Symmetry (including Reflection and Rotation)

186. Reflection in the Y-Axis · Mirrors across the y-axis

The segment PP' is always perpendicular to the y-axis and bisected by it.

Reflection across the y-axis: P(x, y) ↦ P′(−x, y). The y-coordinate stays; x flips sign. Segment PP′ is perpendicular to the y-axis and bisected by it.

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Selina ICSE: Symmetry (including Reflection and Rotation)

What this lesson covers

Try to break it

Drag P around. P' always mirrors across the y-axis — same y-coordinate, opposite x-coordinate. The line PP' stays perfectly horizontal and crosses the y-axis at 90°. Try to break either property — impossible.

How you build it

Reflect a point across the y-axis.

Pick the Point tool. Click on the y-axis to drop point A — anywhere on it. Use a grid intersection for a clean coordinate.
Click again on the y-axis, at a different spot, to drop point B. A and B together identify the mirror line.
Click anywhere to the right of the y-axis to drop point P — the point we will reflect.
Pick the Arc tool. Click A for the centre, then click P to set the radius. The arc passes through P and curves across the y-axis to the left. Every point on this arc is exactly the distance AP from A.
Stay on the Arc tool. Click B for the centre, then click P for the radius. This second arc also passes through P. To the left of the y-axis, the two arcs cross at exactly one point — that will be P'.
Switch to the Point tool. To the left of the y-axis, the two arcs cross at exactly one point. Click that intersection — this is P', the mirror image of P. Because AP' = AP and BP' = BP, the y-axis is the perpendicular bisector of PP'. If P = (x, y) then P' = (−x, y).
Pick the Line tool. Click P, then P'. The line PP' crosses the y-axis at a right angle and the y-axis bisects it exactly in half — the y-axis is the perpendicular bisector of PP'. This confirms x flips sign while y stays the same.

The proof, step by step

Prove that the y-axis perpendicularly bisects the segment PP prime.

Draw PB perpendicular to the y-axis, meeting it at B.
In the coordinate system, P has coordinates (x, y), so PB = x.
By definition of reflection, P'B = PB = x, and P' lies on the opposite side.
Therefore, the x-coordinate of P' is -x, while the y-coordinate remains y. Reflection in y-axis changes the sign of the abscissa.

Worked example

A point A(3, 4) is reflected in the y-axis to get point A'. What are the coordinates of A'?

Reflection in the y-axis changes the sign of the x-coordinate (abscissa) while keeping the y-coordinate (ordinate) the same. Thus, A(3, 4) becomes A'(-3, 4).

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