l3 rigid motion transformations...

L3 – Rigid Motion Transformations Name _____________________________ 3.1 - Pre-Assessment Per _____ Date ______________________

Geometry Q1 L3 – Rigid Motion Transformations Handouts

Which of the following could represent a translation using the rule T (x, y) = (x , y + 4), followed by a reflection over the given line? (The pre-image is the shaded circle.) Note: there could be more than one correct answer. Which of the following could represent the figure, after a vertical translation by a non-zero value followed by a reflection over a vertical line? (The pre-image is the shaded triangle.)

1. Today we will be learning more about rigid motion transformations, which are sequences of basic transformations. The two exercises above are examples of sequences of rigid motion transformations. Explain what you think sequences of rigid motion transformations are.

a) b) c) d)

d) c) b) a)

L3 – Rigid Motion Transformations Name _____________________________ 3.2 - Sequences of Transformations Per _____ Date ______________________


Grandma decides she wants to rearrange her room and wants to try it out on paper before asking her grandchildren to help her move everything. Identify the objects in the room that could be moved to their new position as a result of a single transformation. Original Room Layout

Rearranged Room Layout In this lesson we will explore sequences of transformations, also called composite transformations. Later we will return to Grandma’s room and identify the transformations required to complete the rearrangement of her room.



One of the fundamental questions in Geometry is “How can we tell if Figures A and B have the same shape and size, without measuring them?” Intuitively, this could be answered “yes” if it were possible to “Cut out Figure A, move it around and possibly flip it over, and place it onto Figure B so that it matches perfectly.” In the last lesson we learned how to translate, reflect, and rotate figures in the plane, which clearly resulted in images that retained the same size and shape as their pre-image. But, not all images with the same size and shape result from using only a single simple transformation; sometimes moving Figure A onto Figure B requires using a sequence of simple transformations. For example, we may need to translate Figure A, then rotate it, and if the orientation is still off reflect it as well. Such a sequence of translations, reflections, and rotations is referred to as a rigid motion transformation. Why do you think they call these “Rigid Motion?” A Rigid Motion Transformation (#VOC) is a composition (or sequence) of one or more translations, rotations, and reflections. Rigid motion transformations preserve distances (i.e. size) and angle measures (i.e. shape). Note: a single translation, rotation, or reflection is a simple example of a rigid motion transformation. 1. For each of the following pairs, the left pre-image was transformed into the right image.

Which CANNOT be achieved with a single translation, rotation, or reflection? Try to argue why it is not possible. You do not need to identify a transformation that moves the pre-mage to the image.



2. Translate ∆ABC to the right 9 units and down 7 units and label the image ∆A’B’C’. Translate the image triangle ∆A’B’C’ you just drew to the left 2 units and up 3 units, label this image ∆A’’B’’C’’.

Describe the single translation that would carry ∆ABC onto ∆A’’B’’C’’.

The above example can be summarized with one single translation. Therefore, this is not considered a sequence of transformations.

3. Translate ∆DEF to the right 4 units and down 2 units. Label the image ∆D’E’F’.

Reflect ∆D’E’F’ over the horizontal line and label the image ∆D’’E’’F’’.

Notice that this transformation CANNOT be achieved with a single transformation. Therefore this is considered a sequence of transformations, and ∆D’’E’’F’’ is congruent to ∆DEF.

The angle measurements and side lengths of the image ∆D’’E’’F’’ are equal to those of the pre-image, ∆DEF.

L3 – Rigid Motion Transformations Name _____________________________ 3.3 Sequence of Transformations Practice Set Per _____ Date ______________________


1. Translate ΔRST using the rule T (x, y) = (x – 4, y – 3), then reflect over the x-axis. Draw and label the image ΔR’S’T’.

2. Reflect ΔRST over the y-axis, then reflect over the line x = 3. Draw and label this image ΔR”S”T”.

3. Point P is in quadrant III and is then reflected over the x-axis followed by a reflection over the y-axis. In which quadrant is the new P’ ?

a) Quadrant I

b) Quadrant II

c) Quadrant III

d) Quadrant IV

4. Which of the following diagrams could represent a reflection over a vertical line followed by a non-zero vertical translation? (The pre-image is the shaded star.)

