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Rigid Motion in Geometry

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  • 0:04 Rigid Transformations
  • 1:48 Translations in Rigid Motion
  • 2:26 Rotations in Rigid Motion
  • 3:29 Reflections in Rigid Motion
  • 4:18 Lesson Summary
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Lesson Transcript
Instructor: Stephanie Matalone

Stephanie taught high school science and math and has a Master's Degree in Secondary Education.

In this lesson, we will define rigid motion and go over the three types of rigid motion seen in geometry: translations, rotations, and reflections. We will define each and look at examples.

Rigid Transformations

Let's say you're sitting in class and decide to move to the desk in front of you to see the board better. You just participated in rigid motion, specifically a translation. You participated in rigid motion because you did not change size or shape, just shifted spots. It was a translation specifically because you still faced forward, maintaining your orientation.

So, what is a rigid motion in geometry? Rigid motion is otherwise known as a rigid transformation and occurs when a point or object is moved, but the size and shape remain the same. This differs from non-rigid motion, like a dilation, where the size of the object can increase or decrease.

When working with rigid motion, you will typically see two objects that show the transformation. The original object is called the pre-image, and its points will normally be marked with capital letters. After it's been moved, the transformed image is simply called the image. The points of the transformed image will be marked with an apostrophe. In order for the movement to be rigid, the pre-image and image must be congruent. If you want to keep moving the same image, add more apostrophes to the letters naming the points after each movement.


Rigid Motion of a Triangle
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In this picture, you can see the pre-image of the triangle on the top left. The three points are named A, B, and C. In the first image on the top right, you can see the triangle has been turned, but the lengths of each side of the triangle have been maintained. Since this is the first motion that has occurred, the points are named A', B', and C'. These points would be called ''A prime, B prime, and C prime.'' For each additional movement, another apostrophe is added. These would be called ''double prime'' and ''triple prime.''

Translations in Rigid Motion

The translation - the first specific type of rigid motion we will examine - is the type you performed when you moved forward in the classroom. A translation is a type of rigid motion that occurs when the object shifts or slides without changing its orientation. You translated in your classroom because you did not get bigger or smaller, but also because you still faced forward in the same direction.

Translation
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This picture shows a translation of the pre-image in blue to the image in pink. This object is being translated by shifting its position up and to the right. You can see that the size, shape, and orientation of the image all stay the same.

Rotations in Rigid Motion

The second type of rigid motion is called a rotation. Rotations occur when an object moves around a certain point. The key with rotations is that all of the points will maintain their distance from the center point. If we are rotating an object with point D and that point is 3 inches from the center, point D' must also be 3 inches from the center after the object is rotated.

When rotating an object around a center point, we typically see 90, 180, and 270 degree rotations, although rotations can occur at any number of degrees. When we say we are rotating an object 90 degrees, we mean 90 degrees around that center point and to the right. You can see a 90 degree rotation in this image. The pre-image on the top left has been rotated 90 degrees to produce the image on the top right. As we keep going, the bottom right triangle has been rotated another 90 degrees, or 180 degrees from the original. The triangle on the bottom left has been moved another 90 degrees, or a total of 270 degrees from the original.


Rotation
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