Central Symmetry (Point Symmetry)
Big Idea: Central symmetry is a transformation that maps every point of a figure to another point through a fixed center, creating a "mirror image" rotated by $180°$.
📐 Formal Definition
A point $A'$ is the symmetric image of point $A$ with respect to a center $O$ if and only if:
- $O$ is the midpoint of segment $[AA']$
- This means: $\overrightarrow{OA'} = -\overrightarrow{OA}$
Equivalently, the center $O$ divides $AA'$ such that:
$$OA = OA' \quad \text{and} \quad A, O, A' \text{ are collinear}$$🔑 Key Properties of Central Symmetry
| Property | Description |
|---|---|
| Isometry | Central symmetry preserves distances: if $A \to A'$ and $B \to B'$, then $A'B' = AB$ |
| Preserves angles | All angles are preserved in the image figure |
| Lines → Lines | The image of a line is a line parallel to the original (or the same line if it passes through $O$) |
| Segments → Segments | The image of a segment is a segment of equal length |
| Fixed Point | The center $O$ is the only fixed point: $O$ maps to itself |
| Involution | Applying central symmetry twice returns every point to its original position |
📊 Coordinate Formula
If the center of symmetry is $O(a, b)$, then the image of any point $M(x, y)$ is:
$$M'(x', y') = (2a - x, \; 2b - y)$$Special case: When the center is the origin $O(0,0)$:
$$M'(x', y') = (-x, \; -y)$$This is equivalent to a rotation of $180°$ about point $O$.
💡 Remember!
Central symmetry is the same as a rotation of $180°$ about the center point.
Don't confuse it with:
- Axial symmetry (reflection over a line)
- Translation (sliding without rotation)
🌍 Real-World Examples
- Playing cards (King, Queen)
- The letter S, N, Z
- A parallelogram has central symmetry about the intersection of its diagonals
- Yin-Yang symbol (approximately)