A B C D

L3 – Rigid Motion Transformations Name _____________________________ 3.3 Sequence of Transformations Practice Set Per _____ Date ______________________


Assume all of the points below are graphed on a Cartesian Coordinate Plane. Give the new coordinates after performing the rigid motion transformation.

5. Point A (2, 6) is translated horizontally 4 units and vertically – 6 units, then reflected over the y-axis. Give the coordinates for A’.

6. Point B (– 3, 7) is reflected over the x-axis, then translated vertically 8 units. Give the coordinates for B’.

7. Point C (– 22, 17) is translated using the rule: T (x, y) = (x – 13, y + 4), then reflected over

the y-axis. Give the coordinates for C’.

8. Point X (1, – 2) was transformed to point X’ (– 1, 2). What type of transformation could NOT have occurred?

a) a 180˚ rotation about the origin

b) a reflection over the y-axis followed by a reflection over the x-axis

c) a vertical translation of 4 followed by a reflection over the x-axis

d) a horizontal translation – 2 followed by a reflection over the x-axis

e) none of these – all transformations would give the same result

L3 – Rigid Motion Transformations Name _____________________________ 3.4 Sequence of Transformations Homework Per _____ Date ______________________


1. Reflect ΔDEF over the y-axis, then translate horizontally – 5 and vertically 6 units.

2. Describe how the coordiantes for ΔDEF changed to D’,E’, and F’.

3. What type of symbolic representation could be written for this type of sequence of transformations?

T (x, y) = ( , )

Each of the pairs of figures below have the same size and shape. Describe a sequence of transformations that may have occurred. Be specific. If the image reflected over a line, draw the line of reflection.

Could the transformations above have more than one correct answer?

P1

image

pre-image

L3 – Rigid Motion Transformations Name _____________________________ 3.5 - Composite Transformations Per _____ Date ______________________


Indicate for each of the three following pairs of transformations, whether the two composite transformations would sometimes, always, or never give the same result when applied to a figure. Justify your answer by explaining in words or drawing diagrams.

1. Would the two composite transformations give the same result always, sometimes, or never? Defend your answer. • A reflection over the y-axis followed by a vertical translation of 6. • A vertical translation of 6 followed by a reflection over the y-axis.

2. Would the two composite transformations give the same result always, sometimes, or never? Defend your answer. • A translation, T (x, y) = (x, y + 2) followed by a reflection over the x-axis. • A reflection over the x-axis followed by another reflection over the x-axis.

3. Would the two composite transformations give the same result always, sometimes, or never? Defend your answer. • A reflection over the x-axis followed by a reflection over the y-axis. • A reflection over the y-axis followed by a reflection over the x-axis.

L3 – Rigid Motion Transformations Name _____________________________ 3.6 - Scavenger Hunt Per _____ Date ______________________


How it works: posters will be provided by the teacher.

1. At your first poster, ignore the composite transformation and letter (these go with another description).

2. Write down the description/rule from the bottom of the poster in the first box below. 3. Draw a sketch below of the composite transformation described by in the poster. 4. Look around the room for a diagram that matches your sketch and go to that poster. 5. Write the letter from the top right corner of the new poster. 6. Find your next description/rule at the bottom of the poster. 7. Repeat this process until all 10 boxes are complete. 8. Write out the letters in order to reveal a new description in how to transform a figure for your

last exercise! BOW!/RAIN = RAINBOW!

1) Description/Rule:

Sketch: Letter/#:


Sketch: Letter/#:


Sketch: Letter/#:


Sketch: Letter/#:

L3 – Rigid Motion Transformations Name _____________________________ 3.6 - Scavenger Hunt Per _____ Date ______________________



Sketch: Letter/#:


Sketch: Letter/#:


Sketch: Letter/#:


Sketch: Letter/#:


Sketch: Letter/#:


Sketch: Letter/#:

L3 – Rigid Motion Transformations Name _____________________________ 3.7 - Composite Transformations Exit Pass Per _____ Date ______________________


Use the transformation rule you uncovered during the scavenger hunt activity to transform the figure below about point X. Be sure to label your image (A’, B’, C’, etc.).

L3 – Rigid Motion Transformations Name _____________________________ 3.8 – Composite Reflections over Parallel Lines Per _____ Date ______________________


1. Reflect the figure over the x-axis then reflect over the line y = 5. Label the new image using A’, B’, C’, etc.

2. How could you describe the two lines of reflection from question #1?

a) the two lines are perpendicular c) the two lines are parallel

b) the two lines are intersecting d) none of the statements are true

3. Describe how the figure changed after the composite transformation (double reflection).

4. What single transformation results in the same image?

5. Regardless of the figure and the two parallel lines chosen, if a figure is reflected first over one line and then over the second line, this same rigid motion transformation can be achieved by a single __________________________.

