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GEOMETRY UNIT 3

WORKBOOK

CHAPTER 9 Transformations & Symmetry

SPRING 2017

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Geometry Section 9.1 Notes: Reflections Transformation: Isometry: Ex. Circle the isometries (if any). 1) 2) 3) Draw the reflected image of each figure below. 1) 2)

*Pre-image: *Image:

Example 1: Suppose that you must bounce the cue ball off side A before it rolls into the pocket at B. Locate the point C along side A that the ball must hit to ensure that it will roll directly toward the pocket.

Reflection

• Reflects a point so the line of reflection is the ___________________________ ________________________ of the segment connecting the two points.

• If a point is on the line of reflection, then . • It uses the line like a mirror to reflect a point/figure. A reflection is a(n) _____________________________.

l l

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Example 2: Quadrilateral JKLM has vertices J(2, 3), K(3, 2), L(2, –1), and M(0, 1). a) Graph JKLM and its image over x = 1. a) b) b) Graph JKLM and its image over y = –2 . Example 3: Quadrilateral ABCD has vertices A(1, 1), B(3, 2), C(4, –1), and D(2, –3). a) Graph the image reflected in the x – axis. b) Graph the image reflected in the y – axis. Name the new ordered pairs. Name the new ordered pairs. Example 4: Quadrilateral ABCD has vertices A(1, 1), B(3, 2), C(4, –1), and D(2, –3). Graph the image under reflection of the line y = x. Name the new ordered pairs.

Reflection in x-axis

Reflection in y-axis

Reflection in line y = x

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Geometry Name _____________________________________________ Section 9.1 Practice Worksheet For numbers 1 and 2, use the figure and given line of reflection. Then draw the reflected image in this line using a ruler.

1. 2.

For numbers 3 – 6, graph each figure and its image under the given reflection. Then, name the new ordered pairs.

3. quadrilateral ABCD with vertices 4. △FGH with vertices F(–3, –1), G(0, 4) A(–3, 3), B(1, 4), C(4, 0), and and H(3, –1) in the line x = -2 D(–3, –3) in the line y = x

5. rectangle QRST with vertices Q(–3, 2), 6. trapezoid HIJK with vertices H(–2, 5), R(–1, 4), S(2, 1), and T(0, –1) I(2, 5), J(–4, –1), and K(–4, 3) in the x-axis in the y-axis

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7. Vincent is making a star. Complete the star by drawing the reflected image of the figure in line m. 8. Maria placed a paper cutout on a table and then left to get some glue. Her friend James flipped over the cutout without telling Maria. Still, when Maria came back it was impossible for her to be able to see that anything had been changed. Draw the line through the figure that represents the line over which James must have flipped the figure. 9. Tylia drew a shape on a piece of paper and set up a mirror up next to it on her desk. The shape and its image in the mirror are shown below. Draw a line to show where Tylia’s mirror must have been set up. 10. Wilfred hired an interior designer to layout the furniture in his bedroom. The designer produced the plan shown in the figure. Unfortunately, Wilfred’s window is located on the opposite wall from the plan. Wilfred decided to just reflect the plan over the vertical line through the center of the room. Draw the reflected plan. 11. Casey drew this triangle on the coordinate plane: a) What are the vertices of the image of this triangle if it is reflected over the y – axis? b) What are the vertices of the image of this triangle if it is reflected over the line y = x?

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Geometry Section 9.2 Notes: Translations Translations: The following are examples of translations. What do you think a translation is? 1) 2) 3) Properties of Translations: Function Notation

K (-8, 5) ____ (___, ___)

A (-10, 2) ____ (___, ___) T (-6, -1) ____ (___, ___)

So to generalize, (x, y) (______, ______) Example 1: Describe the translation used to map B(-3, 2) onto B’(4, -5).

Example 2: What is the image of D (4, -3) using the translation (x, y) (x – 8, y + 4)? Example 3: What is the pre-image of D’ (4, -3) using the translation (x, y) (x – 8, y + 4)? Component Form/Vector Notation

S’

P’ A’

Z’

S

P A

Z

K’

A’

T’

K

T

A

P

I

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Example 4: a) Graph ∆TUV with vertices T(–1, –4), U(6, 2), and V(5, –5) along the vector 3,2− . Then, name the new

points. b) Graph pentagon PENTA with vertices P(1, 0), E(2, 2), N(4, 1), T(4, –1), and A(2, –2) along the vector 5, 1− − . Then,

name the new points. Example 5: The graph shows repeated translations that result in the animation of the raindrop. a) Describe the translation of the raindrop from position 2 to position 3 in function notation and in words. b) Describe the translation of the raindrop from position 3 to position 4 using a translation vector.

