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8.10: Defining Rotation

  • Page ID
    6027
  • Transformations by which a figure is turned around a fixed point to create an image.

    Rotations

    A transformation is an operation that moves, flips, or otherwise changes a figure to create a new figure. A rigid transformation (also known as an isometry or congruence transformation) is a transformation that does not change the size or shape of a figure.

    The rigid transformations are translations, reflections, and rotations. The new figure created by a transformation is called the image. The original figure is called the preimage. If the preimage is \(A\), then the image would be \(A′\), said “a prime.” If there is an image of \(A′\), that would be labeled \(A′′\), said “a double prime.”

    A rotation is a transformation where a figure is turned around a fixed point to create an image. The lines drawn from the preimage to the center of rotation and from the center of rotation to the image form the angle of rotation.

    f-d_5bd378816240175b57aafb19bd4153808d8ceb9185448d5a41cca5ad+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{1}\)

    While we can rotate any image any amount of degrees, \(90^{\circ}\), \(180^{\circ}\) and \(270^{\circ}\) rotations are common and have rules worth memorizing.

    Rotation of \(180^{\circ}\): \((x,y)\rightarrow (−x,−y)\)

    f-d_b4ef9052919a49e54ea44a8eb70c9e1607220f7859dbcb521864e127+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{2}\)

    Rotation of \(90^{\circ}\): \((x,y)\rightarrow (−y,x)\)

    f-d_03349ba80ca27af7bf6de529ca4dfdb711df230232521052374e7a05+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{3}\)

    Rotation of \(270^{\circ}\): \((x,y)\rightarrow (y,−x)\)

    f-d_628df37afd1b6174fac4ff2f47c50f0bb169c69b8630f6509d8c1eee+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{4}\)

    What if you were given the coordinates of a quadrilateral and you were asked to rotate that quadrilateral \(270^{\circ}\) about the origin? What would its new coordinates be?

    Example \(\PageIndex{1}\)

    A rotation of \(80^{\circ}\) clockwise is the same as what counterclockwise rotation?

    Solution

    There are \(360^{\circ}\) around a point. So, an \(80^{\circ}\) rotation clockwise is the same as a \(360^{\circ}−80^{\circ}=280^{\circ}\) rotation counterclockwise.

    f-d_454fbd0676a577134f216708838b2ad4fbd71ae22d11307c36b209c8+IMAGE_TINY+IMAGE_TINY.pngFigure \(\PageIndex{6}\)

    Example \(\PageIndex{2}\)

    A rotation of \(160^{\circ} counterclockwise is the same as what clockwise rotation?

    Solution

    \(360^{\circ}−160^{\circ}=200^{\circ}\) clockwise rotation.

    f-d_141eb82986cf4fdd2ec570705378edc76ecdf6a274ec0c0a54f9e366+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{7}\)

    Example \(\PageIndex{3}\)

    Rotate \(\overline{ST} 90^{\circ}\).

    f-d_c9a0dead78ab17b5167785f09d241c12dfe66c9d20afe577f027d1ce+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{8}\)

    Solution

    f-d_4649371723391b5c19a7056c9a92e621b32a11e898fb055dea4bef8d+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{9}\)

    Example \(\PageIndex{4}\)

    The rotation of a quadrilateral is shown below. What is the measure of \(x\) and \(y\)?

    f-d_986c5537b9fbb3570f61219d5dcfb529d2a6cb0550f23996f8ba0c88+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{10}\)

    Solution

    Because a rotation produces congruent figures, we can set up two equations to solve for \(x\) and \(y\).

    \(y=4\begin{array}{rr}
    2 y=80^{\circ} & 2 x-3=15 \\
    y=40^{\circ} & 2 x=18 \\
    & x=9
    \end{array}\)

    Example \(\PageIndex{5}\)

    Rotate \(\Delta ABC\), with vertices \(A(7,4)\), \(B(6,1)\), and \(C(3,1)\), \(180^{\circ}\) about the origin. Find the coordinates of \(\Delta A′B′C′\).

    f-d_2106b0082b257143fbc4a52ce940af65c8fb7396b520f6824b28c411+IMAGE_TINY+IMAGE_TINY.png
    Figure \(\PageIndex{11}\)

    Solution

    Use the rule above to find \(\Delta A′B′C′\).

    \(\begin{aligned}A(7,4)&\rightarrow A′(−7,−4) \\ B(6,1)&\rightarrow B′(−6,−1) \\ C(3,1)&\rightarrow C′(−3,−1)\end{aligned}\)

    Review

    In the questions below, every rotation is counterclockwise, unless otherwise stated.

