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Rigid Transformations

Identify transformations, translations, reflections and rotations.

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Rigid Transformations

Source: https://farm4.staticflickr.com/3073/2297342276_1cbf1d2b9b.jpg
License: CC BY-NC 3.0

Newton is planning to build a tiny house as a music studio in his back yard. But when he shows his dad the site plans, his dad points out a few problems with the design. As it's drawn, the tiny house will get all the run-off from the house roof, and it's oriented with the long axis and most of the windows facing East-West which means that the house will almost always be too hot. How should Newton change his design to fix these problems?

In this concept, you will learn how to recognize different transformations of rigid shapes on a coordinate plane.

Recognizing Rigid Transformations

The x-axis is the bold line running from left to right on the coordinate plane, and it is usually labeled with an "x". The x-coordinate of an ordered pair is found with relation to it.

The y-axis is the central line that runs up-down and is labeled with a "y". Y-coordinates are plotted in reference to this axis. 

The  vertex of a shape is the place where two sides of the shape come together. 

A transformation is the movement of a figure on the coordinate grid. There are three different types of transformations: translation, reflection, and rotation.

A translation is when a figure appears to slide from one place to another, but its orientation remains the same. 

Here is a translation.

You can see here that the figure was not changed at all. It simply slid from one point to another point.

A reflection is a flip of a figure. A mirror image is one common example. Another is the image you see in the lake or a spoon. When a figure is reflected on the coordinate grid, it is flipped across an axis or a line of symmetry. Here is a reflection.

In this example, the y-axis is the line of symmetry because the shape is a mirror image on the other side of that line.

A rotation is a turn on the coordinate grid. Here is an example.

The easiest way to identify what kind of transformation has taken place is to draw both the original and the transformed shapes on a grid and to compare them. If the transformed shape has the same orientation as the original, then it is a translation. If the orientation has changed and it's a mirror image, then it's a reflection. And if it has turned on its side, then it is a rotation.

Here is an example.

In this picture, triangle \begin{align*}ABC\end{align*} is the original figure. What kind of transformation is \begin{align*}A'B'C'\end{align*}?

First, determine whether the shape has changed its orientation.

In this case, it has not. So this is an example of a translation.

Examples

Example 1

Earlier, you were given a problem about Newton and his back yard tiny house.

His father told him that his house would get leaked on and would overheat if he put it where he planned to. So he has to figure out how to move the house so that it isn't under the house eaves and so that the long axis of the house isn't facing East-West.

Newton breaks his problem into two parts: the water problem and the orientation problem.

First, he focuses on the water problem. In order to get it out from under the eaves of the big house roof, he just needs to move the tiny house away from the building. That is easy. He just translates the tiny house several feet away from the big house on his site plans.

Next, he turns to the overheating problem. He concludes that this is a problem of orientation. As long as the house is oriented East-West, it will get too much sun, so he has to change the orientation. But if he were to just flip the house, it would still be oriented East-West. So the only option is for him to rotate the plan until the long axis of his house is oriented North-South.

The answer is Newton must first translate and then rotate his tiny house.

Example 2

License: CC BY-NC 3.0

In the photo above, tell what kind of transformation this is, and explain why.

First, describe what the picture is about.

In this picture a group of carved chess pieces are seen standing up and then, again, upside down in the glass tabletop..

Next, determine whether the orientation of anything in the photo has changed.

The pieces are upside down, so their orientation has changed.

Then, determine whether there is a line of symmetry across which the pieces are flipped.

In this case, there is. The line of symmetry is the line between the board and the water. The pieces appear the same on the other side but flipped.

The answer is reflection.

Identify each of the following examples as a rotation, translation or reflection.

Example 3

First, determine whether the orientation of the shape has changed.

In this case, it has, so it is not a translation.

Next, determine how the orientation has changed. Is the shape on its side or upside down?

In this case, it is on its side. So it is not a reflection.

The answer is rotation.

Example 4

First, determine whether the shape has changed its orientation.

It has not. So it is a translation.

The answer is translation.

Example 5

First, determine what the congruent shape in the image is.

In this case, the shape in question is the wing of the butterfly.

Next, determine if both butterfly wings are the same orientation.

They are not.

Then, determine how the orientation has changed. Is the other wing flipped or is it laid on its side?

In this case, it is flipped across an axis of symmetry--the butterfly body--so it is reflected.

The answer is reflection.

Review

Identify each image as a translation, rotation or reflection.

For numbers 6 – 12 Draw your own figures, and demonstrate three different translations, three different rotations and three different reflections.

Complete the following reflections according to the directions.

  1. Reflect this image across the \begin{align*}y\end{align*} axis.

  1. Rotate the following image.

15 – 20 Go on a scavenger hunt and find examples of each of the different types of transformations in real world contexts.

Review (Answers)

To see the Review answers, open this PDF file and look for section 11.18. 

Resources

 

Vocabulary

Angle

A geometric figure formed by two rays that connect at a single point or vertex.

Reflections

Reflections are transformations that result in a "mirror image" of a parent function. They are caused by differing signs between parent and child functions.

Transformation

A transformation moves a figure in some way on the coordinate plane.

Transformations

Transformations are used to change the graph of a parent function into the graph of a more complex function.

Translation

A translation is a transformation that slides a figure on the coordinate plane without changing its shape, size, or orientation.

Rigid Transformation

A rigid transformation is a transformation that preserves distance and angles, it does not change the size or shape of the figure.

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