l2 translations, reflections, rotations...

L2 – Translations, Reflections, and Rotations Name ________________________________ 2.1 - Pre-Assessment Per _____ Date _________________________

Geometry Q1: L2 Translations, Reflections, and Rotations Handouts

Have you ever wanted to rearrange the furniture in your room? First you might want to make sure that the furniture would fit in the new location. You could draw a scale model of your room and each piece of furniture, and then move the pieces around. Mathematically this act of rearranging the furniture is referred to as performing a series of Rigid Motion Transformations. Determine how each figure was transformed (translated, reflected, or rotated).

(Hint: there could be more than one correct answer.)

1. Draw the lines of reflection and centers of rotation in the above figures, when applicable.

2. Define translation, in your own words.

3. Explain the similarities and differences between reflection and rotation.

L2 – Translations, Reflections, and Rotations Name ________________________________ 2.2 – Introduction and Definitions Per _____ Date _________________________


One of the main goals of Geometry is to determine when two objects have the same shape and/or the same size. For example, if we move a couch across the room and then rotate it a little to better fit the shape of the room, the couch still has the same shape and size. This is an example of what we call a transformation of the couch from its former position to a new position. A transformation (#VOC) is a function that assigns to each point in the plane a point in the plane that may be the same or different. We refer to the input of a transformation (i.e. the object to be transformed) as its Pre-Image (#VOC) and the resulting output (i.e. the transformed object) as its Image (#VOC). We begin our study of transformations by investigating some special types of transformations that serve as basic building-blocks. Below are 9 pairs of shapes, each labeled A and B. Arrange them into groups based on the types of transformations that you might use to move shape A onto shape B. In this lesson we will formally define three of the four types of building-block functions used to define transformations. For now, we can think of them as:

1) Move A to the left or right, and/or up or down. 2) Reflect A over a line. 3) Rotate A about a point. 4) Resize A, keeping the shape but increasing or decreasing the size.



We will now formally define the first three types of transformations. Translations The first building-block transformation is referred to as a translation, which is a function that moves each point in the plane a fixed value left or right, and/or up or down.

Translate the triangle (on the left) to the right 8 units and down 5 units.

This is the same as a horizontal shift _______ units, and a vertical shift _______ units.

Translations move horizontally, vertically, and diagonally. A translation that appears to move diagonally is a combination of both horizontal and vertical movement.

Grandma is considering moving a non-traditional shaped picture a little up and to the left from its current placement on the wall. The following figure shows the translation she used from point A to point A’. Connect each of the remaining points in the Pre-image of this translation with its corresponding point in the Image.

What do you notice about the lines connecting the points in the Pre-image to their corresponding points in the Image? The pre-image was shifted horizontally _______units and vertically _______units. Now we are ready to use this information to define a translation.



A translation (#VOC) is a function that moves all points on a figure a fixed distance “a” units in the x-direction and “b” units in the y-direction. This can be represented by: T(x, y) = (x+a, y+b) for real values of a and b. When the x-coordinate is changed, the figure moves horizontally. When the y-coordinate is changed, the figure moves vertically. We can describe the translation in words such as, “to the left 3 units and up 2 units” or “horizontally – 3 and vertically 2”. The translation of Grandma’s picture can also be given by the rule: T(x, y) = (x _____, y _____). Note that this is equivalent to defining a translation in terms of moving all points on a figure a fixed distance along parallel paths to the image. More specifically, a translation r-units in the

direction of ray AB! "!!

is a function that takes each point P to the point S, where PS = r, 𝑃𝑆 || 𝐴𝐵

and 𝐴𝑃 || 𝐵𝑆. Here it is important to note that direction AB! "!!

is not the same as direction BA! "!!

. A translation can change the image’s (check all that apply):

! Position ! Size ! Shape

Reflections

The next building block is a reflection. Grandma is driving through a neighborhood and noticed that there is a similarity between the houses. She noticed that the houses all look the same except some of them are “facing” the opposite direction. She drew a sketch of what she saw so she could further explore it.

What type of transformation do you think Grandma noticed?

What is special about the Property Line?

A reflection over the line l (#VOC) is a function that moves each point on a figure to its mirror image over line l.



