Isometries-100

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Isometries Rotations Translations Compositions
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Isometries-100
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What is the definition of an isometry? Give three
examples of isometries.
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Isometries-100Answer
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• Isometry a transformation that perseveres
  length, angle measure, parallel lines, etc.
• ex. Reflections
• Rotations
• Translations

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Isometries-200
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• Which of the following is not a rotation of
• ?

c)
a)
b)
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Isometries-200Answer
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• c)

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Isometries-300
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• True or false?
• 1) Transformations that are not isometries are
  called rigid transformations.
• 2) Flips, turns and slides are nicknames for
  reflections, rotations and translations
• 3) Isometries preserve angle measures and
  parallel lines

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Isometries-300Answer
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1. False
2. True
3. True

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Isometries-400
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• Find the value of each variable if the given
  transformation is an isometry

50
4d
7
a
50
b
12
2c3
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Isometries-400Answer
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• a90 c2
• 2c37
• 2c4
• c2
• b40 d3
• 180-90-50 4d12
• 90-50 d3
• 40

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Isometries-500
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• Is the given transformation an isometry?
• ABC XYZ
• A(-4,2) X(2,2)
• B(-1,4) Y(4,-1)
• C(-1,1) Z(1,-1)

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Isometries-500Answer
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• Yes
• Use the distance formula to compare the side
  lengths
• ABv(-41)²(2-4)² BCv(-11)²(4-1)²
  ACv(-41)²(2-1)²
• v(-3)²(-2)² v(0)²(3)²
  v(-3)²(1)²
• v94 v9 v91
• v13 3 v10
• XYv(2-4)²(21)² YZv(4-1)²(-11)²
  XZv(2-1)²(21)²
• v(-2)²(3) ² v(3)²(0)²
  v(1)²(3)²
• v49 v9 v19
• v13 3 v10
• ABXY BCYZ
  ACXZ

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Rotations-100
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Does this figure have rotational symmetry? If so,
describe the rotation that maps the figure onto
itself.
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Rotations-100Answer
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• Yes, the star does have rotational symmetry. To
  map the figure onto itself, you could rotate the
  object 72 or 144.

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Rotations-200
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• A(2,-3) Al(-3,-2)
• If A was rotated clockwise around the origin,
  what was the angle of rotation?

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Rotations-200Answer
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• 90
• In a 90 clockwise rotation, (x,y) (y,-x)
• If you use that information, you can substitute
  in (2,-3) to get (-2,-3), which are the
  coordinates of the given pre-image and image

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Rotations-300
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m
K
What is the measure of the angle of rotation?
.Al
.A
.All

138
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Rotations-300Answer
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• 84
• When you reflect a figure over line k then over
  line m, the angle of rotation is 2x (xthe
  measure of the acute angle formed by k and m)
• So, x180-138
• x42
• 2(42)84

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Rotations-400
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• Rotate (7,-2) 90clockwise around the origin.
  Name the point of the image. Do the same for 180
  and 270 clockwise.

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Rotations-400Answer
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• 90(-2, -7) because (x,y) (y,-x)
• 180(-7,2) because (x,y) (-x,-y)
• 270(2,7) because (x,y) (-y,x)

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Rotations-500
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5c
Find the values of all the variables
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4b
65
a
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2d2
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Rotations-500Answer
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• a130 c1
• a2(65) 5c5
• a130 c1
• b2 d4
• 4b8 2d210
• b2 2d8
• d4

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Translations-100
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• Reflect AB, A(3,-3) B (2,-4), over y1. What
  are the coordinates of Al and Bl

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Translations-100Answer
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• Al(3,5) Bl(2,-6)

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Translation-200
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• Find the other endpoint using the following
  vectors.
• 1.(-4,0) vector lt2,-3gt
• 2. (5, -2) vector lt5,1gt

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Translation-200Answer
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• (-2,-3)
• (-42,0-3)
• (-2,-3)
• 2. (10,-1)
• (55,-21)
• (10,-1)

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Translation-300
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• Use the following coordinate notation to find the
  other endpoint.
• (x, y) (x2, y-3)
• (1,4)
• 2. (-3, -1)

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Translation-300Answer
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• (3,1)
• (12,4-3)
• (3,1)
• (-1,-4)
• (-32,-1-3)
• (-1,-4)

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Translation-400
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• A translation of AB is described by vector
  PQlt2,-5gt. Find the value of each variable.
• A(w-5,-3)Al(10,x-1)
• B(z,3y1)Bl(5,5)

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Translation-400Answer
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• w10 y3
• w-5210 3y1-55
• w-310 3y-45
• w13 3y9
• x-7 y3
• -3-5x-1 z3
• -8x-1 z25
• -7x z3

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Translation-500
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• Write the equation for the line of reflection
• A(2,3) B(6,-1)

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Translation-500Answer
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• y x-3
• Explanation (2,3) (6,-1)
• Slope (31)-1 midpoint(62 3-1) (4,1)
• (2-6) 2 , 2
• Perp. Line slope1
• y1xb
• 11(4)b
• 14b
• -3b
• yx-3

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Compositions-100
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• What is a composition? What is a glide
  reflection?

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Compositions-100Answer
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• A composition is when 2 or more transformations
  are combined to form a single transformation
• A glide reflection is a transformation in which
  every point P is mapped onto Pll by the following
  2 steps
• -a translation that maps P onto Pl
• -a reflection in line k such that the line of
  translation is parallel to reflection line k

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Compositions-200
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• When you switch the order of transformations,
  does it affect the final image? In what cases?

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Compostitions-200Answer
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• In a composition, it does affect the final image,
  but it does not in a glide reflection.

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Compostitions-300
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• Rotate A(3,2) 90 about the origin and reflect
  over the x-axis.

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Compositions-300Answer
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• Al (2,-3)
• All(2,3)

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Compositions-400
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• Sketch the image of AB, A(4,2) B(7,0), after a
  composition using the given transformations (in
  the given order)
• Translation
• (x,y) (x-4,y2)
• Rotation
• 270 clockwise about the origin

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Compositions-400 Answer
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• Translation
• A(0,4) B(3,2)
• Rotation
• A(-4,0) B(-2,3)

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Compositions-500
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• Sketch the image of A (-5,2) after translating it
  using vector lt3,-4gt and reflecting over x4

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Compositions-500Answer
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• After translation Al(-2,-2)
• After reflection All(10,-2)