Rules for Reflections Over Y=X Line

📚 Definition of Reflection Over the Y=X Line

Reflection over the line $y=x$ is a transformation that creates a mirror image of a point or shape, with the line $y=x$ acting as the mirror. In this specific reflection, the x and y coordinates of a point are interchanged.

📜 History and Background

The concept of reflections has been a cornerstone of geometry for centuries. The specific study of reflections across defined lines, like $y=x$, became formalized with the rise of coordinate geometry, allowing mathematicians to precisely describe and analyze these transformations using algebraic methods. René Descartes' work in the 17th century greatly contributed to this field, bridging algebra and geometry.

🔑 Key Principles

• 🔄 Coordinate Swap: The fundamental principle is the interchange of x and y coordinates. If a point is $(a, b)$, its reflection over $y=x$ is $(b, a)$.
• 📐 Perpendicular Distance: The distance from the original point to the line $y=x$ is the same as the distance from the reflected point to the line $y=x$. The line connecting the point and its image is perpendicular to $y=x$.
• ♾️ Invariant Points: Points that lie on the line $y=x$ remain unchanged after reflection. These are invariant points of the transformation.

✍️ Step-by-Step Guide to Reflecting Over Y=X

1. 📍 Identify the Point: Determine the coordinates of the point you want to reflect. Let's say it's $(x, y)$.
2. 🔀 Swap Coordinates: Interchange the x and y coordinates. The new point will be $(y, x)$.
3. 📈 Plot the New Point: Plot the new point $(y, x)$ on the coordinate plane. This is the reflection of the original point over the line $y=x$.

➕ Real-World Examples

Let's illustrate with a few examples:

✍️ Practice Quiz

Reflect the following points over the line $y=x$:

1. Point A: (3, 7)
2. Point B: (-1, 4)
3. Point C: (0, -5)
4. Point D: (6, 6)

✅ Answer Key

1. Point A': (7, 3)
2. Point B': (4, -1)
3. Point C': (-5, 0)
4. Point D': (6, 6)