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Translations Lesson 6-1

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Vocabulary Start-Up A transformation is an operation that maps an original geometric figure, the preimage, onto a new figure called the image. A translation slides a figure from one position to another without turning it.

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**Vocabulary Start-Up List 3 Characteristics Shape stays the same**

Define in Your Own Words List 3 Characteristics Shape stays the same Size stays the same Faces the same way a slide without turning or flipping Translations Draw an Example Draw a Non-example

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**Translations in the Coordinate Plane**

Words: When a figure is translated, the x-coordinate of the preimage changes by the value of the horizontal translation a. The y-coordinate of the preimage changes by a vertical translation b. Symbols: (x, y) → (x + a, y + b)

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**Translations in the Coordinate Plane**

When translating a figure, every point of the preimage is moved the same distance and same direction. Congruent figures have the same shape and size. So the preimage and image are congruent.

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Example 1 Graph JKL with vertices J(-3, 4), K(1, 3), and L(-4, 1). Then graph the image of JKL after a translation of 2 units right at 5 units down. Write the coordinates of its vertices. From the graph, the coordinates of the vertices of the image are J’(-1,-1), K’(3, -2), and L’(-2, -4).

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Got it? 1 Graph ABC with vertices A(4, -3), B(0, 2), and C(5,1). Then graph the image of ABC after a translation of 4 units left at 3 units up. Write the coordinates of its vertices. A’(0,0), B’(-4, 5), C’(1, 4)

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Example 2 Triangle XYZ has vertices X(-1,-2), Y(6, -3) and Z(2, -5). Find the vertices triangle X'Y'Z' after a translation of 2 units left and 1 unit up. So, the vertices of XYZ are X’(-3, -1), Y’(4, -2), Z’(0, -4) X’(-3, -1) Y’(4, -2) Z’(0, -4)

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Got it? 2 Quadrilateral ABCD has vertices A(0, 0), B(2, 0) C(3, 4), and D(0, 4). Find the vertices quadrilateral A‘B‘C‘D’ after a translation of 4 units right and 2 units down. A’(4, -2), B’(6, -2) C’(7, 2), D’(4, 2)

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Example 3 A computer image is being translated to create the illusion of movement. Use translation notation to describe the translation from point A to point B. Point A is located at (3, 3). Point B is located at (2, 1). (x, y) → (x + a, y + b) (3, 3) → (3 + a, 3 + b) → (2, 1)

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Example 3 A computer image is being translated to create the illusion of movement. Use translation notation to describe the translation from point A to point B. 3 + a = b = 1 a = -1 b = -2 So, the translation is (x – 1, y – 2), 1 unit to the left and 2 units down.

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Got it? 3 Refer to the figure in Example 3. If point A was at (1, 5), use translation notation to describe the translation from point A to point B. (x + 1, y – 4)

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Reflections Lesson 6-2

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**Reflections in the coordinate plane**

OVER THE X-AXIS: Words: To reflect a figure over the x-axis, multiply the y-coordinate by -1. Symbols: (x, y) → (x, -y) Model:

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**Reflections in the coordinate plane**

OVER THE Y-AXIS: Words: To reflect a figure over the y-axis, multiply the x-coordinate by -1. Symbols: (x, y) → (-x, y) Model:

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**Reflections in the coordinate plane**

A reflection is a mirror image of the original figure. It is the result of a transformation of a figure over a line called the a line of reflection. In a reflection, each point of the preimage and its image are the same distance from the line of reflection. So, in a reflection, the image is congruent to the preimage.

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Example 1 Triangle ABC has vertices A(5, 2), B(1, 3), C(-1, 1). Graph the figure and its reflected image over the x-axis. Then find the coordinates of the vertices of the reflected image. The coordinates are A’(5, -2), B’(1, -3), C’(-1, -1).

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Example 2 Quadrilateral KLMN has vertices K(2, 3), L(5, 1), M(4, -2), and N(1, -1). Graph the figure and its reflection over the y-axis. Then find the coordinates of the vertices of the reflected image. The coordinates are K’(-2, 3), L’(-5, 1), M’(-4, -2), N’(-1, -1)

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Got it? 1 & 2 Triangle PQR has vertices P(1, 4), Q(3, 7), and R(4, -1). Graph the figure and its reflection over the y-axis. Then find the coordinates of the reflected image. P’(-1, 4), Q’(-3, 7), R’(-4, -1)

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Example 3 The figure is reflected over the y-axis. Find the coordinates of point A’ and point B’. Then sketch the figure and its image on the coordinate plane. A(1, 4) → A’(-1, 4) B(2, 1) → B’(-2, 1)

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Got it? 3 The figure is reflected over the x-axis. Find the coordinates of point A’ and point B’. Then sketch the image of the coordinate plane. A’(-2, -2) B’ (2, -2)

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Rotations Lesson 6-3

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Real-World Link 1. A spin can be clockwise or counterclockwise. Define these words in your own words. Clockwise _________________________ Counterclockwise __________________ 2. If the section 8 on the left part of the wheel spins 90 clockwise, where will it land? At the top Rotating to the right Rotating to the left

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Real-World Link 3. If one of the sections labeled 4 makes three complete turns counterclockwise, how many degrees will it have traveled? 1,080 4. Are there any points on the wheel that stay fixed? yes; the center 5. Does the center of the wheel change position? no 6. Does the distance from the center to the edge change as it spins? no

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**Rotate a Figure About a point**

A rotation is a transformation in which a figure is rotated, or turned about a fixed point. The center of rotation is the fixed point. A rotation does not change the size or shape of the figure. So, the pre-image and image are congruent.

