12-2 translations holt geometry i can i can - translate figures on the coordinate plane -...
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Example 1: Identifying Translations Tell whether each transformation appears to be a translation. Explain. No; the figure appears to be flipped. Yes; the figure appears to slide. A. B.TRANSCRIPT
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12-2 Translations
Holt Geometry
I CANI CAN
- Translate figures on the coordinate plane- Translate figures on the coordinate plane-Can convert between vector notation andcoordinate notation
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A translation is a transformation where all the points of a figure are moved the same distance in the same direction. A translation is an isometry, so the image of a translated figure is congruent to the preimage.
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Example 1: Identifying Translations
Tell whether each transformation appears to be a translation. Explain.
No; the figure appears to be flipped.
Yes; the figure appears to slide.
A. B.
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Check It Out! Example 1
Tell whether each transformation appears to be a translation.a. b.
No; not all of the points have moved the same distance.
Yes; all of the points have moved the same distance in the samedirection.
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Translations using vector notation
A vector is a set of directions telling a point howto move. - Denoted with angle brackets < x, y > -Tells movement in x direction, then y direction
Example: < -2, 4 > means to move 2 units to the leftand 4 units up.
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Example: Sketch the segment AB with A(1,-3) andB(-2,0). Translate this segment along the vector< -3, 5 >
A
B
x movement is 3 units to the lefty movement is 5 units up
A'
B'
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Translations using arrow notation<-3, 5> in arrow notation would say(x, y) (x – 3, y + 5)
Examples:Turn < 2 , -1> into arrow notation1.(x ,y) (x + 2, y – 1)
2. Turn (x, y) (x + 3, y) into vector notation< 3, 0 >
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Example: Drawing Translations in the Coordinate PlaneTranslate the triangle with vertices D(–3, –1), E(5, –3), and F(–2, –2) along the vector <3, –1>.
The image of (x, y) is (x + 3, y – 1).D(–3, –1) D’(–3 + 3, –1 – 1)
= D’(0, –2)E(5, –3) E’(5 + 3, –3 – 1)
= E’(8, –4)F(–2, –2) F’(–2 + 3, –2 – 1)
= F’(1, –3)Graph the preimage and the image.
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Check It Out! ExampleTranslate the quadrilateral with vertices R(2, 5), S(0, 2), T(1,–1), and U(3, 1) along the vector <–3, –3>. The image of (x, y) is (x – 3, y – 3).R(2, 5) R’(2 – 3, 5 – 3)
= R’(–1, 2)S(0, 2) S’(0 – 3, 2 – 3)
= S’(–3, –1)T(1, –1) T’(1 – 3, –1 – 3)
= T’(–2, –4)U(3, 1) U’(3 – 3, 1 – 3)
= U’(0, –2)Graph the preimage and the image.
R
S
T
UR’
S’
T’
U’
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12-3 Rotations
Holt Geometry
I CANI CAN
Rotate 90º, 180º, and 270º around the originRotate 90º, 180º, and 270º around the originDetermine the angle of rotationDetermine the angle of rotation
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Remember that a rotation is a transformation that turns a figure around a fixed point, called the center of rotation. A rotation is an isometry, so the image of a rotated figure is congruent to the preimage.
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Example 1: Identifying Rotations
Tell whether each transformation appears to be a rotation. Explain.
No; the figure appearsto be flipped.
Yes; the figure appearsto be turned around a point.
A. B.
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Check It Out! Example 1
Tell whether each transformation appears to be a rotation.
No, the figure appears to be a translation.
Yes, the figure appears to be turned around a point.
a. b.
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We will only be dealing with right-angle rotations.(90º, 180º, 270º, and 360º)
Unless otherwise stated, all rotations in this book are counterclockwise.
Helpful Hint
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Easy way to do a rotation on a coordinate plane:
- Actually TURN the paper the number of degrees you require
-IGNORE the old numbering of the axes. Countout to your new coordinates, and write them downsomewhere.
-Return paper to original orientation and plot thosenew points.
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Do the following example on your white board
Rotate triangle ABC with verticesA(2,-1), B(4,1), and C(3,3)by 90º about the origin.
A(2,-1) A'( , )
B(4, 1) B'( , )
C(3, 3) C'( , )
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Do the following example on your white board
Rotate triangle DEF with verticesD(2,3), E(-1,2), and F(2,1)by 180º about the origin.
D(2,3) D'( , )
E(-1,2) E'( , )
F(2, 1) F'( , )
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Do the following example on your white boardGraph the triangle A(2,-4), B(3,5),C(6,1); Then rotate it 270º.
A(2,-4) A'( , )
B(3, 5) B'( , )
C(6, 1) C'( , )
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By 270º, (x, y) (y, –x)
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Example : Drawing Rotations in the Coordinate Plane
Rotate ΔJKL with vertices J(2, 2), K(4, –5), and L(–1, 6) by 180° about the origin.
The rotation of (x, y) is (–x, –y).
Graph the preimage and image.
J(2, 2) J’(–2, –2)
K(4, –5) K’(–4, 5) L(–1, 6) L’(1, –6)
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Check It Out! Example
Rotate ∆ABC by 180° about the origin.
The rotation of (x, y) is (–x, –y).A(2, –1) A’(–2, 1)B(4, 1) B’(–4, –1)C(3, 3) C’(–3, –3)
Graph the preimage and image.