Symmetry (including Reflection and Rotation)

184. Mirror Image at Zero · Reflection of a point in the origin

P' is always the exact reflection of P across the origin O.

Reflection through the origin: P(x, y) ↦ P′(−x, −y). Both coordinates flip sign. The origin O is the midpoint of segment PP′. Equivalent to a 180° rotation about O.

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Selina ICSE: Symmetry (including Reflection and Rotation)

What this lesson covers

Try to break it

Drag P anywhere on the grid. P' always lands on the opposite side of the origin O, with both coordinates flipped in sign. Drag P across the x-axis or y-axis and P' tracks across the opposite axis at the same moment. Try to drag P so PP' doesn't pass through O — impossible.

How you build it

Reflect a point in the origin.

Mark point O at the origin. Point tool: click the origin where the axes cross — call it O.
Mark point P anywhere on the grid. Point tool: drop P somewhere away from O.
Draw the line through P and O. Line tool: click P, then O — the line extends through both.
With centre O and radius OP, swing an arc. Arc tool: click O for the centre, then P for the radius.
Mark P' where the arc meets the line on the opposite side of O. Point tool: click where the arc crosses the line — P' has coordinates (−x, −y), the exact reflection of P.

The proof, step by step

Prove that P prime is the reflection of P through the origin.

Let P have coordinates (x, y) and its reflection in the origin be P'(x', y').
By definition of reflection in the origin, O is the midpoint of the segment PP'.
Using the midpoint formula: (x + x')/2 = 0 and (y + y')/2 = 0.
Solving these equations gives x' = -x and y' = -y. Thus, P' is (-x, -y).

Worked example

A point A(3, -4) is reflected in the origin. What are the coordinates of its image A'?

Reflection in origin changes the sign of both coordinates. So, A(3, -4) becomes A'(-3, 4).

(3, 4)
(-3, 4) — correct
(-3, -4)
(3, -4)