Reflect the polygon over the given line. You can use a mira to help you, or fold patty paper along the line of reflection to match all vertices. A reflection over the line l is a function that takes each point P not on the line l to the point S, where l is the perpendicular bisector of 𝑃𝑆. Note, it leaves points on the line fixed. A reflection shows a mirror image of a figure over a given line. It is important to identify the line of reflection. This line also becomes the line of symmetry as the pre-figure and figure are exact mirror images of one another. Images can be reflected in the Cartesian Coordinate Plane. The x- and y-axes may be used as the lines of reflection. The figure was reflected over the x -axis in the diagram at the right.

Figures could be reflected over other lines in the Coordinate Grid, such as the linear function: y = 2x + 1.

If the line of reflection lies within the pre-image, its reflection will overlap with the original, as in the example below where the pentagon reflects over the x-axis. A reflection can change the image’s:




Rotations

The last rigid motion transformation is the rotation about a point. Grandma has decided that she wants to rotate her triangular table 90° so that the shortest side is against the wall. The diagram shows the rotation.

What is the point that Grandma rotated the table around? A clockwise/counter-clockwise rotation X ° about the point C (#VOC) is a function that moves each point in a figure to the point located X-degrees in the CW/CCW direction along the circle with center C and radius equal to the distance from the point to C. In other words, a rotation turns a figure about a central point C, preserving each point’s distance from the center. Using patty paper, trace and rotate the points A, B, and C around point P. Rotate the points 90° clockwise (CW) about point P and label the new points A’, B’, and C’. Rotate points A, B, and C 180˚ CW about point P and label the new points A’’, B’’, and C’’.

Wall



Trace the trapezoid and begin rotating the figure around point P. How many degrees does the trapezoid need to be rotated until it is back in its original position?

Draw the figure after a 90° counter-clockwise (CCW) rotation about point P.

A rotation of x-degrees CW/counter-clockwise (CCW) about a fixed point P is a function that takes each point R to a point S, where R and S are located on the circle centered at P, and S is located x-degrees CW/CCW from R. In this diagram, the pre-image was rotated. It is critical to describe the direction, degrees, and center of rotation. This particular pre-image was rotated 90° CW about the origin (0, 0).

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The same pre-image could be rotated 90° counter-clockwise about one if its vertices, point (-2, -1). This image looks very different, as seen below.

Rotations are based on circles. If a figure is rotated 360˚, the image matches the exact location of its pre-image. Similarly, a 180° rotation CW would result in the same image as a 180° CCW rotation. As you explore rotations further, you will notice other patterns when rotating images. A rotation can change the image’s:


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L2 – Translations, Reflections, and Rotations Name ________________________________ 2.3 – Practice Set Per _____ Date _________________________


a) b) c) d) e) none of these

1. Translate ΔRST horizontally 4 units and vertically – 3 units. Label the new image R’S’T’.

2. Translate ΔRST using the

rule: T (x, y) = (x – 4, y + 3). Label the new image R’’S’’T’’.

3. Describe how the pre-image

(ΔRST) moved differently in exercise #1 versus exercise #2.

4. Which of the following diagrams could show a translation using the rule: T(x, y) = (x, y – 4)?

5. Describe the translation in words then give the rule.

describe:

rule: T (x, y) = ( , )

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Pre-image: Image:

A (3, 5) A’ ( , )

B (- 1, 6) B’ ( , )

C (- 2, 8) C’ ( , )

6. Reflect ΔJKL below, over the y-axis. Label the new image J’K’L’. 7. Draw and label (J’’K’’L’’), the image of ΔJKL after a reflection about the line y = x.

8. ΔABC is reflected over the y-axis. Give the new coordinates for its image:

9. Describe the pattern you notice in the coordinates from exercise #8, when reflecting an image over the y-axis. Create a generalization about this pattern.

R (x, y) = ( , )

10. Identify and write the equation representing the line of reflection for each of the diagrams below.

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11. Rotate rectangle ABCD 90˚

clockwise about the origin. Label the image A’, B’, C’, and D’.

12. Describe a counter-clockwise

rotation that would result in the same image as exercise #1.