    1. If you rotated the letter p 180^{\circ}\) counterclockwise, what letter would you have?
    2. If you rotated the letter p \(180^{\circ}\) c\(ockwise, what letter would you have?
    3. A \(90^{\circ}\) clockwise rotation is the same as what counterclockwise rotation?
    4. A \(270^{\circ}\) clockwise rotation is the same as what counterclockwise rotation?
    5. A \(210^{\circ}\) counterclockwise rotation is the same as what clockwise rotation?
    6. A \(120^{\circ}\) counterclockwise rotation is the same as what clockwise rotation?
    7. A \(340^{\circ}\) counterclockwise rotation is the same as what clockwise rotation?
    8. Rotating a figure \(360^{\circ}\) is the same as what other rotation?
    9. Does it matter if you rotate a figure \(180^{\circ}\) clockwise or counterclockwise? Why or why not?
    10. When drawing a rotated figure and using your protractor, would it be easier to rotate the figure \(300^{\circ}\) counterclockwise or \(60^{\circ}\) clockwise? Explain your reasoning.

    Rotate each figure in the coordinate plane the given angle measure. The center of rotation is the origin.

    1. \(180^{\circ}\)
      f-d_63703c4cb8ab951d54bf6940341c64c94d7e9589ad8b78cbc95c20b5+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{12}\)
    2. \(90^{\circ}\)
      f-d_f6956c0fc08d46105307a1cc543971faa872e2797e56765129be4a71+IMAGE_TINY+IMAGE_TINY.pngFigure \(\PageIndex{13}\)
    3. \(180^{\circ}\)
      f-d_f29843cc3b55e68610128e9eb1804052fd1e81560e203b2e519e4a54+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{14}\)
    4. \(270^{\circ}\)
      f-d_f0d2144f3f77f6bda778881d2e5f468b41890d498b901a16ee0bc6d6+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{15}\)
    5. \(90^{\circ}\)
      f-d_080ebb6ebfa7b98e8e4ed8cf758c8425cc81879f867b367dd32266c2+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{16}\)
    6. \(270^{\circ}\)
      f-d_918ca54d3fc5878587c7d6db07b27008400f4f7ea870e713adf5d742+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{17}\)
    7. \(180^{\circ}\)
      f-d_4a22195243555095a7dd9c998ddbe585ce3fae7ebe9c3f11c88cf301+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{18}\)
    8. \(270^{\circ} \)
      f-d_14a46e8ce6734a72415efe7a341bd337eee72b73dfd34f56862cd65f+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{19}\)
    9. \(90^{\circ} \)
      f-d_1aed1901ce53397ed28e3a3981e737e39d67d0346eef9def43a6bfe8+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{20}\)

    Find the measure of \(x\) in the rotations below. The blue figure is the preimage.

    1. f-d_8b488aea742a64899e3c16443d593b53e46c2258bcdb0d492d658ef2+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{21}\)
    2. f-d_90f54e345893ce27f2d74fa8868cd3f51bcebe26882b84a8cde18739+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{22}\)
    3. f-d_e1f8af28940ed30c5db261b6e93ed6aa7eed6ceccc1a3698c3698a2f+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{23}\)

    Find the angle of rotation for the graphs below. The center of rotation is the origin and the blue figure is the preimage. Your answer will be \(90^{\circ}\), \(270^{\circ}\), or \(180^{\circ}\).

    1. f-d_ebade3d8d191a2ea67f773738adbb08920c5c0f8b1eabdf2278cb508+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{24}\)
    2. f-d_868c7738dc0987a25a2abfd694236ce25435bec832e0e26b1d8f67ee+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{25}\)
    3. f-d_dd16236a956752560b48b506f38423e5fb9c3d43a0b2061bfa032f1f+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{26}\)
    4. f-d_f642fac3bce3972d9816a75e9c538b1b86ed5d89438076777d597a04+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{27}\)
    5. f-d_c2f8a248048e8f8c96c73e82c6f370e0f965961ee4dd1172724713e9+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{28}\)
    6. f-d_1d5326df1b312816970baef3dab5814ed1681d18a35f89df62ba4687+IMAGE_TINY+IMAGE_TINY.png
      Figure \(\PageIndex{29}\)

    Review (Answers)

    To see the Review answers, open this PDF file and look for section 12.4.

    Vocabulary

    Term Definition
    Center of Rotation In a rotation, the center of rotation is the point that does not move. The rest of the plane rotates around this fixed point.
    Image The image is the final appearance of a figure after a transformation operation.
    Origin The origin is the point of intersection of the x and y axes on the Cartesian plane. The coordinates of the origin are (0, 0).
    Preimage The pre-image is the original appearance of a figure in a transformation operation.
    Rotation A rotation is a transformation that turns a figure on the coordinate plane a certain number of degrees about a given point without changing the shape or size of the figure.
    Rigid Transformation A rigid transformation is a transformation that preserves distance and angles, it does not change the size or shape of the figure.

    Additional Resources

    Interactive Element

    Video: Transformation: Rotation Principles - Basic

    Activities: Rotations Discussion Questions

    Study Aids: Types of Transformations Study Guide

    Practice: Defining Rotation

    Real World: Radical Rotations