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GEOMETRY CHAPTER 9 WORKBOOK

Reflection

Translation

Rotation

Dilation

SPRING 2017

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Reflection

Reflection in the x-axis

Reflection in the y-axis

Reflection in the line y=x

Translations

Translation in the Coordinate Plane

Rotation

90°

180°

270°

Compositions of Transformations

Glide reflection

Translation

Rotation

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Symmetry

Line Symmetry

Rotational Symmetry

Three – Dimensional

Dilation

Dilations in the Coordinate

Plane

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Geometry Section 9.1 Notes: Reflections Date: Learning Targets:

1. I can draw reflections in the coordinate plane.

Vocab. and Topics Definitions, Examples and Pictures If the images to the right describe a reflection, in your own words, define reflection:

Reflection in a line

Example 1

Draw the reflected image of quadrilateral WXYZ in line p.

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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. b) Graph JKLM and its image over y = –2 .

Reflection in the x-axis Reflection in the y-axis

Reflections on a Coordinate Plane

(𝒙,𝒚) → ( , )

(𝒙,𝒚) → ( , )

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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.

Reflection in the line 𝒚 = 𝒙

(𝒙,𝒚) → ( , )

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.

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You Try!

1. Graph the image of the figure below using the transformation given. a) Reflection across the x-axis b) reflection across y = 3

2. Given the coordiante points of an image, graph the points below. Then use the transformaiton given to plot the new figure. c) reflection across the x-axis d) reflection across y = −2

T(2, 2), C(2, 5), Z(5, 4), F(5, 0) H(−1, −5), M(−1, −4), B(1, −2), C(3, −3)

Summary

Reflection over x-axis Reflection over y-axis Reflection over 𝒚 = 𝒙

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9-1 Reflections Exercises Graph △FGH and its image in the given line.

1. x = –1 2. y = 1 Graph quadrilateral ABCD and its image in the given line.

3. x = 0 4. y = 1 Graph each figure and its image under the given reflection.

5. △DEF with D(–2, –1), E(–1, 3), 6. ABCD with A(1, 4), B(3, 2), C(2, –2),

F(3, –1) in the x-axis D(–3, 1) in the y-axis

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Geometry Section 9.2 Notes: Translations Date: Learning Targets: 1. I can draw translations in the coordinate plane.

If the images to the right describe a translation, in your own words, define translation:

Translation

Translation Vector

Component Form

Translation on a Coordinate Plane

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Example 1

Example 1

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

Example 2

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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Example 3: Given the rule, ( ) ( )5,2, +− yxyx , describe in component form. Then transform the figure given

the vector.

You Try!

Example 4: Use the translation (x, y) → ⟨−𝟓,𝟖⟩.

1. What is the image of ( )4,2B ? 2. What is the image of (21,5)D ?

3. What is the preimage of ’(23, 24)F ? 4. What is the preimage of ’(7,25)H ?

5. What is the image of ( )0,2J ? 6. What is the preimage of ’(24,6)K ?

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9-2 Translation Graph each figure and its image along the given vector.

1. quadrilateral TUVW with vertices T(–3, –8), U(–6, 3), V(0, 3), and W(3, 0); ⟨4, 5⟩

2. △QRS with vertices Q(2, 5), R(7, 1), and S(–1, 2); ⟨–1, –2⟩

3. parallelogram ABCD with vertices A(1, 6), B(4, 5), C(1, –1), and D(–2, 0); ⟨3, –2⟩

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4. Rectangle RECT has vertices R(–2, –1), E(–2, 2), C(3, 2), and T(3, –1). Graph the figure and its image along the vector ⟨2, –1⟩.

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Geometry Section 9.3 Notes: Rotations Date: Learning Targets: 1. I can draw rotations in the coordinate plane.

If the images to the right describe a Rotation, in your own words, define rotation:

Center of Rotation

Angle of Rotation

Direction of

Rotation

Rotations on a Coordinate Plane

Describe in your own words, how

will you help yourself remember

this?

90˚ Rotation 180˚ Rotation 270˚ Rotation

(𝒙,𝒚) → ( , )

(𝒙,𝒚) → ( , ) (𝒙,𝒚) → ( , )

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Example 1 Triangle DEF has vertices D(–2, –1), E(–1, 1), and F(1, –1). Graph ΔDEF and its image after a rotation of 270˚ about the origin?

Example 2 Hexagon DGJTSR is shown below. What is the image of point T after a 90° counterclockwise rotation about the origin?

Example 3 Triangle PQR is shown below. What is the image of point Q after a 180° counterclockwise rotation about the origin?

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Example 4: Find the image that represents the rotation of the polygon about the origin. Then graph the polygon and its image.

a) Rotate

( )( )( )

1, 2

4, 1 90

3, 4

A

B

C

−

− °

−

Find the image that represents the rotation of the polygon about the origin. Then graph the polygon and its image.

b) Rotate

( )( )( )( )

1, 2

2,1 270

3, 1

1, 3

A

B

C

D

− −

°

−

−

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9-3 Rotations Graph each figure and its image after the specified rotation about the origin.

