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Reflection in Mathematics

The Mirror Image Transformation

In mathematics, a Reflection is a type of rigid geometric transformation that creates a mirror image of a shape. It involves flipping a figure over a line, known as the Line of Reflection.

Think of it like looking into a mirror or seeing a mountain reflected in a calm lake. In geometry, every point on the original shape (pre-image) has a corresponding point on the reflected shape (image) that is equidistant from the reflection line.

1. Key Properties of Reflection

  • Isometry: Reflection is a "rigid" transformation. This means the shape does not change size or structure; it only changes orientation.
  • Equidistance: Every point and its reflection are the same distance from the line of reflection.
  • Orientation Reversal: If you read the points of a shape clockwise (A → B → C), the reflected image will read counter-clockwise (A' → B' → C').

2. Reflection over the X-Axis

[Image of reflection over x-axis graph]

When you reflect a point across the horizontal x-axis, the x-value stays the same, but the y-value flips its sign (positive becomes negative, and vice versa).

Rule: (x, y) → (x, -y)

Example: Reflect point P(3, 4) over the x-axis.
New P' = (3, -4).

3. Reflection over the Y-Axis

[Image of reflection over y-axis graph]

When you reflect a point across the vertical y-axis, the height (y-value) stays the same, but the x-value flips its sign (left becomes right).

Rule: (x, y) → (-x, y)

Example: Reflect point A(-2, 5) over the y-axis.
New A' = (2, 5).

4. Reflection over the Line y = x

[Image of reflection across line y=x]

The line y = x is a diagonal line passing through the origin at a 45-degree angle. Reflecting over this line swaps the x and y coordinates.

Rule: (x, y) → (y, x)

Example: Reflect point B(1, 4) over y = x.
New B' = (4, 1).

5. Reflection over the Line y = -x

This is the diagonal line going downwards. Reflecting over this line swaps the coordinates and changes their signs.

Rule: (x, y) → (-y, -x)

Example: Reflect point C(2, 3) over y = -x.
New C' = (-3, -2).

Conclusion

Mastering Reflection is essential for understanding symmetry and coordinate geometry. By simply applying these coordinate rules, you can predict exactly where a shape will land after a flip.