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Geometry Name _____________________________________ Section 9.2 Practice Worksheet For numbers 1 and 2, use the figure and the given translation vector to draw the translation of the figure along the given translation vector. 1. 2. For numbers 3 and 4, graph each figure given the following translations. Then, name the new points. 3. quadrilateral TUWX with vertices 4. pentagon DEFGH with vertices D(–1, –2), T(–1, 1), U(4, 2), W(1, 5), and X(–1, 3); E(2, –1), F(5, –2), G(4, –4), and H(1, –4); Use translation vector ⟨–2, –4⟩ Use (x, y) (x – 1, y + 5)

5. Describe the translation used to map B(-1, 4) onto B’(4, -5) using function notation. 6. Describe the translation used to map A(0, -2) onto A’(-2, -2) using a translation vector. 7. What is the pre-image of C’ (-3, 7) using the translation (x, y) (x – 3, y + 4)? 8. Wynette wants to see the translation of the five-sided figure shown below. Draw the translation so that the indicated vertex is translated to the location of the dot.

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9. A wallpaper design consists of repeated translations of a single isosceles triangle. The pattern is shown overlaid on a coordinate plane. The space above the triangle around the coordinate (5, 1) should be filled with a missing triangle. What are the coordinates of the vertices of the triangle that fill this space consistently with the rest of the pattern? 10. Gus reflects an object twice. The first step is to reflect it over the line y = –1 . Then Gus completes the composite reflection by reflecting it over the line y = 1. The net effect is a translation of the object. Describe this translation. 11. Lacy performs the translation (x, y) (x + 5, y + 3) to an object in the coordinate plane. Kyle performs the translation (x, y) (x – 4, y + 2) to the same object after lacy. What single translation could have been done to achieve the same effect as Lacy and Kyle’s combined translations? Would the result have been different if Kyle did his translation first? 12. Use the coordinate plane to the right to answer parts a and b. a) The image of square S under a translation is square S’. Describe the translation. b) Draw the image of the circle C under the same translation that you described in part a.

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Geometry Section 9.3 Notes: Rotations To rotate a figure you must know: In a coordinate plane, sketch the quadrilateral whose vertices are A(2, -2), B(4, 1), C(5, 1), and D(5, -1).

ABCD Image Coordinates 900 CCW 180o CCW 270o CCW

A (2, -2) A’ ( , ) A’’ ( , ) A’’’( , )

B (4, 1) B’ ( , ) B’’ ( , ) B’’’( , )

C (5, 1) C’ ( , ) C’’ ( , ) C’’’( , )

D (5, -1) D’ ( , ) D’’ ( , ) D’’’( , )

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90o CCW 180o CCW 270o CCW

Example 1: Hexagon DGJTSR is shown below. What is the image of point T after a 270° counterclockwise rotation about the origin? Multiple Choice: a) (5, –3) b) (–5, –3) c) (–3, 5) d) (3, –5) Example 2: Triangle PQR is shown below. What is the image of point Q after a 90° counterclockwise rotation about the origin?

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Geometry Name _____________________________________ Section 9.3 Practice Worksheet For numbers 1 and 2, use the figure and the point of rotation to draw the rotation of the figure. 1. 110° 2. 280° For numbers 3 – 6, graph each figure and its image as a rotation about the origin. Then, name the new ordered pairs. 3. ∆PQR with P(1, 3), Q(3, –2), and R(4, 2); 90° 4. ∆ABC with A(–4, 4), B(–2, –1), and C(2, –4); 270° 5. quadrilateral WXYZ with W(1, 3), X (3, 1), 6. trapezoid FGHI with F(8, 7), G(5, 8), Y(–5, 6), and Z(–6, 5); 180° H(–3, –7), and I(–7, –2); 90° 7. A damaged compass points northwest. If you travel west by the compass, what is your angle of rotation to true north? 8. Nicki is making a flyer that contains a large capital “M”. She decides that she needs to rotate the “M” clockwise by 60°. Draw the rotated image.

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9. Mariah is looking at her sink. The mark on the faucet for the cold water is shown below. What angle does she need to rotate the faucet so that its mark has the correct orientation? 10. A circular dial with the digits 0 through 9 evenly spaced around its edge can be rotated clockwise 36°. How many times would you have to perform this rotation in order to bring the dial back to its original orientation? 11. A tessellation is when a single shape is repeated to tile a plane with no gaps or overlaps. Below is an example of ∆RST rotated around the point R. This forms a tessellation. Determine if the following shapes can form a tessellation by rotation around the given point. a) b) c)

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Geometry Section 9.4 Notes: Compositions of Transformations For 1-3, perform the stated transformations on the pre-image, ∆ABC in different colors. Give the coordinates of the image, ∆A’B’C’. A=(____,____) B=(____,____) C=(____,____)

