Introduction to rigid transformations

To see what a translation is, please grab the point and move it around.

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Nice! You translated the point. In geometry, a translation moves a thing up and down or left and right.

Here, try translating this segment by dragging it from the middle, not the endpoints:

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Notice how the segment's direction and length stayed the same as you moved it. Translations only move things from one place to another; they don't change their size, arrangement, or direction.

Now that we've got a basic understanding of what translations are, let's learn how to use them on the coordinate plane.

Translations on the coordinate plane

Coordinates allow us to be very precise about the translations we perform.

Without coordinates, we could say something like, "We get $\maroonD{B'}$ by translating $\blueD{B}$ down and to the right."

[[☃ image 2]]

But that's not very precise. If we use a coordinate grid, we can say something more exact: "We get $\maroonD{B'}$ by translating $\blueD{B}$ by 5 units to the right and 4 units down."

[[☃ image 1]]

More compactly, we can describe this as a translation by $\langle 5,-4 \rangle$.

The negative sign in front of the 4 tells us the vertical shift is downwards instead of upwards. Similarly, a translation to the left is indicated by the first value being negative.

###Pre-images and images

For any transformation, we have the pre-image figure, which is the figure we are performing the transformation upon, and the image figure, which is the result of the transformation. For example, in our translation, the pre-image point was $\blueD{B}$ and the image point was $\maroonD{B'}$.

Note that we indicated the image by $\maroonD{B'}$, pronounced B prime. It is common, when working with transformations, to use the same letter for the image and the pre-image, simply adding the "prime" suffix to the image.

Let's try some practice problems

Problem 1

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[[☃ explanation 1]]

Problem 2

[[☃ graded-group 1]]

[[☃ explanation 1]]

Challenge problem

[[☃ graded-group 1]]

[[☃ explanation 1]]

##What is a rotation?

In the figure below, one copy of the trapezoid is rotating around the point.

[[☃ image 1]]

In geometry, rotations make things turn in a cycle around a definite center point. Notice that the distance of each rotated point from the center remains the same. Only the relative position changes.

In the figure below, one copy of the octagon is rotated $22\degree$ around the point.

[[☃ image 2]]

Notice how the octagon's sides change direction, but the general shape remains the same. Rotations don't distort shapes, they just whirl them around. Furthermore, note that the vertex that is the center of the rotation does not move at all.

Now that we've got a basic understanding of what rotations are, let's learn how to use them in a more exact manner.

The angle of rotation

Every rotation is defined by two important parameters: the center of the rotation—we already went over that—and the angle of the rotation. The angle determines by how much we rotate the plane about the center.

[[☃ image 1]]

For example, we can tell that $\maroonD{A'}$ is the result of rotating $\blueD{A}$ about $P$, but that's not exact enough.

In order to define the measure of the rotation, we look at the angle that's created between the segments $\overline{PA}$ and $\overline{PA'}$.

[[☃ image 2]]

This way, we can say that $\maroonD{A'}$ is the result of rotating $\blueD{A}$ by $45\degree$ about $P$.

Clockwise and counterclockwise rotations

This is how we number the quadrants of the coordinate plane.

[[☃ image 2]]

The quadrant numbers increase as we move counterclockwise. We measure angles the same way to be consistent.

Conventionally, positive angle measures describe counterclockwise rotations. If we want to describe a clockwise rotation, we use negative angle measures.

[[☃ image 1]]

For example, here's the result of rotating a point about $P$ by $-30\degree$.

###Pre-images and images

For any transformation, we have the pre-image figure, which is the figure we are performing the transformation upon, and the image figure, which is the result of the transformation. For example, in our rotation, the pre-image point was $\blueD{A}$, and the image point was $\maroonD{A'}$.

Note that we indicated the image by $\maroonD{A'}$—pronounced, "A prime". It is common, when working with transformations, to use the same letter for the image and the pre-image; simply add the prime suffix to the image.

Let's try some practice problems

[[☃ graded-group-set 1]]

###Challenge problems

[[☃ graded-group-set 3]]