# How to Preserve Length & Angle Measurements

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will discuss transformations that can be used to preserve length and angle measurement when performed on a shape. We will look at real-world and mathematical examples of preserving length and angle measurement.

## Let's Get Moving

Suppose you are about to sit down at your desk to get some studying done for the math class you're in. You sit down and move a piece of paper by sliding it about 8 inches to the left to make room for your laptop.

Guess what? Before even starting to study, you've already performed something mathematical! By simply moving a piece of paper in the way you did, you performed a mathematical feat called a transformation.

## Preservation of Length and Angle Measurement

Interesting! When we are going to perform a transformation on a shape, we call that shape the pre-image, and after we've performed the transformation on the shape, we call it the image. In your paper scenario, the paper in its original position would be called the 'pre-image', and after you moved it, we would call the paper the 'image'.

## Transformations

Transformations involve moving a shape in specific ways. There are four different types of transformations that we can perform on a shape:

1. Translations involve sliding a shape. Sliding your piece of paper 8 inches to the left is an example of performing a translation on that piece of paper.
2. Rotations involve turning a shape about a point. For instance, if you placed a star sticker on a bicycle tire and then turned the tire, you would rotate the star about the center of the tire.
3. Reflections involve flipping a shape over a line. For example, when you turn a page in a book, you are reflecting the page over the line connecting the pages.
4. Dilations involve resizing a shape to make it bigger or smaller. An example of this could be a reel of movie film being projected onto a big screen. Everything on the film is enlarged to fit the screen.

### Rigid Transformations

When a transformation preserves length and angle measurement, we call it a rigid transformation. Let's determine which of the transformations are rigid transformations, and observe how they preserve length and angle measurement.

#### Translation

Notice that when you translated the piece of paper, you didn't change the piece of paper in any way other than the place in which it was sitting. In other words, all of the sides of the pre-image have the same length as the sides of the image, and all of the angles of the pre-image have the same measurement as the angles of the image. We call this preservation of length and angle measurement.

#### Rotation

The next transformation is rotation. Our example of a rotation was the star sticker on the bicycle tire. Let's think about this. As the sticker rotates around the center of the tire, its shape does not change, so the star's side lengths and angle measurements are unchanged. In general, when we rotate a shape about a point, we preserve length and angle measurement, so rotation is a rigid transformation.

#### Reflection

Okay, moving on to reflections! Picture yourself turning the page of a book, so you're reflecting the page over the center line of the book. Assuming the page turn goes normally, and no pages are torn or anything, the page's side lengths and angle measurements remain the same.

Ah! We've found another rigid transformation since reflections preserve length and angle measurement.

#### Dilations

The last transformation is a dilation. Hmmâ€¦to figure this one out, suppose we are projecting a triangle onto the big screen in a movie theater.

Well, it looks like the angles still have the same measurement, but obviously, the side lengths of the image are much longer than those on the pre-image. Since side length is not preserved, dilations are not rigid transformations. We call transformations that don't preserve length and angle measurement (as in a dilation) a non-rigid transformation.

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