Name: ______________________________

Geometry Period _______

Unit 1: Rigid Motions

In this unit you must bring the following materials with you to class every day:

Straightedge (this is a ruler)

Pencil

Different colored pen & highlighter

This Booklet

A device

Headphones

Please note:

You may have random material checks in class

Some days you will have additional handouts to support your understanding of

the learning goals in that lesson. Keep these in a folder and bring to class every

day.

All homework for this unit is in this booklet.

Answer keys will be posted as usual for each daily lesson on our website,

Save it as a favorite!

1-1 Notes: Congruency and Rigid Motions

Learning Goal: What does it mean for two figures to be congruent? What is a rigid motion?

Think-Pair-Share:

Consider the pairs of shapes in each example below. Would you agree or disagree that the two figures within

each example are the same? Why or why not? If you argue that they are “the same,” what movements can we

do to show this? (Hint: can you perform any changes that result in the two figures overlapping?)

a. b. c.

Let’s discuss: 1. What did we mean when we referred to the figures being “the same” or not? 2. We will refer to these “movements” as __________________.

3. When two geometric figures are said to have the same side lengths and angle measures (shape and size),

they are said to be __________________.

*For this unit, we will consider congruency to be shown when two figures coincide exactly when they are

____________________________________________.

4. Let’s come to an agreement on some common language for these types of transformations:

Slide translation Flip reflections Turn Rotation

1-1 Notes

Is anything missing? Why?

`

`

Identifying Transformations:

Consider the pairs of shapes in each example below. Identify which transformations, if any, map one of the

figures onto the other figure and discuss whether or not the figures are congruent. (What types of movements

would allow us to show that these figures are congruent?)

*These changes or movements can be referred to as Rigid Motions.

Rigid motion: a transformation that preserves length and angle measure. A rigid motion maps lines to lines,

rays to rays, and segments to segments.

When performing rigid motions, we can refer to the original figure as the ______________ and the resulting figure as the ______________.

Quick Write!

Do all rigid motions produce congruent figures? [Hint: use our new vocabulary words (congruent,

transformation, rigid motion) to describe the how they are related?]

Transformation: is a function that moves or changes a figure in some way to produce a new figure called an

image. Another name for the original figure is the preimage. The points on the preimage are the inputs for the

transformation, and the points on the image are the outputs.

Summary:

1. What was a new concept that you learned today? ______________________________________________

2. On a scale of 1-3, with 3 = “I’m completely comfortable” and 1 = “I feel confused and need practice,” how

well do you feel you can explain this new concept to one of your peers?

Circle One: 1 2 3

1-1 Homework 1) Go to our Google classroom and work on your assignment for tonight. (Student info sheet) Be sure to submit.

2) Answer the following question.

Be sure to check your answer with the key online.

WHEN YOU MAKE CORRECTIONS, USE A DIFFERENT COLOR THAN YOU DID YOUR HW IN!

In the diagram below, and are graphed.

Use the properties of rigid motions to explain why .

Flipped lesson video next side. You will be quizzed on this tomorrow in class

3. Directions:

Watch the assigned video and take all notes on this page as you watch. Mastery of the

content of this video is essential for our next lesson in class.

This page will be checked tomorrow in class and an entrance ticket into class will be assigned to prove your mastery

of the concept.

IF YOU DO NOT WATCH THIS VIDEO, YOU WILL BE REQUIRED TO COMPLETE THE ASSIGNMENT AT LUNCH OR AFTER

SCHOOL WITH ME THAT DAY AND YOU WILL NOT BE ABLE TO PARTICIPATE IN THE FOLLOWING DAY’S LESSON.

Go to EdPuzzle and watch the “Translations” video! Don’t forget – log in with your student Google account!

Video Lesson Goal: What is a translation? How do you perform it?

Translations

A ________________________ that shifts a figure by adding a value to the x-coordinates and a (possible different) value

to the y-coordinates

Notation for a Translation

Suppose we move "a" units right or left (think! In which direction can we move on the x axis?)

and "b" units up or down (think! In which direction can we move on the y axis?)

OR

Let’s try it: Translate a figure

∆ABC has coordinates, A (2, 0), B (1, 7), and C (5, 1). On the accompanying set of axes, graph, label, and state the coordinates of ∆A'B'C', the image of ∆ABC after a translation of T1,-4

What does it mean to “State”?

