Reflections Across Specific Lines
Beyond the axes, two diagonal lines appear constantly in geometry: and . Reflecting across them swaps the coordinates — a beautifully simple rule once you see the pattern.
Introduction
In Lesson 4.1.4, you reflected across horizontal and vertical axes. Now you'll reflect across the diagonal lines and . The rules are elegant — they swap the and coordinates — and knowing them unlocks fast solutions to many competition and exam problems.
Past Knowledge
Reflections across axes (4.1.4). The line has slope 1 through the origin.
Today's Goal
Reflect points and figures across and using coordinate rules.
Future Success
Symmetry analysis (4.2.1), inverse functions in Algebra II, and matrix transformations.
Key Concepts
Reflection Rules — All Four Lines
| Line | Rule | Shortcut |
|---|---|---|
| x-axis | Negate | |
| y-axis | Negate | |
| Swap and | ||
| Swap and negate both |
Why Swapping Works
Every point on has equal coordinates, like . When you reflect a point over this line, the and swap because the line acts as a “diagonal mirror” between the two axes. For , points have coordinates like , so you must also negate after the swap.
Connection to Inverse Functions
In algebra, the graph of an inverse function is the reflection of across . This is why we “swap and ” to find inverses!
Worked Examples
Reflecting Across
Reflect across .
Rule:
— the coordinates simply swapped.
Reflecting a Triangle Across
Reflect with , , across .
Rule:
Identifying the Line of Reflection
. Over which line was reflected?
Check the four rules:
- x-axis — no.
- y-axis — no.
- : — no.
- : — yes!
The line of reflection is .
Common Pitfalls
Mixing Up and Rules
For , just swap: . For , swap and negate: . Students often apply the wrong version.
Negating Only One Coordinate for
The rule is , not or . Both coordinates must be negated after the swap.
Real-Life Applications
Cryptography & Data Encoding
Swapping and negating coordinates is a form of simple transformation cipher. In computer graphics, swapping coordinates is used to rotate textures by 90° efficiently — the same rule as reflecting across combined with an axis reflection.
Architecture & Symmetry
Many buildings feature diagonal lines of symmetry. The Louvre Pyramid in Paris, for example, has multiple reflective symmetry planes — including diagonal ones — ensuring the structure looks balanced from every angle.
Practice Quiz
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