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Rotations

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Rotations

Study Guide

Key Definition

A rotation in geometry is a transformation that turns a figure around a fixed point called the center of rotation. The amount of rotation is measured in degrees, such as $90^\circ$ or $180^\circ$.

Important Notes

Rotations can be clockwise or counterclockwise.
A full rotation is $360^\circ$.
A rotation does not change the size of the figure.
Use a protractor to measure angles during manual rotations.
The coordinates of points change according to rotation rules.

Mathematical Notation

$\angle$ represents an angle

$\triangle$ represents a triangle

$90^\circ$ represents a 90-degree rotation

$180^\circ$ represents a 180-degree rotation

$\sqrt{}$ represents a square root

Remember to use proper notation when solving problems

Why It Works

Rotations maintain the shape and size of figures because they are isometries, meaning the distances between points are preserved.

Remember

A $90^\circ$ clockwise rotation swaps and negates the x and y coordinates: $(x, y) \rightarrow (y, -x)$.

Quick Reference

90-degree clockwise Rotation:$(x, y) \rightarrow (y, -x)$

180-degree clockwise Rotation:$(x, y) \rightarrow (-x, -y)$

270-degree clockwise Rotation:$(x, y) \rightarrow (-y, x)$

Understanding Rotations

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A rotation turns a figure around a fixed point called the center of rotation. For example, a $90^\circ$ rotation turns a figure a quarter circle.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What is the result of a $90^\circ$ clockwise rotation of the point $(1, 2)$?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are designing a spinning logo with a key point at $(x, y)$. If the logo is rotated $180^\circ$ clockwise, what will be the coordinates of the point after rotation?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If a figure is rotated $270^\circ$ clockwise, what is the equivalent counterclockwise rotation?

Challenge Quiz

Single Choice Quiz

Advanced

After a $360^\circ$ rotation, what are the coordinates of a point $(a, b)$?

Recap

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Review key concepts and takeaways