Figures Symmetric with Respect to a Line
What is Line Symmetry?
A figure has line symmetry (also called reflective symmetry or bilateral symmetry) if there exists a line that divides the figure into two parts that are mirror images of each other. This line is called the line of symmetry or axis of symmetry.
Formal Definition
Two points $A$ and $A'$ are symmetric with respect to a line $\ell$ if:
- The line $\ell$ is the perpendicular bisector of the segment $\overline{AA'}$.
This means two conditions must hold:
- The line $\ell$ is perpendicular to $\overline{AA'}$: $\ell \perp \overline{AA'}$
- The line $\ell$ passes through the midpoint $M$ of $\overline{AA'}$, so $AM = MA'$
A figure is symmetric with respect to line $\ell$ if for every point $P$ on the figure, its mirror image $P'$ (reflected across $\ell$) is also on the figure.
Key Properties
- Distance Preservation: If $A$ maps to $A'$ and $B$ maps to $B'$, then $AB = A'B'$.
- Angle Preservation: Angles are preserved under reflection. If $\angle ABC = \theta$, then $\angle A'B'C' = \theta$.
- Points on the axis: Any point on the line of symmetry $\ell$ maps to itself.
- Orientation Reversal: Reflection reverses the orientation (clockwise ↔ counterclockwise).
- Self-inverse: Reflecting twice across the same line returns every point to its original position.