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Reflections

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Reflections

Study Guide

Key Definition

In geometry, a reflection is a transformation representing a flip of a figure across a line, known as the line of reflection. The figure and its reflection are $\cong$ (congruent).

Important Notes

Reflections do not change the size of the shape.
A reflection flips a figure over a mirror line.
Reflected figures are congruent to the original.
The line of reflection is equidistant from corresponding points on the figure and its image.
Reflections can be performed over the x-axis, y-axis, or any line.

Mathematical Notation

$\cong$ represents congruence

$\triangle$ represents a triangle

$\angle$ represents an angle

$\overline{AB}$ represents line AB

Remember to use proper notation when solving problems

Why It Works

Reflections work because they map each point of the shape to a corresponding point on the other side of the line of reflection, maintaining equal distance from the line.

Remember

A reflection over the x-axis changes the sign of the y-coordinate: $(x, y) \rightarrow (x, -y)$

Quick Reference

Reflection over x-axis:$(x, y) \rightarrow (x, -y)$

Reflection over y-axis:$(x, y) \rightarrow (-x, y)$

Reflection over line y = x:$(x, y) \rightarrow (y, x)$

Understanding Reflections

Choose your learning level

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Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A reflection flips a shape across a line, like a mirror image, without changing its size.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

What happens to the coordinates of a point $(x, y)$ when it is reflected over the x-axis?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are designing a logo that needs to be symmetrical across the vertical line x = 0 (the y-axis). How would you apply reflection techniques to ensure symmetry?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

If a shape is reflected over both the x-axis and y-axis, what is the resulting transformation?

Challenge Quiz

Single Choice Quiz

Advanced

Reflect the triangle with vertices $(1, 2)$, $(3, 5)$, and $(6, 2)$ over the y-axis. What are the new coordinates?

Recap

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Review key concepts and takeaways