6.15: Recognizing Rotation Transformations
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 \begin{align*}90^\circ\end{align*}
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 \begin{align*}90^\circ\end{align*}
Next, let’s look at rotating figures on the coordinate plane.
Rotate this figure \begin{align*}90^\circ\end{align*}
First, let’s write down the coordinates for each of the points of this pentagon.
\begin{align*}& A (3, 5)\\
& B (4, 4)\\
& C (3, 3)\\
& D (1, 2)\\
& E (1, 4)\end{align*}
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 \begin{align*}90^\circ\end{align*}
Let’s apply this to the coordinates above.
\begin{align*}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)\end{align*}
Now we can graph this rotated figure. Notice that we use \begin{align*}A^\prime\end{align*}
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 \begin{align*}y\end{align*}
Here is what that would look like.
\begin{align*}& 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)\end{align*}
Now we can graph this new rotation.
We can also graph figures that have been rotated \begin{align*}180^\circ\end{align*}
Let’s see what this looks like.
\begin{align*}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)\end{align*}
Now we can graph this image.
Write the three ways to figure out new coordinates for rotating \begin{align*}90^\circ\end{align*}
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 \begin{align*}72^\circ\end{align*}
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 \begin{align*}90^\circ, 180^\circ\end{align*}
90∘,180∘ 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 \begin{align*}90^\circ\end{align*}
Video Review
Transformation: Rotation CK12
Practice
Directions: Answer the following questions about rotations, translations and tessellations.
 What is a translation?
 What is a rotation?
 What is a tessellation?
 True or false. A figure can be translated up or down only.
 True or false. A figure can be translated \begin{align*}180^\circ\end{align*}
180∘ .  True or false. A figure can be rotated \begin{align*}90^\circ\end{align*}
90∘ clockwise or counterclockwise.  True or false. A figure can’t be translated \begin{align*}180^\circ\end{align*}
180∘ .  When rotating a figure \begin{align*}90^\circ\end{align*}
90∘ counterclockwise, we switch the \begin{align*}x\end{align*}x and \begin{align*}y\end{align*}y coordinates and multiply which one by 1?  When rotating a figure \begin{align*}90^\circ\end{align*}
90∘ clockwise, we multiply which coordinate by 1?  True or false. When rotating a figure \begin{align*}180^\circ\end{align*}
180∘ , 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)
 Rotate the figure \begin{align*}90^\circ\end{align*}
90∘ clockwise  Rotate the figure \begin{align*}90^\circ\end{align*}
90∘ counterclockwise  Rotate the figure \begin{align*}180^\circ\end{align*}
180∘
A Triangle with the coordinates (1, 3) (5, 1) (5, 3)
 Rotate the figure clockwise \begin{align*}90^\circ\end{align*}
90∘  Rotate the figure counterclockwise \begin{align*}90^\circ\end{align*}
90∘  Rotate the figure \begin{align*}180^\circ\end{align*}
180∘
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.Coordinate Notation
A coordinate point is the description of a location on the coordinate plane. Coordinate points are commonly written in the form (x, y) where x is the horizontal distance from the origin, and y is the vertical distance from the origin.Image
The image is the final appearance of a figure after a transformation operation.Origin
The origin is the point of intersection of the and axes on the Cartesian plane. The coordinates of the origin are (0, 0).Preimage
The preimage is the original appearance of a figure in a transformation operation.Reflection
A reflection is a transformation that flips a figure on the coordinate plane across a given line without changing the shape or size of the figure.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.Rotational Symmetry
A figure has rotational symmetry if it can be rotated less than around its center point and look exactly the same as it did before the rotation.Transformation
A transformation moves a figure in some way on the coordinate plane.Translation
A translation is a transformation that slides a figure on the coordinate plane without changing its shape, size, or orientation.Rigid Transformation
A rigid transformation is a transformation that preserves distance and angles, it does not change the size or shape of the figure.Image Attributions
Here you'll recognize rotation transformations and rotational symmetry.