 # Translations Lesson 6-1.

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Translations Lesson 6-1

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.

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

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)

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.

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).

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)

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)

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)

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)

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.

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)

Reflections Lesson 6-2

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:

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:

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.

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).

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)

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)

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)

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)

Rotations Lesson 6-3

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

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

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.

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 180and graph M and L.

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)

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.

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

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)

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)

Dilations Lesson 6-4

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

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.

Dilations in the Coordinate Plane
Symbols: (x, y) → (kx, ky) Model:

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)

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)

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)

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.

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.

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