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3. Rigid Motions
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United States of AmericaFL
Grade 912

3.03 Rotations

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Lesson

Concept summary

A rotation is a transformation of a figure by turning it about a point called the point of rotation. The amount of rotation can be expressed in the number of degress. The direction of the rotation for two-dimensional figures can be described in the clockwise or counterclockwise direction. When it is not specified we assume the direction is counterclockwise.

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A rotation can be denoted in functional notation in the form R_{\theta,P}(\text{Shape being rotated}) where \theta is the angle of rotation in the counterclockwise direction, and P is the point of rotation.

When rotated about the origin, the transformation mappings are as follows:

  • Degree of rotation counterclockwise: 90 \degree \, \, \, \qquad Transformation mapping: \left(x,y \right) \to \left(-y,x\right)
  • Degree of rotation counterclockwise: 180 \degree \qquad Transformation mapping: \left(x,y \right) \to \left(-x,-y\right)
  • Degree of rotation counterclockwise: 270 \degree \qquad Transformation mapping: \left(x,y \right) \to \left(y,-x\right)
  • Degree of rotation counterclockwise: 360 \degree \qquad Transformation mapping: \left(x,y \right) \to \left(x,y\right)

If a rotation maps a figure onto itself, then we say that the figure has rotational symmetry about that point. We can call that point the shape's center of rotation.

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For example, if we rotate the square about the origin by 90\degree counterclockwise, we can check that each point on the square will be mapped to another point on the square.

The result is that the image of the rotation will be the same as the pre-image.

We can see that the square also has rotational symmetry about its center for rotations of 180\degree and 270\degree as well.

Worked examples

Example 1

\overline{AB} has been rotated counterclockwise about the origin.

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a

Describe the rotation required to map \overline{AB} to \overline{A'B'}.

Approach

We can identify the rotation that has taken place by looking at the coordinates of A and B in the preimage and comparing them to the coordinates of A' and B' in the image.

Solution

We can see that A\to A' maps \left(3, 3\right) \to \left(-3,-3\right) and B\to B' maps \left(2, 1\right) \to \left(-2,-1\right). Both are of the form\left(x,y\right)\to\left(-x,-y\right) which corresponds to a 180\degree rotation about the origin.

Reflection

We can see that the entire shape has moved from the first quadrant to the third quadrant which, when rotating about the origin, will always coincide with a 180\degree rotation in either clockwise or counterclockwise direction.

b

Write the transformation mapping.

Approach

We know the functional notation mapping is R_{\theta,P}(\text{Shape being rotated}) where \theta is the angle of rotation in the counterclockwise direction, and P is the point of rotation.

We know that the point of rotation is the origin, O\left(0,0\right).

We also found in part (a) A\to A' and B\to B' followed the mapping \left(x,y\right)\to\left(-x,-y\right) which corresponds to a 180\degree rotation about the origin.

Solution

Function notation: R_{180\degree,O}(\overline{AB})=\overline{A'B'}

Example 2

Consider the figure ABCD:

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a

Determine the image of ABCD when rotated about the point A by 180\degree.

Approach

We can determine the image of each corner rotated 180\degree about A by considering their translation from A and then rotating that translation.

Consider that, when rotating 180\degree, we have that:

  • Right \to Left
  • Down \to Up
  • Left \to Right
  • Up \to Down

Solution

In the pre-image, we can get to the other corners from A by the following translations:

  • A\to B: translate 2 units to the right
  • A\to C: translate 2 units to the right and 2 units up
  • A\to D: translate 2 units up

If we apply a rotation of 180\degreeto these translations, we can get the images of the corners:

  • A\to B': translate 2 units to the left
  • A\to C': translate 2 units to the left and 2 units down
  • A\to D': translate 2 units down

We can plot this as:

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b

Determine the image of ABCD when rotated about the point \left(2,0\right) by 90\degree clockwise.

Approach

A rotation of 90\degree clockwise will have the following effect on translations:

  • Left \to Up
  • Up \to Right
  • Right \to Down
  • Down \to Left

Solution

Using the same approach as for part (a), we can note that:

  • \left(2,0\right)\to A: translate 1 unit left and 1 unit down
  • \left(2,0\right)\to B: translate 1 unit right and 1 unit down
  • \left(2,0\right)\to C: translate 1 unit right and 1 unit up
  • \left(2,0\right)\to D: translate 1 unit left and 1 unit up

If we apply a rotation of 90\degree clockwise to these translations, we can get the images of the corners:

  • \left(2,0\right)\to A': translate 1 unit up and 1 unit left
  • \left(2,0\right)\to B': translate 1 unit down and 1 unit left
  • \left(2,0\right)\to C': translate 1 unit down and 1 unit right
  • \left(2,0\right)\to D': translate 1 unit up and 1 unit right

Using these translations, we can see that

  • A'=D
  • B'=A
  • C'=B
  • D'=C

In other words, the rotation maps ABCD onto itself.

Example 3

Identify the rotation(s) that map ABCD onto itself.

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Approach

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If we look at the point of intersection of the axes of symmetry we can find the point of rotation for rotational symmetry.

Solution

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Rotating about the point \left(1,2\right) by 90 \degree clockwise or 270 \degree counterclockwise, we can see that we do not get rotational symmetry.

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Rotating about the point \left(1,2\right) by 180 \degree clockwise or counterclockwise, we can see that we get rotational symmetry.

Rotating about the point \left(1,2\right), we can see that we only get rotational symmetry when we rotate by 180\degree clockwise about the point \left(1,2\right).

This is the function R_{180\degree,\left(1,2\right)}

Outcomes

MA.912.GR.2.1

Given a preimage and image, describe the transformation and represent the transformation algebraically using coordinates.

MA.912.GR.2.3

Identify a sequence of transformations that will map a given figure onto itself or onto another congruent or similar figure.

MA.912.GR.2.4

Determine symmetries of reflection, symmetries of rotation and symmetries of translation of a geometric figure.

MA.912.GR.2.5

Given a geometric figure and a sequence of transformations, draw the transformed figure on a coordinate plane.

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