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## High school geometry

### Course: High school geometry>Unit 1

Lesson 2: Introduction to rigid transformations

# Rigid transformations intro

Transformations in math involve changing a shape's position or which way the shape points. There are three main types: translations (moving the shape), rotations (turning the shape), and reflections (flipping the shape like a mirror image). Rigid transformations keep the shape's size and angles the same. The image is the shape in its new position and direction.

## Want to join the conversation?

• Is a translation and a transformation the same thing? •   No.A transformation includes rotations, reflection, and translations. Translations just slide the figure around the grid.
• what other types of transformations are there besides rigid transformations? •  Nice question!

A common type of non-rigid transformation is a dilation. A dilation is a similarity transformation that changes the size but not the shape of a figure. Dilations are not rigid transformations because, while they preserve angles, they do not preserve lengths.

Have a blessed, wonderful day!
• Hi! Someone down in the thread asked about other ways of translating, rotating, and reflecting points, and I answered their question. Because I think that this information is valuable for the KA community, I also decided to post my answer here. Sorry if this isn't a question, and I hope it helps! :)

There is another way--*algebraic representation
. For the sake of this video, Sal used a graph to translate, rotate, and reflect the quadrilateral.

Translation: Right/Up -> (x + n, y + n) & Left/Down -> (x - n, y - n)
You can translate points and shapes in any direction using this key by adding or subtracting the distance moved.

Example: Pretend that there is a point like (1, 2) and you want to move the point up 1 and to the right 2. In this scenario, you would add 1 and 2 to each of the point's coordinates making it look something like this: (1 + 2, 2 + 1). You would eventually end up with the translated point as (3, 3). If you wanted to translate an entire shape without using a graph, you would do this method with all of the necessary points of the shape.

Rotation: 90 degrees clockwise -> (y, -x), 180 degrees clockwise -> (*-x, -y*), 270 degrees clockwise -> (*-y, x*)
If you do not want to use a graph, using the key above can help you rotate points and shapes in a clockwise direction.

Example: Say there is a point (2, 5) and you wanted to rotate the point 90 degrees clockwise. If you follow the key, the point would become (5, -2).

Reflection: Over the X-Axis -> (x, -y) & Over the Y-Axis -> (*-x, y*)
You can also use this key when reflecting points or shapes across the x-axis or y-axis.

Example: If you wanted to reflect (5, 3) over the y-axis, you would end up with (-5, 3) based on the key.

Regardless of whether you find algebraic representation easier or more efficient than using a graph, I hope that this helps you and gives you a clearer understanding of translating, rotating, and reflecting. :) • Is Dilation a Rigid Transformation? •  Dilations are not a rigid transformations. For something to be a rigid transformation, angles and side lengths need to stay the same. Dilations increase the size of sides.
• At 0.48 seconds, Sal said that there are an infinite number of points along the shape. I am just checking my understanding; I get that there a LOT of points but surely the number is finite as it is along a fixed 2D shape with lines connecting or have I not understood it? • A side of a polygon is a type of line segment. Any line segment has infinitely many points, though its length is finite. Note that for any two distinct points P and Q on a line segment, no matter how close they are together, there are points (besides P and Q) on that line segment that are between P and Q.
• What are the different types of translations? • A translation (or "slide") is one type of transformation.

In a translation, each point in a figure moves the same distance in the same direction.

Example:
If each point in a square moves 5 units to the right and 8 units down, then that is a translation!

Another example:
If each point in a triangle moves 3 units to the left, and there is no up or down movement, then that is also a translation!

Hope this helps!
• Can someone explain rotations. I think I got Translations and Reflections, but not rotations I have always been stuck on it. Thank you! • There are 3 main types of rotations:
1.) 90∘ clockwise - To move a point or shape 90∘ clockwise, simply use this equation: (x, y) → (y, −x).
2.) 90∘ counterclockwise - To move a point or shape 90∘ counterclockwise, simply use this equation: (x, y) → (−y, x).
3.) 180∘ rotation - To move a point or shape 180∘, simply use this equation: (x, y) → (−x, −y).
If a question asks for a 270∘ clockwise rotation, simply change it to a 90∘ counterclockwise, and vice versa. I hope this helped! Thank you for asking! :)
• will this be taught in geometry?   