In geometry, transformation refers to the movement of objects in the coordinate plane. This lesson will define and give examples of each of the four common transformations and end with a quiz to make sure you are moving in the right direction.
Definitions of Transformations
Geometric transformations involve taking a preimage and transforming it in some way to produce an image. There are two different categories of transformations:
- The rigid transformation, which does not change the shape or size of the preimage.
- The non-rigid transformation, which will change the size but not the shape of the preimage.
Types of Transformations
Within the rigid and non-rigid categories, there are four main types of transformations that we’ll learn today. Three of them fall in the rigid transformation category, and one is a non-rigid transformation.
- Rotation: rotating an object about a fixed point without changing its size or shape
- Dilation: expanding or contracting an object without changing its shape or orientation
- Reflection: flipping an object across a line without changing its size or shape
Why is dilation the only non-rigid transformation? Remember that in a non-rigid transformation, the shape will change its size, but it won’t change its shape.
1. Which figure represents the translation of the yellow figure?
The preimage has been rotated around the origin, so the transformation shown is a rotation.
How to Perform Transformations
Most transformations are performed on the coordinate plane, which makes things easier to count and draw. The best way to perform a transformation on an object is to perform the required operations on the vertices of the preimage and then connect the dots to obtain the figure.A translation is performed by moving the preimage the requested number of spaces.
Move the above figure to the right five spaces and down three spaces. If you take each vertex of the rectangle and move the requested number of spaces, then draw the new rectangle. This will be your translated image:
The mathematical way to write a translation is the following: (x, y) → (x + 5, y – 3), because you have moved five positive spaces in the x direction and three negative spaces in the y direction.Rotation of an object involves moving that object about a fixed point. To rotate a preimage, you can use the following rules.
To rotate an object 90° the rule is (x, y) → (-y, x). You can use this rule to rotate a preimage by taking the points of each vertex, translating them according to the rule and drawing the image. For example, if the points that mark the ends of the preimage are (1, 1) and (3, 3), when you rotate the image using the 90° rule, the end points of the image will be (-1, 1) and (-3, 3).
The rules for the other common degree rotations are:
- For 180°, the rule is (x, y) → (-x, –y)
- For 270°, the rule is (x, y) → (y, –x)
Returning to our example, if the preimage were rotated 180°, the end points would be (-1, -1) and (-3, -3). If it were rotated 270°, the end points would be (1, -1) and (3, -3).Here is what all those rotations would look like on a graph:
Reflection of a geometric figure is creating the mirror image of that figure across the line of reflection. To draw a reflection, just draw each point of the preimage on the opposite side of the line of reflection, making sure to draw them the same distance away from the line as the preimage.
Here’s an example:
In this example, the preimage is a rectangle, and the line of reflection is the y-axis. To draw the image, simply plot the rectangle’s points on the opposite side of the line of reflection.
Point (-5, 4) reflects to (5, 4)Point (-5, 2) reflects to (5, 2)Point (-2, 4) reflects to (2, 4)Point (-2, 2) reflects to (2, 2)Then, connect the vertices to get your image.
Images can also be reflected across the y-axis and across other lines in the coordinate plane. Every reflection follows the same method for drawing.The dilation of a geometric figure will either expand or contract the figure based on a predetermined scale factor. To perform a dilation, just multiply each side of the preimage by the scale factor to get the side lengths of the image, then graph.In this example, the scale factor is 1.
5 (since 2 * 1.5 = 3), so each side of the triangle is increased by 1.5. The angle measures stay the same.
Mathematical transformations involve changing an image in some prescribed manner.
There are four main types of transformations: translation, rotation, reflection and dilation. These transformations fall into two categories: rigid transformations that do not change the shape or size of the preimage and non-rigid transformations that change the size but not the shape of the preimage.
After you’ve completed this lesson, you should have the ability to:
- Define mathematical transformations and identify the two categories
- Describe the four types of transformations
- Explain how to create each of the four types of transformations