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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 i t. Vocabulary Start-Up. - PowerPoint PPT Presentation
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Lesson 6-1TRANSLATIONS
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
VOCABULARY START-UP
Translations
List 3 Characteristics
Define in Your Own Words
Draw a Non-example
Draw an Example
a slide without turning or flipping
Shape stays the sameSize stays the sameFaces the same way
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.
TRANSLATIONS IN THE COORDINATE PLANE
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 ofthe image are J’(-1,-1), K’(3, -2), and L’(-2, -4).
EXAMPLE 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)
GOT IT? 1
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)
EXAMPLE 2
X’(-3, -1)
Y’(4, -2)
Z’(0, -4)
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)
GOT IT? 2
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 = 2 3 + b = 1a = -1 b = -2
So, the translation is (x – 1, y – 2), 1 unit to the left and 2 units down.
EXAMPLE 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)
GOT IT? 3
Lesson 6-2REFLECTIONS
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.
REFLECTIONS IN THE COORDINATE PLANE
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 1
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)
EXAMPLE 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)
GOT IT? 1 & 2
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)
EXAMPLE 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)
GOT IT? 3
Lesson 6-3ROTATIONS
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
REAL-WORLD LINK
Rotating to the right
Rotating to the left
3. If one of the sections labeled 4 makes three complete turns counterclockwise, how many degrees will it have traveled?
1,0804. Are there any points on the wheel that stay fixed?
yes; the center5. Does the center of the wheel change position?
no6. Does the distance from the center to the edgechange as it spins?
REAL-WORLD LINK
no
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.
ROTATE A FIGURE ABOUT A POINT
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.
EXAMPLE 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’.
GOT IT? 1
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.
ROTATIONS ABOUT THE ORIGIN
Symbols:
ROTATIONS ABOUT THE ORIGIN
(x, y) (-x, -y) (x, y) (-y, -x)(x, y) (y, -x)
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)
EXAMPLE 2
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)
GOT IT?
Lesson 6-4DILATIONS
VOCABULARY START-UP
enlargementreduction
scale drawing
same size and same shape as original ratio
scale factorgraphing
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:
DILATIONS IN THE COORDINATE PLANE
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)
EXAMPLE 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)
GOT IT? 1
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 . The dilation is (x, y) (x, y).J(3, 8) J’( 3, 8) (1.5, 4)
K(10, 6) K’( 10, 6) (5, 3)
L(8, 2) L’( 8, 2) (4, 4)
EXAMPLE 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.
GOT IT? 2
Though a microscope, the image of a grain of sand with a 0.25-millimeter diameter appears to have a diameter of 11.25 millimeters. What is the scale factor of the dilation?
Write a ratio comparing the diameters of the two images.
= 45 So, the scale factor of the dilation is 45.
EXAMPLE 3
Lucas wants to ensure a 3-by 5-inch photo to a 7 -by 12 -inch photo. What is the scale factor of the dilation?
2.5
GOT IT? 3
