In this article, we will be exploring the concept of isometry, particularly explaining what transformations are and aren't Isometries. The word isometry is a big fancy word and sounds very complicated. However, it is not too bad... and even better, you'll sound really smart whenever you use the term correctly. Knowing whether a transformation is a form of isometry can be extremely useful... it can help us to predict what a shape is going to look like after it has been translated. I know, I bet you're excited now. So, without any further ado, let's define an isometry...
Isometry Meaning
An isometry is a type of transformation that preserves shape and distance. It's important to note that all isometries are transformations, but not all transformations are isometries! There are 3 main types of transformations that fall under isometry: reflections, translations and rotations. Any transformation that would change the size or shape of an object is not an isometry, so that means dilations are not isometries.
An Isometry is a transformation performed on an object that does not change its shape or size.
Properties of Isometry
The three types of isometric transformation that you need to remember are translations, reflections and rotations. To reiterate, an isometric transformation is a transformation that does not change the shape or size of an object, only its location on a grid. If a shape is moved on a grid and the length of each side has not changed, only its location, an isometric transformation has occurred.
Translations
A translation is a type of isometric transformation. When translating an object, the only thing that happens is that the points of the shape will move from their original position to their new position, depending on what the translation states.
Remember! The distance between each point will be exactly the same after the translation has been performed!
Take the pentagon ABCDE, which has a side length of 1 unit, and translate it by (3, 2). In this case, we have been given the pentagon on a diagram already, so we just need to translate it.
The pentagon ABCDE - StudySmarter Originals
Solution:
The question above asks us to translate the shape by (3, 2), which means we need to draw a new image 3 units across and 2 units above the current shape.
The translation we are about to perform - StudySmarter Originals
If we draw the first point, it can help us figure out how the rest of the shape should look. We know that a translation is an isometric transformation, therefore the sides of the shape will be the same, the only thing that will have changed is its location. A' is the bottom left corner of our new shape, directly connected to the original A point of our first shape.
Given this information, we can draw the rest of the pentagon, as it will have sides of length 1 unit because a translation is an isometric transformation.
The completed translation - StudySmarter Originals
Above is how our final transformation looks!
Reflections
A reflection is another type of isometric transformation, where an object is reflected across an axis. The original object and the reflected object will both have the same dimensions, hence reflection is a type of isometry.
Take the square ABCD, with a side length of 1 unit:
The square ABCD - StudySmarter Originals
Solution:
If we want to perform a reflection on the y-axis, we simply need to copy the shape to its corresponding position. In this case, when reflecting on the y-axis, we know the y-coordinates of the shape should not change. On the other hand, we know that the x-coordinates of each point will change, to be the corresponding negative x-coordinate. In this case, the new image will look like this:
The completed transformation - StudySmarter Originals
Point A has been reflected onto point A', point B is reflected onto point B' and so on. You should notice that the distance to the y-axis doesn't change between the preimage and the new, reflected, image. On top of that, the side lengths of each square are the same.
Remember, A' is pronounced "A prime".
Rotations
The final type of isometric transformation is rotation. A rotation is where an object is moved around a point in a circular motion. Again, no resizing of the object takes place, and as such a rotation is a form of isometric transformation.
You are given a triangle ABC and are asked to rotate it 90o clockwise about the origin.
The triangle ABC - StudySmarter Originals
Solution:
Above we can see we have a triangle and a point marked as our center of rotation. If we wish to rotate it clockwise, we should rotate it to the right.
The completed rotation of our original triangle - StudySmarter Originals
There we are! In this case, we can see that rotation is an isometric translation as each length of the original triangle is kept the same, as well as the distance each point of the triangle is from the origin.
You are given the quadrilateral ABCD and are asked to rotate 90 degrees anticlockwise about the origin.
Quadrilateral ABCD- StudySmarter Originals
Solution:
If we wish to rotate it anticlockwise, we should rotate it to the left about the origin. For point A, we can see that it is 15 units along the x-axis and 10 units up the y-axis. Thus, to rotate 90 degrees anti-clockwise, it needs to go 10 units to the left of the origin and 15 units up. We can do the same for points B, C and D. Joining the points together, we get the parallelogram A'B'C'D'.
The completed rotation of our original parallelogram - StudySmarter Originals
In this case, we can see that rotation is an isometric translation as each length of the original shape is kept the same, as well as the distance each point of the triangle is from the origin.
Laws of Isometry
Now that we've broken down what Isometry is, let's look at another aspect of isometry: direct and opposite isometries. Each isometric transformation is either a direct or opposite isometric transformation. But what are direct and opposite isometries? Well, a direct isometry is a type of transformation that preserves orientation, on top of being an isometry requiring it to keep all the sides of a shape the same length. On the other hand, an opposite isometry keeps the side lengths of a shape the same whilst reversing the order of each vertex.
Direct Isometry
Direct isometry retains the length of a shape's size, as well as the order of its vertices.
Two transformations fall under the purview of direct isometry, these are translations and rotations. This is because both of these transformations preserve the order of the vertices of a shape, as well as retain the same side length in the preimage and new image.
An example of direct isometry - StudySmarter Originals
Notice how in the diagram above, the order of the letters around the shape doesn't actually change. This is the main rule that identifies a transformation as being direct isometry.
Opposite Isometry
Opposite isometry also preserves distances, but unlike direct isometry, it reverses the order of its vertices.
There is only one transformation that fits the definition of opposite isometry, and that is reflection. This is because a reflection changes the order that the vertices of a shape are in after it has been performed.
An example of opposite isometry - StudySmarter originals
Notice how in the diagram above, after the triangle has been reflected, the order of the corners has changed! This is because reflection is an opposite isometry, hence why the shape also looks like the opposite version of itself after it has been reflected.
Isometry - Key takeaways
- An isometric transformation is any type of transformation that preserves lengths and the overall shape of an object.
- The three main forms of isometric transformation are translations, rotations, and reflections.
- There are two types of isometric transformation: direct isometry and opposite isometry.
- Direct isometries are translations and rotations, and they retain the order of the corners.
- Opposite isometry is reflection, as this reverses the order of the vertices.
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