Reflection

Geometry
definition

Also known as: flip, mirror image, mirror transformation

Grade 6-8

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A rigid transformation that flips a figure over a line (the mirror line), producing a mirror image. Creates mirror images; forms the basis for symmetry analysis and transformation composition.

This concept is covered in depth in our reflections, symmetry, and congruent figures, with worked examples, practice problems, and common mistakes.

Definition

A rigid transformation that flips a figure over a line (the mirror line), producing a mirror image.

💡 Intuition

Like looking in a mirror—left and right are swapped, but size and shape are perfectly preserved.

🎯 Core Idea

Reflection reverses orientation while preserving size and shape.

Example

Reflect point (3, 2) over the y-axis: (3, 2) \to (-3, 2)—only the x-sign flips.

Formula

Over x-axis: (x, y) \to (x, -y) Over y-axis: (x, y) \to (-x, y) Over y = x: (x, y) \to (y, x)

Notation

r_\ell denotes reflection over line \ell

🌟 Why It Matters

Creates mirror images; forms the basis for symmetry analysis and transformation composition.

💭 Hint When Stuck

Draw the mirror line, then for each point measure the perpendicular distance to the line and plot the same distance on the other side.

Formal View

Reflection over a line \ell through origin with unit direction \hat{u}: r_\ell(\vec{v}) = 2(\vec{v} \cdot \hat{u})\hat{u} - \vec{v}; over x-axis: r(x,y) = (x, -y); over y-axis: r(x,y) = (-x, y); \det(r_\ell) = -1

Related Concepts

Geometric Transformation

🚧 Common Stuck Point

Reflecting over the x-axis negates y; over the y-axis negates x; over y = x swaps coordinates.

⚠️ Common Mistakes

  • Reflecting over the wrong axis — reflecting over the x-axis changes the y-coordinate, not the x-coordinate
  • Changing the distance to the mirror line — reflected points must be the same distance from the line as the originals
  • Forgetting that reflection reverses orientation — a reflected shape is a mirror image, not identical

Frequently Asked Questions

What is Reflection in Math?

A rigid transformation that flips a figure over a line (the mirror line), producing a mirror image.

Why is Reflection important?

Creates mirror images; forms the basis for symmetry analysis and transformation composition.

What do students usually get wrong about Reflection?

Reflecting over the x-axis negates y; over the y-axis negates x; over y = x swaps coordinates.

What should I learn before Reflection?

Before studying Reflection, you should understand: transformation geo.

Prerequisites

transformation geo

How Reflection Connects to Other Ideas

To understand reflection, you should first be comfortable with transformation geo.

Want the Full Guide?

This concept is explained step by step in our complete guide:

Symmetry, Rotational Symmetry, and Congruence →
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Interactive Playground

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