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**Example 1 Step 1: Graph the original triangle.**

Triangle LMN with vertices L(5, 4), M(5, 7), and N(8, 9) represents a desk in Jackson’s bedroom. He wants to rotate the desk counterclockwise 180 about vertex L. Graph the figure and its image. Then give the coordinates of the vertices for L’M’N’. Step 1: Graph the original triangle. Step 2: Use a protractor to measure 180and graph M and L.

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Got it? 1 Rectangle ABCD with vertices A(-7, 4), B(-7, 1), C(-2, 1), and D(-2, 4) represents the bed of Jackson’s room. Graph the figure and its image a clockwise rotation of 90 about vertex C. Then gives the coordinates of the vertices for rectangle A’B’C’D’. A’(1, 6), B’(-2, 6), C’(-2, 1), D’(1, 1)

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**rotations about the Origin**

Words: A rotation is a transformation about a fixed point. Each point of the original figure and its image are the same distance from the center of rotation.

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**rotations about the Origin**

Symbols: (x, y) → (y, -x) (x, y) → (-x, -y) (x, y) → (-y, -x)

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Example 2 Triangle DEF has vertices D(-4, 4), E(-1, 2), and F(-3, 1). What are the coordinates after a rotation clockwise 90 about the origin? clockwise 90 rule: (x, y) → (y, -x) D(-4, 4) → (4, 4) E(-1, 2) → (2, 1) F(-3, 1) → (1, 3)

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Got it? Quadrilaterals MNPQ has vertices M(2, 5), N(6, 4), P(6, 1) and Q(2, 1). Graph the figure and its image after a counterclockwise rotation of 270 M’(5, -2), N’(4, -6), P’(1, 6), Q’(1, 2)

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Dilations Lesson 6-4

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**same size and same shape as original**

Vocabulary Start-Up same size and same shape as original enlargement reduction scale drawing ratio scale factor graphing

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**Dilations in the Coordinate Plane**

Words: A dilation with a scale factor of k will be: an enlargement, or an image larger than the original, if k > 1, a reduction, or an image smaller than the original, if 0 < k < 1, the same as the original figure if k = 1. Each coordinate is multiplied by the scale factor.

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**Dilations in the Coordinate Plane**

Symbols: (x, y) → (kx, ky) Model:

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Example 1 A triangle has vertices A(0, 0), B(8, 0), and C(3, -2). Find the coordinates of the triangle after a dilation with a scale factor of 4. The dilation is (x, y) → (4x, 4y). A(0, 0) → A’(4 0, 4 0) → (0, 0) B(8, 0) → B’(4 8, 4 0) → (32, 0) C(3, -2) → C’(4 3, 4 -2) → (12, -8)

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Got it? 1 A figure with vertices W(-2, 4), X(1, 4), Y(-3, -1), and Z(3, -1). Find the coordinates of the figure after a dilation with a scale factor of 2. W’(-4, 8) X’(2, 8) Y’(-6, -2) X(6, -2)

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Example 2 A figure has vertices J(3, 8), K(10, 6), and L(8, 2). Graph the figure and the image of the figure after a dilation with a scale factor of 𝟏 𝟐 . The dilation is (x, y) → ( 1 2 x, 1 2 y). J(3, 8) → J’( 1 2 3, 1 2 8) → (1.5, 4) K(10, 6) → K’( 1 2 10, 1 2 6) → (5, 3) L(8, 2) → L’( 1 2 8, 1 2 2) → (4, 4)

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Got it? 2 A figure with vertices F(-1, 1), G(1, 1), H(2, -1), and I(-1, -1). Graph the figure and the image of the figure after a dilation with a scale factor of 3.

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Example 3 Though a microscope, the image of a grain of sand with a 0.25-millimeter diameter appears to have a diameter of millimeters. What is the scale factor of the dilation? Write a ratio comparing the diameters of the two images. 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑖𝑛 𝑑𝑖𝑙𝑎𝑡𝑖𝑜𝑛 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 𝑖𝑛 𝑜𝑟𝑖𝑔𝑖𝑛𝑎𝑙 = = 45 So, the scale factor of the dilation is 45.

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Got it? 3 Lucas wants to ensure a 3-by 5-inch photo to a 7 𝟏 𝟐 -by 12 𝟏 𝟐 -inch photo. What is the scale factor of the dilation? 2.5

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