10

8

6

4

2

2

4

6

8

10

15 10 5 5 10 15

E

D

C

BA

L3 – Rigid Motion Transformations Name _____________________________ 3.8 – Composite Reflections over Parallel Lines Per _____ Date ______________________


6. Which statement below is NOT true, concerning images that have undergone a double reflection over distinct parallel lines?

a) The images have the same orientation. b) The images have the same position. c) The images have the same angle measures. d) The images have the same side lengths. e) None. All of the above statements are true.

Use the diagram of rectangle REST (below) to answer the following questions. 7. If the rectangle were to be reflected over a set of vertical, parallel lines, which point could

NOT be the location of S’?

a) S’ (12, – 5) b) S’ (– 4, – 5) c) S’ (– 5, 5) d) S’ (0, – 5)

8. Describe two completely different types of transformations (e.g. translations and reflections) that would produce an image in the exact same location at the pre-image, REST.

a. b.

9. Rotate rectangle REST 90˚CW about the origin then reflect the figure over the x-axis. Draw and label the image on the grid below.

10. Is there a single transformation that would carry the pre-image onto the final image? Why or

why not?

L3 – Rigid Motion Transformations Name _____________________________ 3.9 - Composite Reflections over Parallel Lines Homework Per _____ Date ______________________


1. Determine whether each transformation appears to be a rotation.

2. Reflect the image over line a then line b. (Fold your paper or use patty paper to help!)

b a

3. Reflect the image over line m then line n. (Fold your paper or use patty paper to help!)

m n

4. Lines a and b from exercise 2 are parallel and lines m and n are parallel from exercise 3. Draw a conclusion about reflecting images over parallel lines.

A B C D

L3 – Rigid Motion Transformations Name _____________________________ 3.10 - Composite Reflections over Intersecting Lines Per _____ Date ______________________


1. Reflect the image over line l then reflect that image over line r. 2. Which of the statements below is NOT true about lines l and r from exercise #1?

a) they are intersecting lines c) they share only one point

b) they are parallel lines d) they are perpendicular lines

3. Would a reflection over line r followed by a reflection over line l produce the same result? 4. What other type of transformation would produce an equivalent result?

r



5. Reflect each of the images over the set of intersecting lines. Reflect first over line g, then over line h. Patty paper or a mira might be useful here.

6. Rotate each of the images 180˚ about the given point and compare the results to exercise #5. 7. After performing the same transformations you did above, Anissa concluded that rotating a

figure 180˚ is the same as reflecting over intersecting lines. What important word does Anissa need to add to her statement to make it true?

Indicate if each statement below is Always (A), Sometimes (S), or Never (N) true?

8. ___________ Two reflections over parallel lines produce results equivalent to a translation.

9. ___________ Two reflections over parallel lines move the figure back onto itself.

10. ___________ Reflecting an image over intersecting perpendicular lines is equivalent to a

180° rotation only if the center of rotation is located where the intersecting lines meet.

11. ___________ Two reflections over distinct parallel lines is equivalent to a rotation.

g

h



12. Alex is creating a digital image for his baseball team’s jerseys, the “A’s”. He wants it to look like the diagram below. Which of the following transformations would result in this image?

13. The images below were rotated about the given points. Draw intersecting lines that would give equivalent results.

A’s a) 90˚ rotation about the intersection of the two

lines.

b) 180˚ rotation about the intersection of the two lines

c) composite reflection over the two intersecting lines

d) both a and b

e) both b and c

f) None of the above

L3 – Rigid Motion Transformations Name _____________________________ 3.11 – Congruence Per _____ Date ______________________


We now return to our earlier question: How can we tell if it possible to cut out Figure A, move it around and possibly flip it over, so that it matches Figure B perfectly, retaining the size and shape of Figure A? 1. Working on your own, use patty paper to trace Figure A on the left below (or physically cut

it out if you don’t have patty paper). Then, move it around so that you can place it in a perfectly matching manner on top of Figure B to the right. Keep track of the types of translations, rotations, or reflections that you use. Try to only use at most one of each. Label the vertices of the image appropriately as A’, B’, and C’.

