Axial vs central symmetry: the difference in one image

Two transformations, two logics

Type	Element	Image obtained by
Axial (reflection)	An axis (line)	Folding along the axis (mirror)
Central (point)	A point (center)	180° flip around the center

Constructing the image of a point M

Axial reflection over line (d)

The image $M'$ satisfies: line $(d)$ is the perpendicular bisector of $[MM']$.

Steps:

1. Drop a perpendicular from $M$ to $(d)$.
2. Mark off the same distance on the other side.

Central (point) symmetry with center O

The image $M'$ satisfies: $O$ is the midpoint of $[MM']$.

Steps:

1. Draw the ray $[MO)$.
2. Extend by length $MO$ beyond $O$.

Effect on a figure

Axial: preserves lengths, angles, areas. Reverses orientation (mirror).

Central: preserves everything AND preserves orientation (unlike axial). A 180° flip doesn’t mirror text.

Mnemonic

• F in a mirror becomes reversed F ⇒ axial.
• F after 180° flip becomes upside-down F ⇒ central.

Common mistakes

1. Confusing them: axial = “mirror flip”, central = “upside down”.
2. Forgetting the axis/center is fixed: points on the axis (axial) or the center (central) are invariant.
3. Central = two axial: yes, if the two axes are perpendicular and meet at the center.

?À toi de jouer

Central symmetry with center O is equivalent to:

A.A 180° rotation around OB.A 90° rotation around OC.A translation by OOD.An axial reflection through O

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