Lesson 1-2 Translations

Today’s Learning Goal: What is a translation? How do you perform translations? What is a vector? What is vector

notation? How do you perform a composition of translations?

Ticket in the Door!

Think back to last night’s homework and answer the following questions. When you are done, put your pencil down and wait for further instructions.

1. With respect to rigid motions, interpret T-6, 2. Specifically, what is this notation telling us to do?

2. Using the graph to the right

a. describe how ∆𝐴𝐵𝐶 was changed (give specific detail).

b. write the rigid motion in proper notation.

Back Together! Given the following pre-image and image below, describe

with your neighbor how the pre-image changed position?

What details in the graph helped you reach this conclusion?

Vectors: an arrow that shows ______________ and ______________ when a figure is translated.

Interpret it! 1) A vector is shown right. Name the vector and describe it!

1-2 Notes

Vector Notation 2) The vertices of ∆𝐴𝐵𝐶 are A(0,3), B(2,4), and C(1,0).

a) Translate 5 units right and 1 unit down.

b) Draw the vector that describes this translation.

c) How can we write translation this using proper notation?

You Try: 3) a. Draw the vector that describes this translation to the right.

b. Write a rule for the translation of ∆𝐴𝐵𝐶 to ∆𝐴′𝐵′𝐶′.

Use appropriate notation. Explain how you arrived at your answer.

Composition of Transformations Definition: When two or more transformations are combined to form a new transformation.

Notation: This Says:

Let's Try It! 4) Apply the compositions below. Show your work by writing out the coordinates of the pre-image, intermediate

image and final image. Graph all images!

Think - Pair - Share 5) Yesterday we talked about congruency. Consider the following composition of translations. a) Do you think that the pre-image, intermediate image and final image in the translations below are congruent? Why

or why not?

Practice Complete each of the following problems. Check in with the key for feedback!

6) Name the vector shown and describe it in a sentence:

7) a) Translate the figure below as follows: 2 units down and 3 units left

* don’t forget to label!

b) Draw the vector that defines the translation.

8) Determine the transformation rule that translates A(3,-2) to A’(-1,4). Use appropriate notation.

9) a. Write a rule in appropriate notation for the translation of ∆𝐴𝐵𝐶 to ∆𝐴′𝐵′𝐶′.

b. Describe in words this transformation.

10)

a. Graph and label ∆𝑋𝑌𝑍 with X(2,4), Y(6,0), and Z(7,2) and its image after the composition T-5,-6 ∘ T-4,2.

(Show all work! i.e., which transformation takes place first?)

b. State the coordinates of the image.

c. What is the relationship between the pre-image

and the image? Justify your answer.

11) In chess, the knight (the piece shaped like a horse) moves in an L pattern. The board shows two consecutive moves

of a black knight during a game. Write a composition of translations for the moves. (Hint: Remember, order matters!)

It may help to describe what happens first and what happens second.

Now, rewrite the composition as a single translation that moves the night from its original position to its ending position.

12) A translation maps the figure on the left to the figure on the right.

a. What is the relationship between the two triangles below? Explain your reasoning.

b. Using the relationship you came up with in #5, solve for b and c.

More time? Take out a device and try this! http://www.transum.org/software/SW/Match_fish/matchfish.asp

http://www.transum.org/software/SW/Match_fish/matchfish.asp

1-2 Homework Directions: Complete the following problems to your best ability. Be sure to check your answers with the key!!

1. Name the vector and describe it:

2. a) Describe the translation that the following represents: T-3,4

b) The vertices of DEF are D(2,5), E(6,3) and F(4,0).

Graph, label and state the coordinates of D’E’F’ after the translation T-3,4.

3. Write a rule using appropriate notation for the translation that took place below:

4. a. Describe the composition of translations that take place in the graph below (Hint: Look at the labels)!

b. Represent the rule for the composition above using composition notation.

c. State one translation in coordinate notation that would map ∆ABC to ∆A”B”C”

in one step.

d. What is the relationship between each of the images? Why does that relationship occur?

5. Directions:

Watch the assigned video and take all notes on this page as you watch. Mastery of the

content of this video is essential for our next lesson in class.

This page will be checked tomorrow in class and an entrance ticket into class will be

assigned to prove your mastery of the concept.

