In this section, you will use an interactive sketch to investigate rotations. Click on the following link to open the applet in a new tab/window.

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This sketch allows you to investigate the changes in coordinates when a rotation is applied to a geometric figure.

Rotations

Next let’s investigate rotations about the origin.

  1. Use your notes to record the initial coordinates of the green point, yellow point, cyan point, and black point.
  2. In the Rotate box, set x = 0 and y = 0. Setting x = 0 and y = 0 means the center of rotation will be the origin, (0, 0)
  3. Set degrees to 90. This will give us a 90° counterclockwise rotation.
  4. Click the Rotate button.
  5. Record the new coordinates of each point.
  6. How did the value of the coordinates change?
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    When you apply a 90° counterclockwise rotation about the origin to a point, the x-coordinate and the y-coordinate change places and the y-coordinate becomes its opposite.Close Pop Up

    You can write the rule for a rotation of 90° counterclockwise about the origin in symbolic language.

    R90°(x, y) = (−y, x)

  7. How was the size and shape of the figure affected?
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    The size and shape did not change.Close Pop Up

    8. Repeat the process using a 180° rotation. (Does it matter if the 180° rotation is clockwise or counterclockwise about the origin?)

  8. In general, how are the x and y coordinates affected when a figure is rotated 180° about the origin?
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    When you apply a 180° rotation about the origin to a point, the x-coordinate and the y-coordinate each become its opposite.Close Pop Up

    9. You can write the rule for a rotation of 180° about the origin in symbolic language.

    R180° (x, y) = (−x, −y)

  9. How was the size and shape of the figure affected?
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    The size and shape did not change.Close Pop Up

    10. Repeat the process using a 270° counterclockwise rotation.

  10. In general, how are the x and y coordinates affected when a figure is rotated 270° counterclockwise about the origin?
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    When you apply a 270° counterclockwise rotation about the origin to a point, the x-coordinate and the y-coordinate change places and the x-coordinate becomes its opposite. Close Pop Up

    11. We can write the rule for reflections across the x-axis in symbolic language.

    R270°(x, y) = (y, −x)

  11. In general, how are the size and shape of a figure affected a figure is rotated about the origin?
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    The size and shape did not change.Close Pop Up

Since the size and shape of figures are not affected when a figure is rotated, we say rotations are congruence transformations.

The table below summarizes the effects on x- and y- coordinates for translations, reflections, and rotations.

Algebraic Representations of Transformations
Translation of h, k

Th, k(x, y) = (x + h, y + k)
Reflection across the x-axis

When you reflect a point across the x-axis, the x-coordinate remains the same, but the y-coordinate is transformed into its opposite.

P(x, y) → Pt(x, −y) or rx-axis(x, y) = (x, −y)
Reflection across the y-axis

When you reflect a point across the y-axis, the y-coordinate remains the same, but the x-coordinate is transformed into its opposite.

P(x, y) → Pt(−x, y) or ry-axis(x, y) = (−x, y)
Reflection across y = x

When you reflect a point across the line y = x, the x-coordinate and the y-coordinate change places.

P(x, y) → Pt(y, x) or ry = x(x, y) = (y, x)
Reflection across y = −x

When you reflect a point across the line y = −x, the x-coordinate and the y-coordinate change places and are negated (the signs are changed).

P(x, y) → Pt(−y, −x) or ry = −x(x, y) = (−y, −x)
Rotation* of 90°

R90°(x, y) = (−y, x)
Rotation* of 180°

R180°(x, y) = (−x, −y)
Rotation* of 270°

R270°(x, y) = (y, −x)

*Unless otherwise indicated, in mathematics, a rotation is assumed to be counterclockwise.