13. Rotate rectangle ABCD 180˚ counter-clockwise about point (1, -1), or vertex A.

14. How would exercise 13 look if the

rectangle was rotated 180˚ clockwise about its vertex A?

15. Describe, in detail, how the triangle was rotated.

16. Point X (6, - 3) is rotated 90˚ clockwise about the origin. What are the coordinates for X’?

a) (3, - 6) b) (- 6, 3) c) (- 3, - 6) d) (3, 6)

17. Place a dot ( ⦁ ) on each of the drawings below, indicating the center of rotation.

L2 – Translations, Reflections, and Rotations Name ________________________________ 2.4 – Homework Per _____ Date _________________________


1. Translate trapezoid ABCD using the rule: T (x, y) = (x – 3, y + 2 !

!).

2. What is the rule that moves an image

down 5 units? rule: T (x, y) = ( , )

3. A 90˚ clockwise rotation about the origin would give the same result as what kind of counter-clockwise rotation about the origin?

4. Rectangle RSTU is reflected over the x-axis. Give the new coordiantes.

pre-image image R (3, -5) R’____________

S (3, -6!!) S’____________

T (-2, -6!!) T’____________

U (-2, -5) U’____________

5. Rotate ΔABC 90˚ counter-clockwise about point P.

L2 – Translations, Reflections, and Rotations Name ________________________________ 2.5 – Transformations Warm Up Per _____ Date _________________________


1. Reflect ΔDEF over the line y = x.

Label your image D’, E’, and F’.

2. If Q (4, -5) was translated and mapped to Q’ (-1, 0) what would the coordinates be for R’ if R (6, 2) was translated using the same rule?

R’ ( , )

3. What kind of transformation would be described by the rule:

T (x, y) = (x, – y) ?

4. Rotate pentagon LMNOP 90° clockwise about the origin. Label your new image (L’, M’, etc.).

5. Describe how the pentagon’s coordinates changed under this rotation.

6. Circle the diagram below that shows a vertical translation.

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L2 – Translations, Reflections, and Rotations Name ________________________________ 2.6 – Transformations Per _____ Date _________________________


Describe what kind of transformation could map the pre-image onto the image. Be as specific as possible in your answers.

One of the examples above could have more than one correct answer.

Discuss all of the possible answers with your class.

(Hint: There are at least possible answers! Can you identify them all?)

L2 – Translations, Reflections, and Rotations Name ________________________________ 2.7 – Scavenger Hunt Per _____ Date _________________________


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

1. At your first poster, ignore the 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 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!

4) Description/Rule:

Sketch: Letter/#:


Sketch: Letter/#:


Sketch: Letter/#:


Sketch: Letter/#:

L2 – Translations, Reflections, and Rotations Name ________________________________ 2.7 – Scavenger Hunt Per _____ Date _________________________



Sketch: Letter/#:


Sketch: Letter/#:


Sketch: Letter/#:


Sketch: Letter/#:


Sketch: Letter/#:


Sketch: Letter/#:

L2 – Translations, Reflections, and Rotations Name ________________________________ 2.8 – Transformations Exit Pass/Homework Per _____ Date _________________________


Use the secret message from your scavenger hunt activity to perform a transformation on the image below.

After translating your figure, analyze what might have changed.

Did the figure’s orientation change?

Did the figure’s position change?

Did the figure’s size change?

L2 – Translations, Reflections, and Rotations Name ________________________________ 2.9 – Rigid Motion Transformations Per _____ Date _________________________


Translations, reflections, and rotations are all examples of transformations that preserve size and shape. The image of a figure (e.g. triangles, rectangles, pentagons, and all polygons) transformed using a translation, reflection, and/or rotation has the exact same side lengths and angle measurements as the pre-image.

Analyze the rectangle FGHI:

What are the side lengths?

𝐹𝐺 = 𝐺𝐻 =

𝐼𝐻 = 𝐹𝐼 =

What are the angle measurements?

m∠ F = m∠ G =

m∠ H = m∠ I =

What is the perimeter?

What is the area?

! Translate FGHI using the rule: T (x, y) = (x – 4, y + 3). Discuss the new image side lengths, angle measurements, perimeter, and area after the translation.