1. trapezoid FGHI has vertices F(7, 7), 2. △LMN has vertices L(–1, –1), G(9, 2), H(3, 2), and I(5, 7); 90° M(0, –4), and N(–6, –2); 90°

3. △ABC has vertices A(–3, 5), B(0, 2), 4. parallelogram PQRS has vertices P(4, 7), and C(–5, 1); 180° Q(6, 6), R(3, –2), and S(1, –1); 270°

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5. Parallelogram WXYZ has vertices W(–2, 4), X(3, 6), Y(5, 2), and Z(0, 0). Graph parallelogram WXYZ and its image after a rotation of 270° about the origin.

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Geometry Section 9.4 Notes: Compositions of Transformations Date: Learning Targets:

1. I can draw glide reflections and other compositions of isometries in the coordinate plane.

If the images to the right describe a composition of

transformations, in your own

words, define composition of

transformations:

Composition of Transformations

Glide Reflection

Example 1

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.

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Composition of Isometries

(Transformations)

Example 2

Example 3

Example 4

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. If PQRS is translated along < 3,−2 >, and reflected in 𝑦 = −1about the origin, what are the coordinates of P’’Q’’R’’S’’? If P(3,2), Q(4,1), R(-1,2) and S(3,4). Quadrilateral ABCD with A(1, 5), B(6, 2), C(-1, 3), D(-4, -2) is reflected in the line y = x and then rotated 90°. Find the coordinates of the image.

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Reflections in Parallel Lines

Reflections in Intersecting Lines

Example 6

A triangle is reflected in two parallel lines. The composition of the reflection produces a translation 22 centimeters to the right. How far apart are the parallel lines?

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9-4 Glide Reflections

Triangle XYZ has vertices X(6, 5), Y(7, –4) and Z(5, –5). Graph △XYZ and its image after the indicated glide reflection.

1. Translation: along ⟨1, 2⟩ Reflection: in y–axis 2. Translation: along ⟨2, 0⟩ Reflection: in x = y 3. Translation: along ⟨–3, 4⟩ Reflection: in x–axis

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4. Translation: along ⟨–1, 3⟩ Reflection: in x–axis 5.Triangle ABC has vertices A(3, 3), B(4, –2) and C(–1, –3). Graph △ABC and its image after a translation along⟨–2, –1⟩ and a reflection in the x–axis.

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Geometry Section 9.5 Notes: Symmetry Date: Learning Targets: 1. I can identify line and rotational symmetries in two‐dimensional figures.

2. I can identify plane and axis symmetries in three‐dimensional figures.

If the images to the right describe

symmetry, in your own words, define

symmetry:

Symmetry

Line Symmetry

Line of 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)

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Rotational Symmetry

Center of Symmetry

Order of Symmetry

Magnitude of Symmetry

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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You Try! 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.

Example 3 Determine whether the entire word has line symmetry and whether it has rotational symmetry. Identify all lines of symmetry and angles of rotation that map the entire word onto itself.

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Three Dimensional Symmetries

Plane Symmetry Axis Symmetry

Example 4 State whether the figure has plane symmetry, axis symmetry, both, or neither. a) b)

You Try! State whether the figure has plane symmetry, axis symmetry, both, or neither. 1. 2. 3. 4.

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5. 6.

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9-5 Symmetry State whether the figure appears to have line symmetry. Write yes or no. If so, draw all lines of symmetry and state their number.

1. 2. 3. 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. State whether the figure has plane symmetry, axis symmetry, both, or neither.

1. 2.

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3. 4. 5. 6.

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Geometry Section 9.6 Notes: Dilations Date: Learning Targets: 1. I can draw dilations in the coordinate plane.

If the images to the right describe a

translation, in your own words, define

translation:

Dilations on a

Coordinate Plane

Scale Factor

Example 1

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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Example 2

Find the image of each polygon with the given vertices after a dilation centered at the origin with the given scale factor. a. J(2, 4), K(4, 4), P(3, 2); r = 2 b. D(–2, 0), G(0, 2), F(2, –2); r = 1.5

Example 3

Example 4

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? Find the scale factor from the pre-image to the image for the following dilation. A(2,5), B(3,-1), C(4,2) and A’(3, 7.5), B’(4.5, -1.5), C‘(6, 3).

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Summary!

Nemo is located on the coordinate plane. Write down Nemo’s coordinate points here: Marlin believes Nemo will be 3 times the size he is now. Dilate the Nemo using a scale factor of 3. Write the coordinate points here: Graph the dilated coordinate points and name them accordingly. When you are finished, compare with your neighbor.

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9-6 Dilations Graph the image of each polygon with the given vertices after a dilation centered at the origin with the given scale factor. 1. E(–2, –2), F(–2, 4), G(2, 4), H(2, –2); 2. A(0, 0), B(3, 3), C(6, 3), D(6, –3), E(3, –3); r = 1

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3. A(–2, –2), B(–1, 2), C(2, 1); r = 2 r = 0.5

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4. A(2, 2), B(3, 4), C(5, 2); r = 2.5 5. A(–2, –2), B(1, –1), and C(2, 0). r = 2.