C         Z₃

D         Z₂

E          None of the above

18        One of the symmetries of a shape is a 60° clockwise rotation.  Then it must also have which of the following symmetries

A         a 90° clockwise rotation

B         a translation

C         a 180° rotation

D         a reflection

E         none of the above

19        The result of applying two consecutive reflections with parallel axes is equivalent to a single

A         rotation

B         glide reflection

C         reflection

D         translation

E         none of the above

20        The result of applying a translation followed by a rotation is equivalent to a single

A         rotation

B         glide reflection

C         reflection

D         translation

E         none of the above

21        The complete symmetries of the pattern …. Z  Z  Z  Z  Z  Z  Z  Z  Z  … are

A         translations and horizontal reflections only

B         translations and 45° rotations only

C         translations and vertical reflections only

D         translations and 180° rotations only

E          none of the above

22        In a wallpaper pattern, the only possible angle(s) of rotational symmetry are

A         60°, 90°, 180°

B         180°

C         60°,  90°. 120°. 180°

D         90°, 180°

E         none of the above

The following questions 23 to 30 refer to the diagram of a pentagon below.

23        Which of the following lines is not an axis of symmetry of the pentagon?

A         BP

B         CD

C         AN

D         ME

E         All of the above are axes of symmetry of this pentagon.

24        Which of the following rotations is a symmetry of the pentagon?

A         a 60° clockwise rotation with center O

B         a 90° clockwise rotation with center O

C         a 144° clockwise rotation with center O

D         a 180° clockwise rotation with center A

E         none of the above

25        Which of the following translations is a symmetry of the pentagon?

A         a translation that sends A to B

B         a translation that sends A to O

C         a translation that sends P to Q

D         a translation that sends A to C

E         none of the above

26        The image of A under the reflection across the line through M and E is

A         P

B         N

C         C

D         D

E         none of the above

27        The image of A under a 216° counterclockwise rotation with center O is

A         none of these

B         B

C         C

D         D

E         E

28        A translation sends the point C to the point P.  The image of M under this translation is

A         A

B         O

C         none of these

D         Q

E         E

29        A reflection sends the point L to the point M.  The image of D under this reflection is

A         A

B         P

C         none of these

D         Q

E         E

30        A glide reflection sends the point D to the point Q and the point L to the point C.  The axis of this glide reflection is a line

A         perpendicular to the line through D and Q

B         through points C and D

C         through points A and N

D         perpendicular to the line through A and N

E         none of these

31        A glide reflection with the axis shown at the right sends point P to the point Q.  The image of R under this same glide reflection is

A         A

B         B

C         C

D         D

E         none of these

32        In the figure at the right, the image of the point A under a reflection about line M, followed by a 90° rotation around the center O is 

A         A

B         none of these

C         C

D         G

E         E

33        The Greek letter F has symmetry type

A         D₁

B         Z₁

C         Z₂

D         D₂

E         none of these

34        A shape has a 216° clockwise rotation as one of its symmetries.  It must also have as a symmetry

A         a translation

B         a 45° clockwise rotation

C         a 108° clockwise rotation

D         a 72° clockwise rotation

E         none of these

35        The symmetry type of a border pattern … F F F F F F F … is

A         mm

B         mg

C         1m

D         m1

E         none of these

36        The symmetry type of the border pattern  … Ƚ  ½È½  ½È½  ½È½  ½È½  ½… is

A         mm

B         mg

C         1m

D         m1

E         none of these

37        In a wallpaper pattern

A         there may or may not be translation symmetry

B          there is translation symmetry in exactly one direction

C         there is translation symmetry in at least two different directions

D         there is translation symmetry in at least four different directions

E          none of the above

Several problems use the table of chrystallographic notations below.  You should use it whenever you need it.  (If a space doesn't say "yes", then it's "no".)

38        What is the symmetry type of the border pattern  … J  J  J  J  J  J  …

__________

39        What is the symmetry type of the border pattern  … B  B  B  B  B  B  …

__________

40        What is the symmetry type of the border pattern   … A  A  A  A  A  A  A  …

__________

41        What is the symmetry type of the border pattern  … X  X  X  X  X  X  …

__________

42        What is the symmetry type of the border pattern …  Z  Z  Z  Z  Z  Z  …

__________

43        What is the symmetry type of the border pattern  … p  b  p  b  p  b  p  b  …

__________

44        What is the symmetry type of the border pattern  … W  M  W  M  W  M  W  M  … (assuming that a W is the same as an M, but upside down.)

__________

45        Rigid motions

A         cannot alter the size of an object

B         cannot alter the shape of an object

C         must preserve distance between points

D         must preserve angles between lines

E         all of the above

F         none of the above

46        Saying that a rigid motion is proper means that

A         it is the result of two translations

B         it is the result of a translation and a rotation

C         it preserves the “handedness” of the object

D         it keeps the object in the same plane

E         it preserves lines and points

47        A translation followed by another translation

A         preserves distances and angles

B         is a proper rigid motion

C         is another translation

D         sometimes has no fixed points

E         all of the above

F         none of the above

48        A rotation followed by another rotation

A         never has any fixed points

B          has no fixed points

C         is not a proper rigid motion

D         is always a translation or a rotation

E          is a glide reflection

49        What is the symmetry type of the letter A

A         Z₁ 

B         Z₂ 

C         D₁ 

D         D₂ 

E          none of these

50        What is the symmetry type of the letter Q

A         Z₁ 

B         Z₂ 

C         D₁ 

D         D₂ 

E          none of these

51        What is the symmetry type of the letter B

A         Z₁ 

B         Z₂ 

C         D₁ 

D         D₂ 

E          none of these

52        What is the symmetry type of the letter Z

A         Z₁

B         Z₂

C         D₁

D         D₂

E          none of these

53        What is the symmetry type of the letter S

A         Z₁

B         Z₂

C         D₁

D         D₂

E          none of these

54        What is the symmetry type of the letter X

A         Z₁

B         Z₂

C         D₁

D         D₂

E          none of these

55        What is the symmetry type of the letter symbol

A         Z₁

B         Z₂

C         D₁

D         D₂

E          none of these

The sample questions are now complete.

Created November 4, 2007

Answers

1    C

2    B

3    C and D

4    2

5    1

6    2

7    2

8    A

9    A

10   E

11    B

12    B

13    C

14    C

15    A

16    D

17    A

18    C

19    D

20    A

21    D

22    E

23    B

24    C

25    E

26    D

27    C

28    C

29    E

30    D

31    A

32    E

33    D

34    D

35    A

36    D

37    C

38    11

39    1m

40    m1

41    mm

42    12

43    1g

44    mg

45    A

46    C

47    E

48    D

49    C

50    A

51    C

52    B

53    B

54    D

55    E

End of the answers