About Rotation in Math
- What is Rotation?
- What is the Rotation of a Figure about the Origin?
- What is the Angle of Rotation?
- Identifying Rotation
- What is Clockwise and Counterclockwise Rotation?
- What are Clockwise and Counterclockwise Rotations for the Coordinates (x, y) for a Movement of 90, 180 and 270 Degrees?
- Solved Examples
- Frequently Asked Questions
What is Rotation?
Definition: Rotation is the circular motion of an object around its center. One of the most common examples of rotation is Earth rotating about its axis. Earth rotates on its own axis and creates 24 hours.
Now in mathematical terms, rotation can be defined as the motion of an object around an axis. It is a type of transformation that includes every point of a figure and rotates it by a specific number of degrees around a given point in a particular direction.
What is the Rotation of a Figure about the Origin?
A rotation or turn in geometry happens by transforming every point of a figure through a given angle and direction about a fixed point. This fixed point is called the center of rotation. Rotation of a figure about the origin is a transformation that rotates the figure about the origin point on the coordinate plane.
What is the Angle of Rotation?
It is the number of degrees by which a figure is rotated about the origin.
Identifying Rotation
In the figure below we can see that a scalene triangle is rotated 270 degrees in the clockwise direction about the point O, that is, origin. Here the rotation is shown in steps.
What is Clockwise and Counterclockwise Rotation?
In geometry, we take the help of a coordinate grid to show rotation. An important thing to note is that a figure can be rotated clockwise or counterclockwise. The figure and its rotations are of the same shape and size, but they may be turned in different directions. Clockwise rotations are as per the clock’s hands, turning to the right, whereas counterclockwise rotations are opposite to that and turn to the left.
What are Clockwise and Counterclockwise Rotations for the Coordinates (x, y) for a Movement of 90, 180 and 270 Degrees?
Let us assume that the coordinates (x, y) have their “center of rotation” located at the origin (0, 0). Now when we rotate point A (x, y) 90° clockwise about the origin, the point becomes point A’ (-y, x). As you can see, the y value of point A becomes the x-value of point A’ and the x value of point A becomes the y-value of point A’ but with the opposite sign.
Now, we will assume point A (x, y) to be rotated 180° counterclockwise and clockwise about the origin and that will create A’ (-x, -y), where the x and y values are the same as point A but with opposite signs.
Here is the summary of how rotations will be for 90, 180, and 270 degrees if we assume the origin, (0, 0) as the center of rotation:
90° clockwise rotation | (x, y) becomes (y, -x) |
|---|---|
90° counterclockwise rotation | (x, y) becomes (-y, x) |
180° clockwise rotation | (x, y) becomes (-x, -y) |
180° counterclockwise rotation | (x, y) becomes (-x, -y) |
270° clockwise rotation | (x, y) becomes (-y, x) |
270° counterclockwise rotation | (x, y) becomes (y, -x) |
The following graph shows the rotations for 90, 180 and 270 degrees in the counter clockwise direction.
Solved Rotation Examples
Example 1: Find an image of point B (4, 3) that was rotated in the clockwise direction for:
a) about the origin by 90°
b) about the origin by 180°.
Solution:
a) As we have learned, 90° clockwise rotation about the origin will result in the coordinates (x, y) to become (y, -x). Similarly, for B (4, 3), 90° clockwise rotation about the origin will give B’ (3, -4).
b) For clockwise rotation about the origin by 180°, the coordinates (x, y) become (-x, -y). Thus, for point B (4, 3), 180° clockwise rotation about the origin will give B’ (-4, -3). The images are represented in the following graph.
Example 2: In the following image, turn the shape by 180° in the clockwise direction.
Solution: We know that a clockwise rotation is towards the right. So, for this figure, we will turn it 180° clockwise. It will look like this:
Example 3: In the following graph, a point K (-3, -4) has been plotted. Show the plotting of this point when it’s rotated about the origin at 180°.
Solution: As we know, 180° clockwise and counterclockwise rotation for coordinates (x, y) results in the same, (-x, -y). So, for the point K (-3, -4), a 180° rotation will result in K’ (3, 4). We can show it graphically in the following graph.
Example 4: The following figure shows a triangle on a coordinate grid. Rotate the triangle ABC about the origin by 90° in the clockwise direction.
Solution: As you can see, triangle ABC has coordinates of A(-4, 7), B(-6, 1), and C(-2, 1). Now, we know that 90° clockwise rotation will make the coordinates (x, y) be (y, -x). So, when we rotate triangle ABC, we will get A’B’C’ and the coordinates will be:
A (-4, 7) becomes A’ (7, 4)
B (-6, 1) becomes B’ (1, 6)
C (-2, 1) becomes C’ (1, 2)
The triangle A’B’C’ will look like:
Let us assume the coordinates of a point to be A (x, y) which will be rotated 270° clockwise and counterclockwise about the origin. Then the clockwise rotation will give the coordinates as A’ (-y, x) and the counterclockwise rotation will give the coordinates as A’ (y, -x).
180° clockwise rotation and 180° counterclockwise rotation of a point will give the same coordinates. So, we need not provide the directions when we are rotating a figure by 180°.