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8.11: Rotation Rules

  • Page ID
    6036
  • State rules that describe given rotations.

    Rules for Rotations

    The figure below shows a pattern of two fish. Write the mapping rule for the rotation of Image A to Image B.

    f-d_a7f1ba8ada39c311e4f9e9aa6b3ac389318a62cb1199fe12d1256c1a+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{1}\)

    In geometry, a transformation is an operation that moves, flips, or changes a shape to create a new shape. A rotation is an example of a transformation where a figure is rotated about a specific point (called the center of rotation), a certain number of degrees. Common rotations about the origin are shown below:

    Center of Rotation Angle of Rotation Preimage (Point P) Rotated Image (Point P′) Notation (Point P′)
    (0, 0) \(90^{\circ}\)(or \(−270^{\circ}\)) \((x,y)\) \((−y,x)\) \((x,y)\rightarrow (−y,x)\)
    (0, 0) \(180^{\circ}\)(or \(−180^{\circ}\)) \((x,y)\) \((−x,−y)\) \((x,y)\rightarrow (−x,−y)\)
    (0, 0) \(270^{\circ}\)(or \(−90^{\circ}\)) \((x,y)\) \((y,−x)\) \((x,y)\rightarrow (y,−x)\)

    You can describe rotations in words, or with notation. Consider the image below:

    f-d_d0310294816764c3d3ce57e3229259104c40b549807794247b6e2555+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{2}\)

    Notice that the preimage is rotated about the origin \(90^{\circ}\)CCW. If you were to describe the rotated image using notation, you would write the following:

    \(R_{90^{\circ}}(x,y)=(−y,x)\)

    Let's write the notation to describe the following CCW rotations on the point (3, 2)\) and draw the image:

    1. about the origin at \(90^{\circ}\)

    Rotation about the origin at \(90^{\circ}: \(R_{90^{\circ}}(x,y)=(−y,x)\)

    1. about the origin at \(180^{\circ}\)

    Rotation about the origin at \(180^{\circ}\): \(R_{180^{\circ}}(x,y)=(−x, −y)\)

    1. about the origin at \(270^{\circ}\)

    Rotation about the origin at 270^{\circ}: \(R_270^{\circ}(x,y)=(y,−x)\)

    f-d_a99b12393f981e994fe34976196e95b85d3b88640c3730b622ed7b81+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{3}\)

    Now let's perform the following rotations on Image \(A\) shown below in the diagram below and describe the rotations:

    f-d_cb0d58549e1c250aa4acbe5b27026fc2b3e647ddbd12c9aeb2f54748+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{4}\)
    1. about the origin at \(90^{\circ}\), and label it \(B\).

    Rotation about the origin at \(90^{\circ}\): \(R_{90^{\circ}}A\rightarrow B=R_{90^{\circ}}(x,y)\rightarrow (−y,x)\)

    1. about the origin at \(180^{\circ}\), and label it \(O\).

    Rotation about the origin at \(180^{\circ}\): \(R_{180^{\circ}}A\rightarrow O=R_{180^{\circ}}(x,y)\rightarrow (−x,−y)\)

    1. about the origin at \(270^{\circ}\), and label it \(Z\).

    Rotation about the origin at \(270^{\circ}\): \(R_{270^{\circ}}A\rightarrow Z=R270^{\circ}(x,y)\rightarrow (y,−x)\)

    f-d_5d12dea134422fe8a3b5e9955842b7d42c1af19fcee2d5eadd25129d+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{5}\)

    Finally, let's write the notation that represents the rotation of the preimage A to the rotated image J in the diagram below:

    f-d_772a1703020ba6bb4fac6d9b409e9217f638fcf709cfdbbd818153c5+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{6}\)

    First, pick a point in the diagram to use to see how it is rotated.

    \(E:(−1,2)\qquad E′:(1,−2)\)

    Notice how both the \(x\)- and \(y\)-coordinates are multiplied by -1. This indicates that the preimage \(A\) is reflected about the origin by \(180^{\circ}\)CCW to form the rotated image J. Therefore the notation is \(R_{180^{\circ}}A\rightarrow J=R_{180^{\circ}}(x,y)\rightarrow (−x,−y)\).

    Example \(\PageIndex{1}\)

    Earlier, you were given the figure below which shows a pattern of two fish. Write the mapping rule for the rotation of Image \(A\) to Image \(B\).

    f-d_a7f1ba8ada39c311e4f9e9aa6b3ac389318a62cb1199fe12d1256c1a+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{7}\)

    Solution

    Notice that the angle measure is \(90^{\circ}\) and the direction is clockwise. Therefore the Image \(A\) has been rotated \(−90^{\circ}\) to form Image \(B\). To write a rule for this rotation you would write: \(R_{270^{\circ}}(x,y)=(−y,x)\).

