Geometry is a fascinating branch of mathematics that involves studying shapes, sizes, and their properties in space. One of the fundamental concepts in geometry is rotating figures. However, many students tend to find rotation in geometry challenging.
So, what is a rotation in geometry? In geometry, a rotation is one of the different types of transformation, taking each point in a figure and rotating it a certain number of degrees around a given point. In other words, a rotation refers to the transformation of a figure by turning it around a fixed point called the center of rotation. A figure can be rotated clockwise or counterclockwise by any degree of measure. For instance, imagine a square drawn on a sheet of paper.
By rotating it 90 degrees counterclockwise around its center, the square will take the position of a diamond shape, and vice versa. Read on to find out everything you need to know about rotations in geometry, including definitions, properties, and examples.
If you are interested in learning about reflection, I wrote a whole article on how to do a reflection in geometry with easy steps by steps.
What Is a Rotation in Geometry?
A rotation is a transformation in which a figure is turned about a fixed point, known as the center of rotation, by a certain angle in a given direction. A rotation is a key idea in many geometric concepts, making it a fundamental topic in geometry.
The first important aspect of a rotation is that it preserves the size, shape, and orientation of the figure being rotated. It means that the original figure and its image after the rotation have the same size, shape, and angle measures.
However, a rotation results in a new position for the object. A rotation is achieved by moving each point of the figure in a circle around the center of rotation by the given angle measure. The direction of the rotation depends on the orientation of the angle. The positive angle rotates the figure anticlockwise, while the negative angle rotates the figure clockwise.
Moreover, a rotation is defined by two elements: the center of rotation and the angle of rotation. The center of rotation is a point that remains stationary during the transformation, while the angle of rotation determines the degree of measure of the rotation. It can be positive or negative, depending on the direction of the rotation. The angle of rotation is usually measured in degrees, and it can be any real number between 0 and 360 degrees.
Properties Of Rotations
Rotations have specific properties that govern how the transformation of a figure takes place. For example, a rotation preserves the size, shape, and angles of a figure. It also preserves the distance between the different points of a figure.
Therefore, two figures that are congruent mean that one figure can be obtained from the other through rotation. Also, rotations are commutative, meaning that the order of the rotation does not matter.
Typically, there are two main properties of every rotation:
- The center: Rotations maintain distance; therefore, the center of rotation must be equidistant.
- And the angle: After you find the center of rotation, you have various ways to determine the angle of the rotation.
- You can measure the angle of the rotation with a protractor.
- Try to estimate the angle of the rotation with benchmark angles.
- You can also calculate the angle of the rotation using the law of cosines in trigonometry.
- Another major property is that any rotation can be expressed as a composition of two reflections.
- Another property is that adjacent angles of a figure before and after the rotation are congruent. It means that if a line is drawn in the original figure, and the same line is drawn in the transformed figure, then the angle between the line and the adjacent sides is equal to the original angle.
The table below contains rotation rules.
| Type of Rotation | Before Rotation | After Rotation |
| Rotation of 90-degree Clockwise | (y, x) | (y, -x) |
| Rotation of 90-degree Counter Clockwise | (x, y) | (-y, x) |
| Rotation of 180-degree Clockwise And Counter Clockwise | (x, y) | (-x, -y) |
| Rotation of 270-degree Clockwise | (x, y) | (-y, x) |
| Rotation of 270-degree Counter Clockwise | (x, y) | (y, -x) |
| Rotation of 360-degree | (x, y) | (x, y) |
The Center of Rotation
The center of rotation is a crucial part of understanding rotations in geometry. It is the fixed point around which a figure rotates. The center of rotation determines the direction and angle of rotation. A figure that rotates clockwise around a center is the same as a figure that rotates counterclockwise around the same point but by the same degree of measurement in the opposite direction.
Rotation Notation
To indicate a rotation in geometry, one places an “r” and the degree of measurement in a counterclockwise direction after it. For instance, R90 means a 90-degree counterclockwise rotation around the center of rotation.
When the rotation is in a clockwise direction, a negative sign is added before the degree of rotation. For example, R-90 indicates a 90-degree clockwise rotation.
Applications Of Rotation in Real Life
Rotations have practical applications in the real world. For example, in sports like gymnastics, ice skating, and diving, a performer rotates their body around a fixed point, which creates a visually stunning performance.
Also, in engineering and architecture, rotating a blueprint of a building can help understand how it will look from different angles. In addition, in astronomy, the rotation of a planet causes it to have a day-and-night cycle.
Furthermore, rotations are used in navigation, where ships and airplanes use the concept of rotation to change their direction or orientation.
Also rotation also finds application in computer graphics, where rotations are used to transform computer-generated images to create animations. Similarly, the concept of rotations is used in video games, physics, and engineering.
If you want to learn more about rotation in geometry, I encourage you to check Khan Academy or watch this video or the video below.
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Wrapping Up
Understanding rotations in geometry is a fundamental concept in understanding shapes and their properties in space. A rotation is a transformation of a figure around a fixed center, which preserves its size, shape, and angles.
The center of rotation determines the direction and angle of rotation, and the notation R is used to represent rotations. The properties of rotations make them commutative, meaning that the order of rotation does not matter.
As we have seen, rotations have practical applications in sports, engineering, and astronomy, among other fields.