Geometry

Rigid Transformations

Rotating, reflecting, and translating figures on the coordinate plane.

Intermediate20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Moving a Shape Without Changing It

Slide a triangle across a page, flip it over like a pancake, or spin it around a point — before reading on, think about what stays exactly the same through each of these moves, even though the triangle's position and orientation clearly change.

Definition — Rigid Transformation

A movement of a figure that preserves every distance and angle within it — the figure's size and shape never change, only its position or orientation. The three basic rigid transformations are translation (sliding), reflection (flipping over a line), and rotation (turning around a point).

Every one of these three moves keeps every pair of points on the figure exactly the same distance apart as they were before — which is exactly why they're called 'rigid': the shape genuinely doesn't bend, stretch, or shrink, no matter how it's moved around.

Worked Example — Translating a Point

Translate the point (3, 2) by moving 4 units right and 1 unit down. Add the horizontal shift to the x-coordinate and the vertical shift to the y-coordinate: (3 + 4, 2 − 1) = (7, 1).

Worked Example — Reflecting a Point Over the y-axis

Reflect the point (5, 3) over the y-axis. Reflecting over the y-axis flips the point to the same distance on the opposite side, negating the x-coordinate while leaving y unchanged: (−5, 3).

Geometry Canvas

Point mode (click to add a point)LabelsMeasurementsSnap to grid

Construct

Tool

Objects

1.
Color
P1
P1: a free point, draggable on the plane
2.
Color
P2
P2: a free point, draggable on the plane
3.
Color
P3
P3: a free point, draggable on the plane
4.
Color
poly1
poly1: the polygon through P1, P2, P3

Measurements

poly1area = 15perimeter = 17.66

Tip

A quick way to spot which rigid transformation happened between a figure and its image: translation keeps the same orientation facing the same way, reflection produces a mirror-image (flipped) orientation, and rotation keeps the same size and mirror-handedness but turns the whole figure around a point.

Common Mistakes

Reflecting over the wrong axis, such as negating the y-coordinate instead of the x-coordinate when reflecting over the y-axis.
Reflecting over the y-axis flips left-right, changing the sign of x; reflecting over the x-axis flips up-down, changing the sign of y — match the axis name to the coordinate it affects.
Assuming a rigid transformation could change a figure's size, such as expecting a rotated triangle to look larger or smaller.
Rigid transformations only change a figure's position or orientation — every distance and angle inside the figure, including its overall size, stays exactly the same.

Key Takeaways

A rigid transformation moves a figure while preserving every distance and angle within it.
Translation slides a figure, reflection flips it over a line, and rotation turns it around a point.
A figure's size and shape are unchanged by any rigid transformation.

Summary

Rigid transformations move figures without changing their size or shape. The next lesson uses these exact transformations to define what it formally means for two figures to be congruent — or similar.

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