Go to EdPuzzle and watch the “Reflection Prep” video! Don’t forget – log in with your student Google account!

Reflections and Congruency - Prep for Tomorrow’s Lesson! 1. Graph the line y = 3

What type of line is this?

What type of slope does it have? What type of slope does it have?

3. Sketch the reflection of triangle 𝐴𝐵𝐶 in Line m and name it triangle A’B’C’

What is the relationship between ∆ABC and ∆A′B′C′ ?

2. Graph the line x = 2

What type of line is this?

What type of slope does it have?

1-3 Reflections Learning Goal: What is a reflection? How do you perform various reflections?

Class Discussion Reflection (over a line):

A _________________________ in which each point in the ________________ has an image that is

_________________________________ from the line of reflection, but is on the ___________________________ of

the line

A reflection in line m maps every point P onto P’ so that line m is the ____________________

________________ of 𝑃�́�̅̅ ̅̅

Special case:

If the ________________ point is ON the line of reflection, then the point remains

( or does not change locations)

Notation:

Reflecting in Horizontal and Vertical Lines:

Example 1: Graph △ABC with vertices A(1, 3), B(5, 2), and C(2, 1) and its image after the reflection in the line n

whose equation is 𝑦 = 1

Quick Write!

Point A is _________ units _____ line n, so its reflection A’ is ____ units____ line n.

Points B is _____ units _____ line n, so its reflection 𝐵′ is ____ units line n.

Points C is ______ line n, so ________= ____ .

1-3 Notes

Tip For Success:

Always graph the

line you are

reflecting over

first!

Example 2: Graph △ABC with vertices A(1, 3), B(5, 2), and C(2, 1) and its image after the reflection

in the line 𝑥 = 3

You Quick Write! Explain how you were able to locate B’.

Practice Directions: Complete each of the following problems. After you finish a page check your work using the key!

1. a) Graph and label △ABC with vertices A (1, 3), B(5, 2), and C(3,-1)

b) Graph and State the coordinate of ∆𝑨′𝑩′𝑪′ after a reflection over the y -axis

c) Explain how you found the coordinates of A’:

2. a. What transformation took place to go from ∆𝑨𝑩𝑪 𝒕𝒐 ∆𝑨′𝑩′𝑪′ ? Be specific!

b. What is the relationship between the image and pre-image? Justify your

answer.

3. In the diagram below is graphed. State the coordinates of ∆𝑋′𝑌′𝑍′ after ry = 2. SHOW ALL WORK!

For examples 4-5, choose a column to work on. The column on the right is meant to challenge you. Graph paper is below

for scrap work.

4. What is the image of B(-5,1) under rx=2 ? 4. What is the image of B(-5,1) under ry=0?

5. What is the image of point C (4,3) under a reflection the line y = 5?

6. What is the image of A(1,3) under a rx-axis

5. The point (0,-1) is the midpoint on segment CC’. If point C’ (0,2) is the image of point C, what are the coordinate of point C?

6. What is the image of A(x,y) under a rx-axis?

1-3 Homework Complete the following problems to your best ability. A key will be available for you next class.

1. Graph ABC and its image after a reflection in the given line stated below. State the coordinates of A’ B’ and C’ in

the space provided.

A(4,-1), B(3,7), C(-1,1); reflection on y = -2

2. Write a rule for the translation of ABC to .A B C (Careful look @ the axes)!

Why are the two images congruent?

3. True/False:

a) The image of the point (4,-3) under a reflection across the x-axis is (-4,-3). Circle one: True False

b) The image of the point (-5,4) under a reflection across the y-axis is (5,4). Circle one: True False

Check your work with

a different color!

4. ∆ABC has vertices A(1,4), B(2,7), C(5,4). Its image is

∆A'B'C' with vertices A'(1,0), B'(2,-3), C'(5,0).

Graph the triangles and draw the line of reflection. Write

the equation of the line of reflection.

5. In the figure to the right 𝐺𝐻 ⃡ is a line of reflection. State and justify two conclusions (or true statements) about

distances in this figure.

Lesson 1-4: Special Reflections

Learning Goal: What is a reflection? How do you perform various reflections over lines and points? What is a line of symmetry?