! Reflect FGHI over𝐺𝐻. Again, discuss the figure after the reflection! ! Rotate FGHI 180° about vertex I and check out its dimensions, angle measurements,

perimeter, and area!

I H

GF

L2 – Translations, Reflections, and Rotations Name ________________________________ 2.10 – Rigid Motion Transformations Practice Set Per _____ Date _________________________


1. Draw parallelogram PARL with P (0, 4½), A (5, 4½), R (3, 1) and L (-2, 1) 2. What is the length of PA? 3. What is the area of the pre-image? (Hint: You can count boxes on the grid if you don’t

remember the formula.)

4. Reflect the pre-image over the x-axis. 5. What is the length of P’A’? 6. What is the area of the reflected image?

7. Using the same coordinate grid as above, plot the three coordinates and connect the ordered pairs to create a triangle: R (-3, -2), S (0, -6), T (-5, -6).

8. What is the area of the pre-image?

9. Rotate ∆RST 90⁰ CW about the origin. Label the new image R’, S’, T’. 10. What is the area of the new figure?

11. Notice the pre-image is in Quadrant III and the image is in Quadrant II. Will this always be

the case for a rotation? Explain your answer.

L2 – Translations, Reflections, and Rotations Name ________________________________ 2.10 – Rigid Motion Transformations Practice Set Per _____ Date _________________________


12. What is the perimeter and area of the rectangle, RECT, below?

Perimeter: Area: 13. Translate RECT using the rule: (x, y) → (x + 6, y). Label the new image (R’, E’, etc.). 14. What is the perimeter and area of the new image?

Perimeter: Area: 15. Draw another rectangle that is twice the perimeter of RECT yet similar (ie. the width is half

the length). What do the two figures have in common?

16. : Does this transformation preserve size and shape?

17. What do we call a transformation that preserves side lengths and angle measurements? (Choose all answers that are correct.)

a) a translation b) rigid motion c) angle equivalence d) convenient e) none of these

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L2 – Translations, Reflections, and Rotations Name ________________________________ 2.11 – Rigid Motion Transformations Exit Pass Per _____ Date _________________________


1. Circle the transformations below that are examples of rigid motion transformations.

2. Describe a rotation that would carry the image onto itself, other than a 360° rotation. For example, a square could be rotated 90° CW/270° CCW, 180° CW/CCW, or 90° CCW/270° CW.

3. Describe two completely different transformations that would carry the pre-image onto the image. Draw on the diagram to help explain you answer if you need to.

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L2 – Translations, Reflections, and Rotations Name ________________________________ 2.12 - Rigid Motion Transformations Homework Per _____ Date _________________________


1. horizontally – 6 & vertically 5 units

2. horizontally 5 & vertically 6 units

3. horizontally 6 & vertically – 5 units

4. horizontally – 5 & vertically 6 units

5. horizontally 13 units

6. vertically 13 units

A. up 13 units

B. left 5 units & up 6 units

C. right 5 units & up 6 units

D. right 13 units

E. left 6 units & up 5 units

F. right 5 units & up 6 units

Double match the translation rule to its descriptions.

_____ & _____ T (x, y) = (x – 5, y + 6)

_____ & _____ T (x, y) = (x + 13, y)

_____ & _____ T (x, y) = (x – 6, y + 5)

_____ & _____ T (x, y) = (x, y + 13)

_____ & _____ T (x, y) = (x + 5, y + 6)

1. If point R (1, 4) was rotated 90˚ clockwise about the origin, what would be the coordinates for R’? a) R’ (4, -1) b) R’ (-1, 4) c) R’ (- 4, 1) d) R’ (4, 1) e) none of these

2. What kind of rotation would result in an image mapped onto the pre-image? Be specific in your description.

3. What CCW rotation would result in the same position as a 90⁰ CW rotation about the origin?

Be specific in your answer. 4. What CW rotation would result in the same image as a 112⁰ CCW rotation about the origin?

