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Rotations

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Recognizing Rotation Transformations
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Have you ever seen something rotate? Do you know what it takes to classify a rotation transformation? How about rotational symmetry? Take a look at this dilemma.

Does this figure have rotational symmetry?

To answer this question, you will need to know about rotations and rotational symmetry. This Concept will teach you all you need to know to answer this question successfully.

Guidance

A transformation is the movement of a geometric figure on the coordinate plane.

A rotation is a turn. A figure can be turned clockwise or counterclockwise on the coordinate plane. In both transformations the size and shape of the figure stays exactly the same.

Let's look at rotations or turns.

A rotation is a transformation that turns the figure in either a clockwise or counterclockwise direction.

We can turn a figure 90^\circ , a quarter turn, either clockwise or counterclockwise. When we spin the figure exactly halfway, we have rotated it 180^\circ . Turning it all the way around rotates the figure 360^\circ .

Now if you look at these two triangles, you can see that one has been turned a quarter turn clockwise. If we talk about that turn or rotation in mathematical language, we can describe the turn as 90^\circ clockwise. We could also turn it 180^\circ , which would show a triangle completed upside down.

Next, let’s look at rotating figures on the coordinate plane.

Rotate this figure 90^\circ clockwise on the coordinate plane.

First, let’s write down the coordinates for each of the points of this pentagon.

& A (-3, 5)\\& B (-4, 4)\\& C (-3, 3)\\& D (-1, 2)\\& E (-1, 4)

Now we have the points. The easiest way to think about rotating any figure is to think about it moving around a fixed point. In the case of graphing figures on the coordinate plane, we will be rotating the figures around the center point or origin.

If we rotate a figure clockwise 90^\circ , then we are going to be shifting the whole figure along the x- axis. To figure out the coordinates of the new rotated figure, we switch the coordinates and then, we need to multiply the second coordinate by -1. This will make perfect sense given that the entire figure is going to shift.

Let’s apply this to the coordinates above.

A (-3, 5) &= A^\prime (5, -3) = (5, 3)\\B (-4, 4) &= B^\prime (4, -4) = (4, 4)\\C (-3, 3) &= C^\prime (3, -3) = (3, 3)\\D (-1, 2) &= D^\prime (2, -1) = (2, 1)\\E (-1, 4) &= E^\prime (4, -1) = (4, 1)

Now we can graph this rotated figure. Notice that we use A^\prime to represent the rotated figure. Here is the graph of this rotation.

That is a great question. Let’s think about what would happen to the figure if we were to rotate it counterclockwise. To do this, the figure would move across the y- axis in fact, the x- coordinates would change completely. In actuality, we would switch the original coordinates around. The x- coordinate would become the y- coordinate and the y- coordinate would become the x- coordinate. Then, we need to multiply the new x- coordinate by -1.

Here is what that would look like.

& A (-3, 5) \rightarrow A^\prime (-5, -3)\\& B (-4, 4) \rightarrow B^\prime (-4, -4)\\& C (-3, 3) \rightarrow C^\prime (-3, 3)\\& D (-1, 2) \rightarrow D^\prime (-2, -1)\\& E (-1, 4) \rightarrow E^\prime (-4, -1)

Now we can graph this new rotation.

We can also graph figures that have been rotated 180^\circ too. To do this, we multiply both coordinates of the original figure by -1.

Let’s see what this looks like.

A (-3, 5) &= A^\prime (3, -5)\\B(-4, 4) &= B^\prime (4, -4)\\C (-3, 3) &= C^\prime (3, -3)\\D (-1, 2) &= D^\prime (1, -2)\\E (-1, 4) &= E^\prime (1, -4)

Now we can graph this image.

Write the three ways to figure out new coordinates for rotating 90^\circ clockwise and counterclockwise and for rotating a figure 180^\circ . Put these notes in your notebook.

Now let’s think about symmetry and rotations.

We can call this rotational symmetry .

A figure has rotational symmetry if, when we rotate it, the figure appears to stay the same. The outlines do not change even as the figure turns.

Look at the figure below.

Look at this image. The star will look the same even if we rotate it. We could turn it 72^\circ or 144^\circ clockwise or counterclockwise. It won’t matter. The star will still appear the same.

Do the following figure have rotational symmetry?

Example A

A square.

Solution: Yes, because you can turn it and it will appear exactly the same.

Example B

The letter U.

Solution: No, it will not appear the same if it is turned.

Example C

An octagon.

Solution: Yes, because you can turn it and it will appear exactly the same.

Now let's go back to the dilemma from the beginning of the Concept.

Does this figure have rotational symmetry?

While the outline of this image has rotational symmetry, the design inside prevents it from having rotational symmetry. If we turn the circle, then the design inside will change. Therefore this image does not have rotational symmetry.

Vocabulary

Transformation
moving a geometric figure on the coordinate plane.
Coordinate Notation
using ordered pairs to represent the vertices of a figure that has been graphed on the coordinate plane.
Reflection
A flip of a figure on the coordinate plane.
Translation
A slide – when a figure moves up, down, left or right on the coordinate plane, but does not change position.
Rotation
A turn – when a figure is turned 90^\circ, 180^\circ on the coordinate plane.
Rotational Symmetry
when a figure can be rotated but appears exactly the same no matter how you rotate it.

Guided Practice

Here is one for you to try on your own.

Does a hexagon have rotational symmetry?

Solution

It has rotational symmetry. You can see that because we can rotate it 90^\circ and 180^\circ and it will still look exactly the same. We could rotate it less than 90^\circ too and it still has rotational symmetry. We can also look at the angles to determine rotational symmetry. Each time we turn the figure, it has two parallel sides on the top and bottom and four other sides at the same angles. It has rotational symmetry.

Video Review

Transformation: Rotation CK-12

Practice

Directions: Answer the following questions about rotations, translations and tessellations.

  1. What is a translation?
  2. What is a rotation?
  3. What is a tessellation?
  4. True or false. A figure can be translated up or down only.
  5. True or false. A figure can be translated 180^\circ .
  6. True or false. A figure can be rotated 90^\circ clockwise or counterclockwise.
  7. True or false. A figure can’t be translated 180^\circ .
  8. When rotating a figure 90^\circ counterclockwise, we switch the x and y coordinates and multiply which one by -1?
  9. When rotating a figure 90^\circ clockwise, we multiply which coordinate by -1?
  10. True or false. When rotating a figure 180^\circ , we multiply both coordinates by -1.

Directions: Write the new coordinates for each rotation given the directions.

A Triangle with the coordinates (-4, 4) (-4, 2) and (-1, 1)

  1. Rotate the figure 90^\circ clockwise
  2. Rotate the figure 90^\circ counterclockwise
  3. Rotate the figure 180^\circ

A Triangle with the coordinates (1, 3) (5, 1) (5, 3)

  1. Rotate the figure clockwise 90^\circ
  2. Rotate the figure counterclockwise 90^\circ
  3. Rotate the figure 180^\circ

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