Do-Now: Fill Out Questions #1 and #2:

Reflecting in the line 𝒚 = 𝒙 Reflecting in the line 𝑦 = −𝑥 𝐺𝑟𝑎𝑝ℎ𝑠→

1) State the coordinates for each:

A (-1,5)

B (0, 2)

C (-3, 1)

D (-4, 4)

A’ (5, -1)

B’ ( )

C’ ( )

D’ ( )

A (2, 5)

B (2, 2)

C (8, 2)

A’ (-5, -2)

B’ ( )

C’ ( )

2) Do you notice anything happening between coordinates of the pre-image compared to the image?

Do you notice anything happening between coordinates of the pre-image compared to the image?

3) Notation

Pre-image: P(x,y)𝑟𝑦=𝑥→ Image: P’

Notation

Pre-image: P(x,y)𝑟𝑦=−𝑥→ Image: P’

4) Write the coordinates for the image of point (8,-3) under a 𝑟𝑦=𝑥

Write the coordinates for the image of point (8,-3) under a 𝑟𝑦=−𝑥

1-4 Notes

Reflections and Congruency

What is the relationship between ∆DEF and ∆D′E′F′ ? Explain your answer!!

Reflections across points

Point Reflections: A reflection through a point so that the point P becomes the between the

point A and its image A.

Let’s Try a Point Reflection! Geometry Leap Frog!

Graph triangle ABC. A(1, 2), B(5, 5), C(5, 2) and reflect it through point (-1, 1). State the coordinates of the image

A(1, 2) →

B(5, 5) →

C(5, 2) →

A

B

C

P

Lines of Symmetry

From past lessons: Can one side of the figure below be folded so that it matches the other?

A figure has line symmetry when:

Practice Part 1: Translation Practice

Complete each of the following problems by the end of class:

1. Translate and label the image of RUST after a translation: T-3,-2

Draw and describe the vector the defines the translation.

a. Sketch the line where we would fold the figure. This is called the

.

b. In this particular example, what it the relationship between the line of symmetry and line of

reflection?

2. a) Draw a vector showing the translation in the diagram below:

b) Describe the transformation in words **Careful the axes are tricky!

c) Write a rule in transformation notation for the translation of ∆𝐴𝐵𝐶 to ∆𝐴′𝐵′𝐶′

3. Graph and label ∆𝑋𝑌𝑍 with X(2,4), Y(6,0), and Z(7,2) and its image after the composite on T0,-7∘T-4,2. Show work.

What is the relationship between the pre-image and the image? Justify your answer.

4. Determine the transformation rule that translates A(3,-2) to A’(-1,4). Use appropriate transformation notation.

5. A translation maps the figure on the left to the figure on the right. What is the relationship between these two triangles?

Using the relationship, solve for b and c.

6. In chess, the knight (the piece shaped like a horse) moves in an L pattern. The board shows

two consecutive moves of a black knight during a game. Write a composition of translations for

the moves. (Hint: Remember, order matters!) It may help to describe what happens first and

what happens second.

Now, rewrite the composition as a single translation that moves the night from its original

position to its ending position.

Part 2: Reflection Practice

7. Graph △ABC with vertices A(1, 3), B(5, 2), and C(2, 1) and its image after the reflection over the line y = x.

For examples 8- 10, choose a column to work on. The column on the right is meant to challenge you. Graph paper may be provided for scrap work.

Determine the number of lines of symmetry for the figure: Hint Sketch them in!

11. 12.

8. What is the image of point A (15,3) under rY=X? 8. What is the image of point A (15,3) under ry = 5?

9.What is the image of point B (-5,1) under ry=-x? 9. What is the image of point B (-5,1) under ry= -x?

10. What is the image of point C (2,-1) under a reflection across point M(0,2).

10. What is the image of point C (2,-1) under a reflection across point M(0,2).

1-4 Homework Complete the following problems to your best ability. A key will be available for you next class.

In Exercise 1, graph ABC and its image after a reflection in the given point. State the coordinates of the image.

1. A (4,-1), B (3, 8), C (-1, 1)

Point of reflection: P (4,-1)

What did you notice about point A’?

In Exercises 2 and 3, graph the polygon and its image after a

reflection in the given line. *CAREFUL WATCH AXES!

2. ry=-x 3. ry=x

P( , ) P’ ( , )

S( , ) S’ ( , )

R( , ) R’ ( , )

Q( , ) Q’ ( , )

R( , )R’ ( , )

Q( , ) Q’ ( , )

P( , ) P’ ( , )

S( , ) S’ ( , )

In Exercises 4 and 5, determine the number of lines of symmetry for the figure.

