Reflection (In x-axis, y-axis, x = a, y = a and the origin; Invariant Points)

295. The Mirror’s Rule · Reflection as a perpendicular bisector

The mirror line is always the perpendicular bisector of segment PP′.

Under a reflection, the mirror line is the perpendicular bisector of the segment joining each point to its image. So the mirror lies exactly halfway between P and P′ and at right angles to PP′.

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For subscribers · Selina ICSE: Reflection (In x-axis, y-axis, x = a, y = a and the origin; Invariant Points)

What this lesson covers

Try to break it

Drag P around. Its reflection P′ always lands so that the mirror line is the perpendicular bisector of PP′ — equal distance, perpendicular crossing. Slide P onto the mirror line and P′ coincides with P (that′s an invariant point). Pull P away and P′ separates symmetrically.

How you build it

Reflect a point across the mirror line.

Place a point P on the left side — this is your original point to be reflected.
Place point P′ on the right side — this will be the reflection of P, equally far from the mirror.
Connect P and P′ with a segment — the mirror will be the perpendicular bisector of this segment.
Bisect the segment PP′ — the perpendicular bisector IS the mirror line, exactly halfway and at right angles to PP′.

The proof, step by step

Prove that the mirror line is the perpendicular bisector of P and its image P prime.

The mirror line is the perpendicular bisector of PP′.
This means PM = MP′ and that PM is perpendicular to the mirror line.
Therefore, reflection preserves distance to the mirror.

Worked example

In a reflection of point P(3, 4) across the x-axis, what are the coordinates of the image P′?

Reflection across the x-axis changes the sign of the y-coordinate while keeping the x-coordinate the same. Thus, (x, y) maps to (x, -y). For P(3, 4), the image is (3, -4).

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