Reflection over a Line | Moosa Academy

A reflection over a line is a transformation that produces a mirror image of a figure across a fixed line, called the axis of reflection. It is an isometry: distances and angles are kept, so the image is congruent to the original.

Theorem Reflection over a line

To reflect a point, drop a perpendicular to the axis and extend it the same distance on the other side. A point and its image are equidistant from the axis.

For a point P and its image P′ in the axis L:

d(P,L) = d(P',L)

Note A point on the axis

If a point lies on the axis of reflection, its distance to the axis is zero, so it does not move — the point is its own image.

Example Reflecting a point

Reflect the point $P = (2,0)$ over the y-axis.

Reflecting over the y-axis:

(x,\,y) \longrightarrow (-x,\,y)

P = (2,\,0) \longrightarrow P' = (-2,\,0)

, and

d = 2

on each side.

Reflecting over the x-axis instead:

(x,\,y) \longrightarrow (x,\,-y)

⟹ change the sign of the coordinate across the axis you reflect in.

Example Reflecting a triangle

To reflect a triangle, reflect each vertex separately, then join the new vertices. The image is congruent to the original.

Every vertex keeps its distance to L.

Lengths and angles are unchanged.

But the order of the vertices reverses.

⟹ △A′B′C′ ≅ △ABC, facing the other way.

Example Reflecting y = x²

Reflect the parabola $y = x^2$ over each axis, using the sign rules: replace $x$ with $-x$ for the y-axis, or $y$ with $-y$ for the x-axis.

Over the y-axis:

(-x)^2 = x^2

, so the curve is unchanged.

Over the x-axis:

y = x^2 \longrightarrow y = -x^2

The origin lies on the axis, so it stays fixed.

Summary

A reflection over a line gives a congruent mirror image; the condition is $d(P,L) = d(P',L)$ .
A point on the axis stays fixed; distances and angles are preserved.
Orientation reverses — the image faces the other way.
Coordinate rules: over the x-axis $(x,y) \to (x,-y)$ ; over the y-axis $(x,y) \to (-x,y)$ .