    Example \(\PageIndex{}\)

    Thomas describes a rotation as point \(J\) moving from \(J(−2,6)\) to \(J′(6,2)\). Write the notation to describe this rotation for Thomas.

    \(J:(−2,6)\qquad J′:(6,2)\)

    f-d_97cfacabfebd35f5a14793fff720af2192f0e66e14dd4e5a46a08e44+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{8}\)

    Solution

    Since the \(x\)-coordinate is multiplied by -1, the \(y\)-coordinate remains the same, and finally the \(x\)- and \(y\)-coordinates change places, this is a rotation about the origin by \(270^{\circ}\) or \(−90^{\circ}\). The notation is: \(R_{270^{\circ}}J\rightarrow J′=R_270^{\circ}(x,y)\rightarrow (y,−x)\)

    Example \(\PageIndex{1}\)

    Write the notation that represents the rotation of the yellow diamond to the rotated green diamond in the diagram below.

    f-d_627dc8bf2d5ae0253c263ce22afce36881590bf48246799f13fa0e32+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{9}\)

    Solution

    In order to write the notation to describe the rotation, choose one point on the preimage (the yellow diamond) and then the rotated point on the green diamond to see how the point has moved. Notice that point E\) is shown in the diagram:

    \(E(−1,3)\rightarrow E′(−3,−1)\)

    Since both \(x\)- and \(y\)-coordinates are reversed places and the \(y\)-coordinate has been multiplied by -1, the rotation is about the origin \(90^{\circ}\). The notation for this rotation would be: \(R_{90^{\circ}}(x,y)\rightarrow (−y,x)\).

    f-d_5da2d322d2bf0fed06fe19ca12677fdc501ea3b840f316d1d604a800+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{10}\)

    Example \(\PageIndex{1}\)

    Karen was playing around with a drawing program on her computer. She created the following diagrams and then wanted to determine the transformations. Write the notation rule that represents the transformation of the purple and blue diagram to the orange and blue diagram.

    f-d_c5df849237dd510be82b51e4cf2d0808b03d21df169ee465a51e9be0+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{11}\)

    Solution

    In order to write the notation to describe the transformation, choose one point on the preimage (purple and blue diagram) and then the transformed point on the orange and blue diagram to see how the point has moved. Notice that point \(C\) is shown in the diagram:

    \(C(7,0)\rightarrow C′(0,−7)\)

    Since the \(x\)-coordinates only are multiplied by -1, and then \(x\)- and \(y\)-coordinates change places, the transformation is a rotation is about the origin by \(270^{\circ}\). The notation for this rotation would be: \(R_{270^{\circ}}(x,y)\rightarrow (y,−x)\).

    Review

    Complete the following table:

    Starting Point \(90^{\circ}\) Rotation \(180^{\circ}\) Rotation \(270^{\circ}\) Rotation \(360^{\circ}\) Rotation
    1. \((1, 4)\)
    2. \((4, 2)\)
    3. \((2, 0)\)
    4. \((-1, 2)\)
    5. \((-2, -3)\)

    Write the notation that represents the rotation of the preimage to the image for each diagram below.

    1. f-d_00934570a85fda7834f2986f4eb1e28de93b3eb21528c949e258b140+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{12}\)
    2. f-d_e913dd600b5449bc96cf268de3bb42552af45c7ea3ff96252ce0ce8d+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{13}\)
    3. f-d_daafb2f609c8e2ee9d89d13fd115a45b50db93f782c006d223e9a246+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{14}\)
    4. f-d_6046d7f5943c0c285e8c07980d6f56cac70dc023668d8dfdb12b7e75+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{15}\)
    5. f-d_2821667e67059844c6d0f1be87cafe0d2488a00be28f5c2f818bbd38+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{16}\)

    Write the notation that represents the rotation of the preimage to the image for each diagram below.

    1. f-d_03119d6f29f8ece25533a72ad7b4831736b5cfe16f6d72cb18dc878e+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{17}\)
    2. f-d_6a823428501dc0b30e258ab75936ddfc20ccaf6884d42e3dec9438be+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{18}\)
    3. f-d_1d524692dda0dd4ae9175b8d92067d600e7bb802fb87c14810e1fcd7+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{19}\)
    4. f-d_4211c7e1f7961c0c07c7570fd7ac58dd0163c85aab6c4bfff8862099+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{20}\)
    5. f-d_76b8d5a378ff86504e4829d078c9b428a68cc72650ae8eaa4e2ddf34+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{21}\)

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 10.9.

    Additional Resources

    Video: Rules for Rotations

    Practice: Rotation Rules

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