Describe in words the simple transformations you used. Try to be as explicit as possible. Now compare your work with a partner. Did you use the same rigid motion transformation? If not, could you have used a different rigid motion transformation?

How do you know the side lengths and angles of the image are equal to those of the pre-image?



2. Use a ruler and protractor to confirm that the sizes and shapes of the two triangles shown below are identical.

Since these two triangles have the same size and shape, there should be a way to cut out the one on the left and move it around to match the one on the right. Therefore, there must be a rigid motion transformation that does just that. Describe a rigid motion transformation with pre-image ∆ABC and image ∆A’B’C’. Use patty paper or a cut-out first, if you like.

Work with a partner to try to determine a sequence of steps that will always work for identifying a rigid motion transformation that will move a pre-image triangle to a matching image, as long as the image has the same size and shape. Your steps should work regardless of the location of the image triangle.



3. For this problem you will work both on your own and with a partner. Complete steps a – c and e on your own.

a. Trace or cut out a copy of the triangle below. b. Write down a sequence of translations, rotations, and/or reflections, being sure to

keep your rigid motion transformation secret from your partner. Make sure the resulting image fits on the diagram below. Adjust accordingly. You may use whatever units you like for the grid.

c. Place the triangle copy on the grid below corresponding to the image of your transformation, and trace its location.

d. Exchange your diagram with your partner. Be sure to indicate the units you used. e. Identify a rigid motion transformation that has the image your partner drew. f. Share your results with your partner, and check to see if your partner’s rigid

motion transformation yields the image you traced. g. Did your partner use the same transformation you wrote down in secret? Don’t

worry if they are not the same, there are multiple transformations that yield the same image.



In this lesson we have seen that the image of a rigid motion transformation has the same size (e.g. triangle side lengths are equal) and shape (e.g. angles measures are equal) as its pre-image. We have also seen that if two figures have the same size and shape, then there exists a rigid motion transformation from one onto the other. Thus, we now have an answer to our earlier question: How can we tell if Figures A and B have the same size and shape without measuring (or cutting out and moving)?

Answer: Figure A has the same size and shape as Figure B only if there exists a rigid motion transformation with pre-image Figure A and image Figure B (or vice-versa since translations, rotations, and reflections are all reversible). In such a case we say that Figure A is congruent to Figure B, and we use the notation A B≅ .

Figure A is said to be congruent (#VOC) to Figure B if there exists a rigid motion transformation with pre-image Figure A and image Figure B. This is the formalization of being able to cut out Figure A and place it perfectly onto Figure B. Finding a Rigid Motion Transformation for Triangles Summary The following steps make it easier to find the rigid motion transformation from one triangle to another, assuming they are congruent. (Note: this method can be adjusted for other shapes as well.)

i. Choose a vertex, say A, on the pre-image triangle.

ii. Identify the corresponding vertex, say A’, on the image triangle. (Note: since the vertices are not labeled in most cases, it is up to you to visually determine corresponding vertices by their equal angle measures.)

iii. Translate the pre-image so that A moves to A’.

iv. Rotate around A’ as needed.

v. If the orientation is still off, reflect about the line x = A’ or y = A’ depending on the

orientation. Note: steps iii - v can be switched per your preferences, and you may not need to use all three types of transformations. Let’s try out our new method!



Show that each pair of triangles below are congruent by describing a rigid motion transformation from the left triangle onto the right triangle. Be sure to label corresponding vertices A’, B’ and C’. Try to use no more than one of each type of simple transformation. 4.

5.

L3 – Rigid Motion Transformations Name _____________________________ 3.12 – Congruence Homework Per _____ Date ______________________


Show that each pair of triangles below are congruent by describing a rigid motion transformation from the left triangle onto the right triangle. Try to use no more than one of each type of simple transformation. 1.

2.

L3 – Rigid Motion Transformations Name _____________________________ 3.13 – Grandma’s Furniture Pair & Share Per _____ Date ______________________


Now let’s revisit Grandma’s rearrangement of her room. Identify the sequence of transformations or single transformation that could be used to move each object in the room. Original Room Layout

Rearranged Room Layout

Lamp ______________________________ Chair ________________________________

Couch ______________________________ Table ________________________________

Desk ______________________________ Fan ________________________________

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