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Isometry

Isometry

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.

A graph with the pentagon ABCDE

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.

A Graph with a pentagon on that is about to be translated

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.

A graph with an original preimage of a pentagon and the same pentagon after it has undergone an isometric translation

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:

A diagram of a square ABCD

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:

A diagram with a square that has been reflected in the y-axis.

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.

A diagram with the triangle ABC

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.

A diagram with a triangle rotated about the origin in an isometric transformation.

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.

Rotations, Example of rotation, Jordan Madge

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'.

Rotations, completed rotation, Jordan Madge

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.

A diagram showing two triangles, one before a rotation has been applied and one after a rotation has been applied. This diagram showcases direct isometry

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.

A diagram with two triangles, a preimage before a reflection and a new image after a reflection. This diagram showcases opposite isometry.

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.

Frequently Asked Questions about Isometry

Isometry in geometry is a type of transformation that changes the location of a shape but doesn't change how the shape looks.

The 3 types of isometry are translations, reflections and rotations.

Isometry is done by performing the specified isometric transformation on a given shape.

Isometric transformations are types of transformations that do not change the shape or size of a given shape.

Isometry is composed of translations, reflections and rotations.

Final Isometry Quiz

Question

What is isometry?

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Answer

An isometric transformation is a transformation that does not change the shape or size of an object.

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Question

What are the types of isometric transformations?

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Answer

There are three types of isometric transformation: translations, reflections and rotations.

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Question

What is direct isometry?

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Answer

Direct isometry retains the order of the vertices in a shape, as well as the regular definition of isometry where distances are kept the same.

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Question

What isometric transformations are direct isometric transformations?

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Answer

Translations and rotations are both direct types of direct isometry.

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Question

What is opposite isometry?

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Answer

Opposite isometry preserves the distances as in regular isometry, but the order the vertices are found is reversed.

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Question

Which isometric transformations are opposite isometric transformations?

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Answer

The only opposite isometric transformation is reflection.

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Question

What steps are taken in a translation?

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Answer

We start with a shape and the translation we need to make in the form (x, y). We move each point on the shape x squares along the x-axis and y squares along the y-axis. Then we connect each point and redraw the shape in the new, translated location.

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Question

What is a translation? 

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Answer

A translation is a type of isometric transformation. When translating an object, the only thing that takes happens is that the points of the shape will move from their original position to their new position, depending on what the translation states. 

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Question

What is a reflection? 

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Answer

A reflection is a type of isometric transformation, where an object is reflected across an axis.

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Question

What is a rotation?

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Answer

A rotation is where an object is moved around a point in a circular motion. 

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Question

How to rotate a shape 90 degrees clockwise about the origin? 

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Answer

For each point of the shape, rotate it 90 degrees right from the origin and then connect the points 

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Question

Is a rotation 90 degrees clockwise ever an isometric transformation? 

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Answer

Yes, each length of the orgional shape will aways be the same 

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Question

What information do we need to rotate a shape?

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Answer

Clockwise or anticlockwise and the center of rotation.

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Question

What does it mean when we see A'?

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Answer

A' is pronounced A prime, and it is what happens when a point gets transformed from one point to another. 

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Question

What information do we need to reflect a shape?

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Answer

Where is the shape being reflected? 

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Question

What information do we need to translate a shape? 

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Answer

The vector that we are translating the shape by. 

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