Central and Axial Symmetry

Vocabulary practice cards covering definitions, properties, and constructions related to central symmetry, axial symmetry, and symmetric geometric figures.

Central symmetry (Half-turn)

A transformation in which two figures $\mathcal{F}$ and $\mathcal{F}'$ superimpose onto each other by performing a half-turn around a fixed point $O$.

Center of symmetry (transformation)

The fixed point $O$ around which a half-turn is performed in a central symmetry.

Symmetric of a point $M$ distinct from $O$

The point $M'$ such that the center $O$ is the midpoint of the segment $[MM']$.

Symmetric of the center $O$

The point $O$ itself.

Ruler and compass construction of a point's symmetric

The method of constructing the symmetric $M'$ of a point $M$ with respect to $O$ by drawing the straight line $(OM)$, placing the compass at $O$, and transferring the distance $OM$ onto the line $(OM)$ on the opposite side of $O$.

Invariant properties of central symmetry

The geometric properties preserved under a half-turn: lengths, angle measures, alignment of points, parallelism, and areas.

Symmetric of a line with respect to a point

A straight line that is parallel to the original line.

Symmetric of a line passing through the center $O$

The line itself.

Symmetric of a segment with respect to a point

A line segment that is parallel to the original segment and has the same length.

Symmetric of a segment when $O$ is its midpoint

The segment itself.

Symmetric of a circle with respect to a point

A circle that has the exact same radius as the original circle.

Axis of symmetry of a figure

A straight line $(d)$ such that a figure $\mathcal{F}$ superimposes onto itself after folding along $(d)$.

Center of symmetry of a figure

A point $O$ such that a figure $\mathcal{F}$ superimposes onto itself after performing a half-turn around $O$.

Symmetric of a point with respect to a line (Axial symmetry)

The point $A''$ corresponding to a point $A$ such that the line is the perpendicular bisector of the segment $[AA'']$.