6. Write a rule for the translation of Triangle ABC to Triangle A’B’C’.

7. In the diagram below, ABC and XYZ are graphed.

a) Describe the reflection that occurred. Be specific.

b) Explain why the images are congruent.

**Check your hw!

Today’s Learning Goal: What is special about a composition of reflections over 2 parallel lines?

Exploration #1: What single transformation is equivalent to a composition of reflections over parallel lines? Let’s continue our work with compositions. Work with your partner and answer the following questions. Be prepared to share your thinking!

Task: Perform the following composition on ∆XYZ: 𝑟𝑦=−2 𝜊 𝑟𝑦=2

Pre-Image Image

Discover the Relationship!

1. What is special about the two lines of reflection?

2. Compare the final image with the pre-image. How are they related?

3. Can you describe a single transformation that will map the original triangle onto its final image? What would it be?

Predict:

4. What do you think might happen when we perform a composition of reflections across two lines that intersect?

Name_________________________________ Date______

Class Discussion:

1-5 Notes

TOGETHER! Reflections in Parallel Lines Theorem:

A composition of reflections over two parallel lines

is the same as one single _____________________.

Partner Check-In:

1. The diagram below shows ∆𝐴𝐵𝐶 after a reflection across the x axis,

then a translation 𝑇6,0.

Draw a vector showing the translation.

2. The vertices of ∆PQR are P(2, 1), Q(4, 1), and R(4, 3). Find ∆P’’Q’’R’’, the image of ∆PQR under T0,-5 ∘ ry-axis.

Quick Summary:

3. A composition of reflections over two parallel lines, can also be done as _________________________

Self-Assess! How are you feeling?

I Get it! Give me a quiz today!

I was okay when we did it as a group but would like

guided practice!

I feel confused on how to work through these problems or how

to set them up on my own.

1-5 Practice!

Before you start, write down one thing you are looking forward to in the next week:

Directions: Complete each of the following special composition problems:

Show your work by stating the coordinates of the pre-images, intermediate images and final images for each example

and graph final image.

Graph: Show Work Here:

1. 𝑟(𝑥=−5) ∘ 𝑟(𝑥=4)

2. Which of these is equivalent to a translation?

a. a reflection across one line

b. a composition of two reflections across intersecting lines

c. a composition of two reflections across parallel lines

3. A figure is reflected across the line y = 2, then reflected across the line y = 4. Which single transformation results in the same image?

a. a reflection across the line y = 3

b. a reflection across the line y = 6

c. a translation 2 units up

d. a translation 4 units up

Can you describe a single transformation that will map the original

triangle onto its final image?

4. Given diagram below, with k||m. A translation maps on to which triangle?

Explain your answer!

5. Given triangle A(3, 2), B(5, 7), C(7, 3), perform ry =x ∘ T-3,4.

6. Productive Struggle! Point B’(1,4) is the image of B(3,2) after a reflection over line c. Write an equation for line c.

Hint: think back to what you already know/remember about equations of lines! Sketch to help!

1-5 Homework

Complete the following problems to your best ability. Be sure to check your answers with the key!!

Don’t forget that you can use both graphic and arithmetic methods to answer the questions.

1. The coordinates of the vertices of are , , and .

a) On the graph below, draw and label .

b) Perform the following composition: 𝑟𝑥−𝑎𝑥𝑖𝑠 ∘ 𝑟𝑦=6

c) State one single transformation that would produce the same final image ∆A”B”C”

2. Find the coordinates of the image of under the transformation 𝑟𝑦=𝑥 ∘ 𝑇1,1.

Follow up: Is this composition commutative? Why?

3. a) Error Analysis: Describe and Correct the Error in the student description below:

b) The original composition described was, “Triangle ABC is mapped to A’’B’’C’’ by a translation 3 units down and a

reflection over the y-axis.” Is this a glide reflection? Justify your answer.

4. Josie says the following composition can be rewritten as a composition consisting of only two transformations.

Determine whether she is correct and explain your reasoning:

𝑟𝑦=−2 𝑜 𝑟𝑦−𝑎𝑥𝑖𝑠 𝑜 𝑟𝑥=3

Triangle ABC is mapped to A’’B’’C’’ by a

translation 3 units down and a reflection

over the y-axis.

