The term "transformation" often implies a complete change, but in mathematics, there exists a concept called similarity transformation that challenges this notion. How can something be transformed and yet still maintain its original shape? In this article, we will delve into the idea of similarity transformation through dilations, and briefly discuss rotations, reflections, and translations.
To understand similarity transformations, it is crucial to first grasp the concepts of similarity and transformation. In geometry, a transformation refers to altering the position or form of a geometric figure. On the other hand, similarity is a property of geometric figures where they share the same shape but differ in size.
We say that two figures are similar when one can be obtained from the other by maintaining the ratio of their corresponding sides. For example, consider the figure below:
Example of similar triangles
The image above displays two triangles of different sizes, but their corresponding sides maintain the same ratio, making them similar.
A similarity transformation occurs when a figure is changed into another figure through either enlargement or reduction in size. The original figure is referred to as the pre-image, while the transformed figure is called the image. In similarity transformations, a scaling factor is used to determine the extent of the size change. If the scaling factor is less than 1, the figure is reduced, and if it is greater than 1, the figure is enlarged.
Each point in the figure is dilated from a fixed center called the center of dilation. This can be seen in the following example:
Example of a small triangle transformed into a larger triangle through a similarity transformation
Prime notation (indicated with an apostrophe, ' ) is often utilized to differentiate between the labeling of the image and the pre-image. To better understand this, let's look at an example:
Given a pre-image with coordinates (x,y), a center of dilation (h,k), and a scaling factor of k, we can apply the dilation and plot the image with coordinates (kx,ky), as shown in the solution below.
Solution:
Step 1: Sketch the pre-image.
Step 2: As the scaling factor is 2, we will multiply each x and y coordinate of the pre-image by 2, resulting in the image coordinates of (2x,2y).
Properties of Similarity Transformations
Here are a few key properties of similarity transformations:
Examples of Similarity Transformations
If the center of dilation is (0,0), we can simply multiply each x and y coordinate by the scaling factor k to obtain the coordinates of the pre-image. For instance, if the pre-image has coordinates (x,y), then the image coordinates will be (kx,ky).
Now, let's look at a few examples to determine if two figures are similar or not:
Given the figures shown below with coordinates (1,2) and (2,4), respectively, determine their similarity.
Step 1: Draw a diagram of the figures.
Step 2: To check for similarity, we can compare the coordinates of the figures. In this case, the figures have been dilated, therefore we can match the bottom left of one figure with the bottom left of the other, and so on. We can see that both figures maintain the same proportions, thus making them similar.
In geometry, similarity transformations are a fundamental concept that allows us to transform one figure into another, while still preserving its shape and proportions. Let's explore an example of a similarity transformation to gain a better understanding of this important concept.
To determine if two figures are similar, we can follow a few simple steps:
Step 1: Compare corresponding sides of the figures.
Step 2: If the corresponding sides are equal, then the figures are similar.
Step 3: If the corresponding sides are not equal, we can also substitute the coordinates of the figures into an equation to check for similarity.
Let's apply these steps to an example:
Figure A has been transformed into Figure B through dilation, where the scale factor is maintained for all corresponding sides. This means that Figure A and Figure B are similar figures. By comparing the corresponding sides, we can see that they are indeed equal, confirming their similarity.
Another way to determine similarity transformations is to examine the graph itself and compare corresponding sides. For instance, in Figure C and Figure D, the corresponding sides are not equal. However, when we look at the proportions of these corresponding sides, we see that they are equal, establishing the figures as similar.
In addition to dilation, there are other types of transformations in geometry that can result in similar figures, such as reflection, rotation, and translation.
A reflection is when a figure is flipped to create a mirror image. For example, Figure G is a reflection of Figure H. In this case, the corresponding sides of the figures are equal in length, making them similar.
A rotation is when a figure is turned about a fixed point. For example, Figure I is a rotation of Figure J. The corresponding sides of these figures are equal, making them similar.
A translation is when a figure is shifted across a plane. For example, Figure K and Figure L are translations of each other. While their placement on the plane may differ, the corresponding sides are equal, confirming their similarity.
These types of transformations can also be used to establish similarity, particularly when it may not be immediately apparent through other methods. Let's take a look at an example:
Figure M and Figure N may not seem similar at first glance. However, if we rotate and then translate Figure M, we get Figure O. As you can see, the corresponding sides of these figures are equal, providing evidence of their similarity.
A similarity transformation is a geometric operation that involves transforming one figure into another through dilation.
An example of a similarity transformation is when one figure is enlarged or reduced while maintaining corresponding sides.
Similarity transformations are found by comparing corresponding sides and/or coordinates to see if they are equal.
The types of similarity transformations are dilation, reflection, rotation, and translation.
A dilation transformation makes polygons similar.
