Deck 22: Transformation Geometry

Question 1

"Figure X is congruent to Figure Y." What does that statement mean?

Answer 1

One of the figures is the image of the other for some rigid motion.

Question 2

Name three types of rigid motions.

Answer 2

reflection (in a line), translation, and rotation (about a point) (in any order)

Question 3

Name the type of rigid motion involved in each of the listed motions. Take into account the different types of "eyes."

Answer 3

rotation; reflection; translation

Question 4

Name the type of rigid motion involved in each of the listed motions.

Question 5

For the figures below, what single type of motion would accomplish each?

Question 6

See below. The unlabeled figure at top left is the original figure. Each of the others, labeled a-h, is the result of a rigid transformation of the original. Determine if each is the result of a translation, rotation, reflection, or glide-reflection.

Question 7

Complete each of the following motions, being careful to fit your answer to the grid.

A) reflection

B) translation

C) glide-reflection

Question 8

Find the missing information. (Drawing is alright. If your drawing is "off," write words to communicate your intent.)

A) Missing: the vector for this translation

B) Missing: the angle and direction for this rotation

Question 9

Complete the chart.

Question 10

Select "same" or "reversed" for each item listed in the chart.
$\begin{array} { | l | l | } \hline \text { Type of rigid motion } & \text { Clock orientation of image compared } \text { with original } \\\hline \end{array}$
$\begin{array} { | l | l l | } \hline \text { A) Reflection}\quad\quad\quad & \text { A) same \quad reversed } \\\hline \text { B) Rotation } & \text { B) same \quad reversed } \quad\quad\quad\quad\quad\quad\quad\\\hline \text { C) Translation } & \text { C) same \quad reversed } \\\hline \text { D) Glide-reflection } & \text { D) same \quad reversed } \\\hline\end{array}$

Question 11

For each pair of figures, determine which single-step transformation might take the shaded figure to the other. If a rotation, find the center and angle of rotation. If a reflection, show the line of reflection. If a translation, give an arrow (vector).
a.

b.

c .

A) $\underline{\quad\quad}$
B) $\underline{\quad\quad}$
C) $\underline{\quad\quad}$

Question 12

Accurately draw each of the following.
A. the vector for this translation


B. the line of reflection involved here

Question 13

Complete each statement.

A) The composition of ten different reflections and five different rotations could possibly be a _____ or a _____.
B) A rotation always has _____ fixed point(s). (How many?)
C) A translation with a vector 50 centimeters long can be expressed by the composition of two reflections, where the reflecting lines are (i) _____ centimeters apart, (ii) _____, and (iii) _____.
D) If P' is the image of P for a clockwise rotation of 140° with center X, then key relationships tell us that _____ and _____.
E) The composition of 200 reflections could be achieved most efficiently with _____ reflection(s).

Question 14

(Note to the instructor: To make the tests different, different shapes and locations were drawn on the test copies.)

A) Use the MIRA to find the image of the given shape, for the composition
(reflect in line k)° (reflect in line x).

B) Name the single rigid motion equal to the composition in part A, and provide in words or in a drawing all the important information about that motion (e.g., if reflection, then the line of reflection; if rotation, then the center and angle; if translation, etc).

Question 15

A) Give the final image of X for (translation, vector v)° (reflection in line m).
B) Name the single rigid motion equal to the composition in part A. You do not have to give all the important information, just the type of motion.
(translation, vector v)° (reflection in line m) is a _____.

Question 16

A) Give an argument which shows this statement is true: Every rigid motion can be expressed with, at most, three reflections. (Note: Please write full sentences.)
B) Illustrate the statement given in part A by showing the line(s) of reflection that, with composition if necessary, give(s) the image below.

Question 17

Experiment with the composition of three reflections in three different parallel lines to predict what single motion the composition is. Describe the single motion as completely as possible, relating the description to the relative positions of the original lines. (This means if you think the composition is a reflection, tell where the line of reflection will be with respect to the original three lines; if you think the composition is a rotation, tell where the center will be and the angle and direction; if you think the composition is a translation, etc.)

Question 18

Complete each statement.

A) The composition of 15 reflections in different lines with 13 rotations with different centers might be a _____ or a _____.
B) A reflection has _____ fixed point(s). (How many?)
C) If two polygons are similar with a scale factor of 2, then a 20° angle in the smaller polygon will correspond to an angle of _____ in the larger polygon.

Question 19

A) Show the image of shape S for the composition given by (reflection in m)° (translation with vector v).

B) What single type of rigid motion accomplishes the same thing as the composition in part A?

Question 20

Two lines intersect at point Q, making an angle of 60°. What single rigid motion equals the composition of reflections in those two lines? Describe this single rigid motion as completely as you can from the given information.

Question 21

Which of the following would be more valuable if you could have only one of them (and composition)? (Hint: theory)

A) all possible rotations and translations
B) all possible reflections

Question 22

Name the kind(s) of symmetry this figure has. (The dots mean it continues forever.)

Question 23

Joe and John were arguing about glide-reflections. Joe says that the order that you do the steps of a glide-reflection makes a difference in your answer. John says that it doesn't.
Who is correct about whether the order matters? Why? (What was done wrong?)