Guiding questions:

1) Which

transformation

occurs 1st? 2nd?

2) Where is the

pre-image?

Where is the

final image?

http://www.bing.com/images/search?q=thinking&qpvt=thinking&FORM=IGRE&adlt=strict#view=detail&id=43856601B2061B01F2D6DE355ABEFB9914C9AE3E&selectedIndex=29

1-7 Notes: Rotations and More Special Compositions

Learning Goals: What is a rotation? How do you perform a rotation about point p? What is special about a composition of

reflections over 2 intersecting lines?

Exploration #1: What single transformation is equivalent to a composition of reflections over perpendicular lines?

Task: △ABC with vertices A(1, 3), B(5, 2), and C(2,1) Perform the

following composition: 𝑟𝑦=−2 𝜊 𝑟𝑥=1

Discover the Relationship!

1) What is special about the two lines of reflection?

2) Compare the final image with the pre-image. How are they related?

3) Can you describe a single transformation that will map the original triangle onto its final image? What would it be?

Key Fact:

*A composition of reflections over intersecting lines is equivalent to one ________________________

*A composition of reflections over intersecting AND perpendicular lines is equivalent to _____________.

Rotation: a rigid motion in which a figure is turned about a fixed point called the _____________________.

We can rotate along a circular arc either clockwise or counterclockwise. **For this unit, unless otherwise stated, positive rotations are always counterclockwise.** Notation: Rangle measure

1-7 Notes

Analyzing the Angle of Rotation of a Figure About a Point P:

4) Describing a rotation. When describing a rotation you always need 2 details:

a. What direction do we assume the figure was rotated?

b. What is the center of rotation?

*You can choose whichever method you like best: 1. MEMORIZE the rules Or 2. Rotate your page the given amount and read new coordinates

6) Graph quadrilateral RSTU with vertices R(3, 1), S(5, 1), T(5, −3), and U(2, −1) and its image after a 270° rotation about the origin.

Complete the following and then check in!

7. Graph △JKL with vertices J(3, 0), K(4,3) L (6,0) and its image after a 90° rotation about the origin. Show your work! Raise your hand and check in!

Pro-Tip:

Write all of the coordinates first THEN plot the image

State the coordinates of the image:

Practice

1. Graph: Show Work Here: Consider ∆𝐴𝐵𝐶.On the coordinate axes below,

show the composition 𝑟(𝑦=2) ∘ 𝑟(𝑥=1)

Can you describe a single transformation that will map the original triangle onto its final image? Be specific!

9. a) In each figure below, the triangle on the left has been mapped to the one on the right by a rotation about point

P. Identify all six pairs of corresponding parts (vertices and sides). Mark the corresponding parts on the triangles?

b) What rigid motion mapped △ABC onto △XYZ.

In Exercises 10-12, tell whether the statement is always, sometimes, or never true. Explain your reasoning.

10. A composition of reflections over 2 intersecting lines is a rotation.

11. If two figures are congruent, then there is a rigid motion or a composition of rigid motions that maps one figure onto the other. 12. A composition of reflections over 2 intersecting lines is the same thing as a rotation of 1800.

Corresponding Vertices Corresponding Sides B corresponds to _______ AB corresponds to _______

A corresponds to _______ BC corresponds to _______

C corresponds to _______ AC corresponds to _______

1-7 Homework 1. Graph ∆𝐴𝐵𝐶 with endpoints A(−4, 4), B(−1, 7), and C(-1,4) and its image after the composition: T-2,-1 ° R270 about the origin. State the coordinates of the final image.

2. Figure ABCD was rotated about point E. Are the pre-image and image congruent? Justify your answer using properties of rigid motions.

3. Error Analysis: Kim answered the following question explain why the two triangles are congruent. Correct the errors in describing the congruence transformation. Kim’s Response: “△ABC is mapped to △A’’B’’C’’ by R180°. So the triangles are obviously congruent”

4. Graph △CAT with vertices at C(-3,2), A(2,4), T(3,1) and its image after a rotation of 180° about the origin. Label the image △C’A’T’. How did you get the vertices for △C’A’T’?

