Reflection Over the Line y = x

Reflection Over the Line y = x — Moosa Academy

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

Theorem The rule

(x,\, y) \rightarrow (y,\, x)

No minus signs appear anywhere. A negative coordinate stays negative — it simply moves to the other position. So $(-2,\, 4)$ becomes $(4,\, -2)$ , not $(4,\, 2)$ .

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

y = x

at a right angle and is cut in half there.

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

Example Four quick reflections

(3,\, 1) \rightarrow (1,\, 3)

— an ordinary swap

(-2,\, 4) \rightarrow (4,\, -2)

— the minus travels with its number

(0,\, 3) \rightarrow (3,\, 0)

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

(5,\, 5) \rightarrow (5,\, 5)

— 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 $x = y$ :

(a,\, a) \rightarrow (a,\, a)

These are precisely the points of the line $y = x$ 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:

P(x_1,\, y_1)

and

Q(x_2,\, y_2)

are

\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}

apart

P'(y_1,\, x_1)

and

Q'(y_2,\, x_2)

are

\sqrt{(y_1 - y_2)^2 + (x_1 - x_2)^2}

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.