1) Rotate ∆ABC 90o clockwise about the origin

A’=(____,____) B’=(____,____) C’=(____,____)

2) Reflect ∆A’B’C’ in x = 3

A’’=(____,____) B’’=(____,____) C’’=(____,____)

3) Translate ∆A’’B’’C’’ (x, y) (x – 3, y – 2)

A’’’=(____,____) B’’’=(____,____)C’’’=(____,____) COMPOSITION: THEOREM: The composition of 2 or more isometries is an ____________________. Example 1: ΔTUV has vertices T(2, –1), U(5, –2), and V(3, –4). Graph ΔTUV and its image after a translation along ⟨–1 , 5⟩ and a rotation 180° about the origin. SPECIAL COMPOSITIONS 1) GLIDE REFLECTION: a composition made up of ___________________________ & _________________________ Example 2: Quadrilateral BGTS has vertices B(–3, 4), G(–1, 3), T(–1 , 1), and S(–4, 2). Graph BGTS and its image after a translation along ⟨5, 0⟩ and a reflection in the x-axis.

Transformations Warm Up

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2) REFLECTION IN PARALLEL LINES: The composition of two reflections in parallel lines can be described by a __________________________________________ that is ____________________ the distance between the two lines. Example 3: If the distance between the two lines to the right is 2 cm, how could you describe the translation? 3) REFLECTION IN INTERSECTING LINES: The composition of two reflections in intersecting lines can be described by a __________________________________________ through an angle that is _______________________ the measure of the angle formed by the lines. Example 4:

a) If m TRY = 42o, then ''m ARA = b) If ''m ARA = 56o, then m TRY =

Y

T

R

A''C''

B''

B'A'

C'

B

AC

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Geometry Name _____________________________________ Section 9.4 Practice Worksheet For numbers 1 – 4, ∆ABC has vertices A(1, 3), B(–2, –1), and C(3, –2). Complete the given composition of transformations by graphing ∆ABC, ∆A’B’C’, and ∆A’’B’’C’’. Then, name the new ordered pairs of ∆A’’B’’C’’. 1. Translation: along ⟨2, 0⟩ 2. Rotate: 180o CCW Reflection: in y – axis Reflection: in y = x 3. Rotate: 270o CCW 4. Reflection: in y = 1 Translation: along ⟨–1, 2⟩ Rotate: 90o CCW For numbers 5 – 8, describe a single transformation that maps the pre-image to the image. Make sure to state the exact angle of rotation or length of the translation. 5. 6. 7. 8. 20o

3 cm

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9. To make a bridge structurally sound, its base must be a reflection along the center line of the river. Given the base on one side of the river, draw the other base. 10. Describe the transformations that are combined to create the carpet pattern. 11. Surish says a reflection in the x-axis and then a reflection in the y-axis is the same as rotating by 180° about the origin. Thomas

says a reflection in the x-axis and then a reflection in the y-axis is the same as reflecting in the line y = x. Is either of them correct? Explain your reasoning.

12. When a rotation and a reflection are performed as a composition of transformations on an image, does the order of the

transformations affect the location of the final image sometimes, always or never? Explain. 13. Describe the transformations that are combined to create each border. a) b) c)

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Geometry Section 9.5 Notes: Symmetry Symmetry: A figure has symmetry if there exists a rigid motion – reflection, translation, rotation, or glide reflection – that maps the figure onto ___________________. Line Symmetry: Example 1: State whether the object appears to have line symmetry. Write yes or no. If so, draw all lines of symmetry, and state their number. a) b) c) Rotational Symmetry: *Order: the number of times a figure ________________ itself as it rotates from 0o to 360o * Magnitude (angle of rotation): the _____________________ _____________________ through which a figure can be rotated so that it maps onto _______________________ Example 2: State whether the figure has rotational symmetry. Write yes or no. If so, locate the center of symmetry, and state the order and magnitude of symmetry. a) b) c)

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Three-Dimensional Symmetry

1) Plane Symmetry: A three-dimensional figure has plane symmetry if the figure can be _________________________ onto itself by a _____________________ in a plane.

2) Axis Symmetry: A three-dimensional figure has axis symmetry if the figure can be ______________________ onto itself by a _________________________ between 0o and 360o in a line.

Example 3: State whether the figure has plane symmetry, axis symmetry, both, or neither. a) b)

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Geometry Name _____________________________________ Section 9.5 Practice Worksheet For numbers 1 – 3, state whether the figure has line symmetry. Write yes or no. If so, draw all lines of symmetry and state their number. 1. 2. 3. For numbers 4 – 6, state whether the figure has rotational symmetry. Write yes or no. If so, locate the center of symmetry and state the order and magnitude of symmetry.

