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Transformations and Symmetry
Introduction
Rigid Transformations
Congruence
Symmetry
Symmetry Groups and Wallpapers
Symmetry in Physics
Dilations
Similarity

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Transformations and SymmetryRigid Transformations

Lugemise aeg: ~30 min

A rigid transformation is a special kind of transformation that doesn’t change the size or shape of a figure. We could imagine that it is made out of a solid material like wood or metal: we can move it, turn it, or flip it over, but we can’t stretch, bend, or otherwise deform it.

Which of these five transformations are rigid?

It turns out that there are just three different types of rigid transformations:

A transformation that simply moves a shape is called a translation.

A transformation that flips a shape over is called a reflection.

A transformation that spins a shape is called a rotation.

We can also combine multiple types of transformation to create more complex ones – for example, a translation followed by a rotation.

But first, let’s have a look at each of these types of transformations in more detail.

Translations

A translation is a transformation that moves every point of a figure by the same distance in the same direction.

In the coordinate plane, we can specify a translation by how far the shape is moved along the x-axis and the y-axis. For example, a transformation by (3, 5) moves a shape by 3 along the x-axis and by 5 along the y-axis.

Translated by (, )

Translated by (, )

Translated by (, )

Now it’s your turn – translate the following shapes as shown:

Translate by (3, 1)

Translate by (–4, –2)

Translate by (5, –1)

Reflections

A reflection is a transformation that “flips” or “mirrors” a shape across a line. This line is called the line of reflection.

Draw the line of reflection in each of these examples:

Now it’s your turn – draw the reflection of each of these shapes:

Notice that if a point lies on the line of reflection, it when being reflected: its image is the same point as the original.

In all of the examples above, the line of reflection was horizontal, vertical, or at a 45° angle – which made it easy to draw the reflections. If that is not the case, the construction requires a bit more work:

To reflect this shape across the line of reflection, we have to reflect every vertex individually and then connect them again.

Let’s pick one of the vertices and draw the line through this vertex that is perpendicular to the line of reflection.

Now we can measure the distance from the vertex to the line of the reflection, and make the point that has the same distance on the other side. (We can either use a ruler or a compass to do this.)

We can do the same for all the other vertices of our shape.

Now we just have to connect the reflected vertices in the correct order, and we’ve found the reflection!

Rotations

A rotation is a transformation that “turns” a shape by a certain angle around a fixed point. That point is called the center of rotation. Rotations can be clockwise or counterclockwise.

Try to rotate the shapes below around the red center of rotation:

Rotate by 90° clockwise.

Rotate by 180°.

Rotate by 90° anti-clockwise.

It is more difficult to draw rotations that are not exactly 90° or 180°. Let's try to rotate this shape by ${10*ang}° around the center of rotation.

Like for reflections, we have to rotate every point in a shape individually.

We start by picking one of the vertices and drawing a line to the center of rotation.

Using a protractor, we can measure an angle of ${ang*10}° around the center of rotation. Let’s draw a second line at that angle.

Using a compass or ruler, we can find a point on this line that has the same distance from the center of rotation as the original point.

Now we have to repeat these steps for all other vertices of our shape.

And finally, like before, we can connect the individual vertices to get the rotated image of our original shape.

Transformations are an important concept in many parts of mathematics, not just geometry. For example, you can transform functions by shifting or rotating their graphs. You can also use transformations to determine whether two shapes are congruent.

Archie