5. A reflection in the x-axis followed by a reflection in the y-axis is equivalent to a rotation of ___________°, because the lines are ________________________________ 6. Use figure A to answer the following questions: a) Which figure is a reflection of Figure A in the line x = a?

(a can be any Integer) .

b) Which figure is a reflection of Figure A in the line y = b?

( b can be any integer).

c) Push your thinking! Which figure is a reflection of Figure A in the

line y = x?

Lesson 1-8-Rotational Symmetry

Learning Goal: What is rotational symmetry?

Recently, we’ve looked at Lines of Symmetry...

Which of the following have multiple lines of symmetry?

Today, we'll talk about something called

Rotational Symmetry

Predict: What do you think rotational symmetry might be like?

Rotational Symmetry: exists when a figure can turn onto _______________ when turned o° to 180°.

Rotational symmetry Order: the number of times a figure _________ onto

itself in one full revolution.

Partner discovery:

1. What are some special characteristics of the polygon shown here?

2. What is the measure of an interior angle?

What is the measure of an exterior angle shown?

Predict- What would be a method of calculating the measure of one exterior

angle? Hint: The whole “way around” is 360°. Doesn’t this polygon have 8 exterior angles?

So how could we calculate the exterior angle?

3. AS A CLASS! We call these special polygons ________________polygons because

they have __________ sides and ________ interior angles.

1-8 Notes

Note! The Exterior

Angle is the angle

between any side of a

shape, and a line

extended from the

next side.

Class discussion:

How many degrees of rotation would guarantee that I will map the original

figure onto itself?

We don’t want to rotate 360° because they we won’t have rotational

symmetry ! We would like to rotate “a little less”but how do we

calculate “ a little less”?

Let’s use technology to help us!

Conclusion! The measure exterior angle of a regular polygon is __________________ to the angle measure

that we need to rotate a regular polygon so that it maps onto itself.

Big take away!

If a polygon is regular, the number of degrees it must be rotated onto itself is found by calculating the exterior angle of

the Polygon:

Formula for calculating the exterior angle:

Let's put the two ideas together! Consider the regular pentagon ABCDE:

1. Draw all lines of symmetry. Locate the center of the figure for rotational symmetry.

2. a) What is the Rotational Symmetry Order?

b) How many are lines of symmetry are there?

3. Look at the regular pentagon shown: What is the least amount of

degrees pentagon ABCDE must be rotated to carry the pentagon onto

itself?

1-8 Practice Answer the following problems completely. Note that this includes mixed review!

1. For a-d, determine whether the figure has rotational symmetry. If so, what is the rotational symmetry order?

a)

b)

c)

d)

2. Consider the regular hexagon shown below. What is the measure of an exterior angle? What is the minimum

number of degrees it must be rotated to carry the hexagon onto itself?

3.

4. Consider the figure shown right:

a. What is the rotational symmetry order?

b. How many lines of symmetry are there?

c. How could you shade the figure so that the resulting figure has

a rotational symmetry order of 3.

5. a) Show your work by writing out the pre-image, intermediate image and final image for the following example

b) Graph final image

Graph: Show Work Here:

𝑅180° ∘ 𝑅90°

6. You walk up to a revolving door to enter a building.

a. You rotate the door 180 degrees. Where are you after this rotation in the

context of this problem? Explain.

b. Then, you rotate the door 360 degrees. Where are you after this rotation in

the context of this problem? Explain

1-8 Homework Directions: Answer the following questions to the best of your ability. Don’t forget that you can use both graphic and

arithmetic methods to answer the questions.

1. If the coordinates of point P are , then is

1)

2)

3)

4)

2. Find the image of point under the composition of translations .

3. What are the coordinates of , the image of , after a counterclockwise rotation of 90° about the origin?

1)

2)

3)

4)

4. Consider the regular dodecagon shown below. What is the minimum number of degrees it must be rotated to carry the

dodecagon onto itself?

5. Which regular polygon has a minimum rotation of 45 degrees to carry the polygon onto itself?

6. Push yourself! Below is the final image, the result of a composition on ∆𝐴𝐵𝐶 where A(-5, 0), B(-5,-5) and C(-2,-4). The

composition is noted as the following:

𝑅𝑛° ∘ 𝑅180°

Math-Hoo thinks that “n” (the last degree of rotation in the composition) is 3600. His friend SLephant thinks that it’s 0o.

Who’s right? Explain.