Reflection Over the Line y = x

Reflecting over an axis flips one sign. Reflecting over the diagonal line does something different and even simpler: it swaps the two coordinates, leaving both signs untouched.

Theorem The rule

No minus signs appear anywhere. A negative coordinate stays negative — it simply moves to the other position. So becomes , not .

Concept What the picture shows

The point and its image sit on opposite sides of the diagonal, the same distance from it. The dashed segment joining them meets the line

at a right angle and is cut in half there.

As with any reflection, the mirror is the perpendicular bisector of the segment .

Example Four quick reflections

— an ordinary swap

— the minus travels with its number

— a point on the y-axis lands on the x-axis

— unchanged, because it lies on the line

The third line is worth noticing: this reflection exchanges the two axes. Everything on the y-axis moves onto the x-axis and back.

Note The fixed points

A point is unmoved exactly when swapping its coordinates changes nothing — that is, when :

These are precisely the points of the line itself, which is what the name of the mirror already tells us.

Note Why lengths and angles survive

Take two points and their images, and compare the distances:

and

are

apart

and

are

apart

The two expressions are the same sum, just written in the other order. Distances are unchanged, so the shape keeps its size and its angles — the reflection is a congruence.

Summary

Reflecting over y = x sends (x, y) to (y, x).
Only the positions swap — no sign changes.
The line y = x is the perpendicular bisector of the segment joining a point to its image.
The fixed points are those with x = y, which form the mirror line itself.
The transformation swaps the two axes.
Distances and angles are preserved, so the image is congruent to the original.