4. 5. 6. For numbers 7 and 8, state whether the figure has plane symmetry, axis symmetry, both, or neither.

7. 8. 9. A paddle wheel on a steamboat is driven by a steam engine that rotates the paddles attached to the wheel to propel the boat through the water. If a paddle wheel consists of 18 evenly spaced paddles, identify the order and magnitude of its rotational symmetry. 10. Examine each capital letter in the alphabet. Determine which letters have 180° rotational symmetry about a point in the center of the letter. 11. A regular polygon has rotational symmetry with an order of 5 and a magnitude of 72°. What is the figure?

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12. Steve found the hubcaps shown below at his local junkyard. Does each hubcap appear to have line symmetry? 13. Martha made the figure shown. How many lines of symmetry does the figure have? 14. Kelly is designing how she wants to put the place settings for her party. She wants the tables to be set up symmetrically with the

design that is on the table. First she places plates shown. a) What is the order and magnitude of the rotational symmetry of the figure? b) Make the least possible number of the additions of plates to the figure so that the table has rotational symmetry of order 4 around its

center. c) Is it possible to rearrange the locations of the 4 plates in the original figure so that (1) their distance from the center of the table does

not change,(2) their centers remain somewhere on the rectangle design, and (3) the resulting figure has a rotational symmetry of order 4? If so, draw the figure. If not, explain why.

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Geometry Section 9.6 Notes: Dilations

Warm-Up: In each of the following, the original picture is labeled by . Label which is the image and which is the pre-image.

Example A Example B

______________ ______________

______________ ______________

Find the scale factor from to . (Advice: before you proceed, be sure the units are the same!!!)

What if the scale factor was 1? Example 1: The shaded triangle is the image and the unshaded triangle is the pre-image.

a) Is the dilation an enlargement or reduction?

b) What is the scale factor of the dilation? Example 2: Determine whether the dilation from figure W to W’ is an enlargement or reduction. Then find the scale factor of the dilation. a) b)

4” 1” 11” 0.5”

EXA EXB

DILATIONS!! The preimage & image are ____.

___________________ ___________________

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Example 3: To create the illusion of a “life-sized” image, puppeteers sometimes use a light source to show an enlarged image of a puppet projected on a screen or wall. Suppose that the distance between a light source L and the puppet is 24 inches (LP). To what distance PP' should you place the puppet from the screen to create a 49.5-inch tall shadow (I'M') from a 9-inch puppet? Example 4: Your fabulous geo teacher draws a perfect free-hand circle on her tablet computer and then projects it through the overhead projector. The projector then displays an enlargement of the perfect circle on the board. The circle drawn on her tablet has a radius of 3 inches. The circle on the board has a diameter of 4 feet. What is the scale factor of the enlargement?

Dilations in a Coordinate Plane Centered at the Origin

Example 5: Trapezoid EFGH has vertices E(–8, 4), F(–4, 8), G(8, 4) and H(–4, –8). Graph the image of EFGH after a

dilation centered at the origin with a scale factor of 14

.

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Geometry Name _____________________________________ Section 9.6 Practice Worksheet For numbers 1 and 2, use a ruler to draw the image of the figure after a dilation with center C and scale factor r as indicated.

1. r = 2 2. r = 14

For numbers 3 and 4, determine whether the dilation of K to K’ is an enlargement or reduction. Then find the scale factor of the dilation and solve for x. 3. 4. For numbers 5 and 6, find the image of each polygon with the given vertices after a dilation centered at the origin with the given scale factor. Name the new ordered pairs. 5. J(2, 4), K(4, 4), P(3, 2); r = 2 6. D(–3, 0), G(0, 6), F(3, –3); r = 1/3 7. Margo has a square picture of her sister that fits in her wallet. Each side of the square is 3 inches long. She decides to take this picture and use a dilation to hang it up on her wall at home so each side of the square is 2 feet long. What scale factor did Margo use?

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8. Determine if the following dilation is a reduction or an enlargement from A to B. Then determine the scale factor from A to B.

9. The pre-image is B and the image is A. Determine whether the dilation from Figure B to Figure A is a reduction or an enlargement. Then, determine the scale factor of the dilation from B to A. 10. Leila drew a polygon with coordinates (–1, 2), (1, 2), (1, –2), and (–1, –2). She then dilated the image and obtained another polygon with coordinates (–6, 12), (6, 12), (6, –12), and (–6, –12). What was the scale factor and center of this dilation? 11. Fred drew the footprint of a stage he was planning to build for his band on a coordinate plane. He decided he wanted to make it smaller because he wanted to make sure it fit at every venue. a) Graph the image of Fred’s state after a dilation centered at (0, 0), with scale factor 0.5. b) The perimeter of the image is 26 unites. What is the perimeter of the original figure?