L2 – Translations, Reflections, and Rotations Name ________________________________ 2.13 – Transformations Summary Warm Up Per _____ Date _________________________


1. If point P (5, – 6 ) was rotated 90° clockwise about the origin, what would be the coordinates for P’?

2. If point X (- 8, 12) was rotated 180˚ about the origin, what would be the coordinates for X’?

a) X’ (8, 12) b) X’ (- 8, - 12) c) X’ (8, - 12) d) X’ (- 8, 12) e) none of these

3. What is the rule to map point T (18, - 14) to its image T’ (6, - 10!!) ?

4. Reflect the image below over the line y = x. Label the new coordinates.

5. Create and complete a pre-image/image table to list the ordered pairs in the space next to the

grid.

6. Do you notice a pattern in the values?

7. Make a generalization about what happens to a figure and its coordinates when reflected over the line y = x.

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L2 – Translations, Reflections, and Rotations Name ________________________________ 2.14 – Transformations Summary Per _____ Date _________________________


Use the space below to draw the rectangle in each of the following ways. You will creating 5 distinct transformations, each starting from the original rectangle. Label each with its corresponding number.

1. Reflect it over the line given. 2. Translate horizontally 6 and vertically – 2 units. 3. Rotate it 180˚ about its vertex Z. 4. Enlarge the rectangle, make it double in size (use the center of the rectangle as the center of

the enlargement). 5. Reduce the rectangle, make it half its original size (use the center of the rectangle as the

center of the reduction).

6. Which of the above exercises were examples of rigid motion? Explain how you know.

7. Exercises 4 and 5 above are examples of dilation. Dilation will be covered in a later unit. Describe what you think dilation means.

L2 – Translations, Reflections, and Rotations Name ________________________________ 2.14 – Transformations Summary Per _____ Date _________________________


The pre-image coordinates are input into a function (either a translation, reflection or rotation) and the resulting image has coordinates that are the output points.

Translation:

Notice how each of the x coordinates changed by adding 6. Each of the y coordinates changed by subtracting 2. Given the function rule for a translation, you can determine the new ordered pairs without needing to graph either the pre-image or the image.

What happens to the coordinates when a figure is reflected over the x-axis?

Reflect the pre-image over the x-axis and compare the inputs and outputs. Can you determine a generalization or function rule to describe this change?

inputs outputs

A A’

B B’

C C’

D D’

Function Rule:

T (x, y) = ___________________

input coordinates:

A (- 4, 5)

B (- 3, 2)

C (- 6, 5)

D (- 5, 2)

translation rule or function:

T (x, y) = (x + 6, y – 2)

output coordinates:

A (2, 3)

B (3, 0)

C (0, 3)

D (1, 0)

→ →

L2 – Translations, Reflections, and Rotations Name ________________________________ 2.15– Transformations Summary Practice Set Per _____ Date _________________________


Explore what happens to the coordinates when a figure is reflected over the y-axis.

1. Reflect the polygon ABCD over the y-axis and compare the inputs and outputs. Input Output A A’ B B’ C C’ D D’

2. What generalization describes a reflection over the y-axis?

T (x, y) = ____________________

3. If the point (27, - 82) was reflected over the x-axis, what would be the image coordinates?

4. If point F (0, 56) was reflected over the y-axis, what would be the coordinates for F’?

5. If a point in Quadrant I is rotated 90˚ CW about the origin, what Quadrant will its new point be in?

6. If point X (- 2, 3) was rotated 90˚ clockwise about the origin, where would X’ be?

a) (- 2, - 3) b) (- 2 , 3) c) (2, - 3) d) (2, 3) e) none of these

7. What type of generalization could you conclude about points and figures rotated 90˚ CW about the origin?

L2 – Translations, Reflections, and Rotations Name ________________________________ 2.15– Transformations Summary Practice Set Per _____ Date _________________________


input T (5, - 12) R (0, 13) E (- 18, 24) H (- 4, - 36)

output T’ R’ E’ H’

8. Translate the input points using the function: T (x, y) = (x + 3!!, y – 7)

9. Rotate the figure below 180˚ about point X.

10. Describe how the figure changes when rotating 180°.

11. Where would W’ be if W (29, - 13) was rotated 180° about the origin?

a) W’ (29, 13) b) W’ (29, - 13) c) W’ (- 29, - 13) d) W’ (- 29, 13) e) none of these

l2 translations, reflections, rotations...

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