Transformations

Translations, Reflections & Rotations

Performing and describing rigid transformations on the coordinate plane.

Intermediate20 min lesson4 min readUpdated August 12, 2026Author not yet attributed

Moving a Figure Exactly

Slide a triangle 4 units right, or flip it over a line, or spin it 90° around a point. Before reading on, think about what stays provably the same through each of these moves — not just visually, but as a fact you could argue for.

Definition — Isometry

A transformation that preserves every distance and angle in a figure — its image (the result) is exactly congruent to its preimage (the original). Translations, reflections, and rotations are all isometries, together called rigid motions.

A translation slides every point of a figure the same fixed distance in the same direction — described as a vector, (a, b), added to every coordinate: (x, y) → (x + a, y + b). Since every point shifts by the identical vector, the distance between any two points of the figure is completely unchanged, which is exactly why a translation is an isometry: two points that started a certain distance apart are still separated by that same vector difference afterward.

Worked Example — Translating a Figure

Translate triangle with vertices (1, 2), (4, 2), (1, 5) by the vector (3, −1). Add 3 to every x-coordinate and subtract 1 from every y-coordinate: (4, 1), (7, 1), (4, 4). The triangle's shape and size are unchanged — only its position moved.

A reflection flips a figure over a line, called the line of reflection, so that the line acts as a perpendicular bisector between every original point and its image. Reflecting over the y-axis negates the x-coordinate, (x, y) → (−x, y), and reflecting over the x-axis negates the y-coordinate, (x, y) → (x, −y) — reflecting over the line y = x swaps the coordinates entirely, (x, y) → (y, x).

Worked Example — Reflecting a Point

Reflect the point (3, 7) over the line y = x. Swapping the coordinates gives (7, 3). Reflect the same original point (3, 7) over the x-axis instead: negate the y-coordinate, giving (3, −7).

A rotation turns a figure a fixed angle around a center point. A 90° counterclockwise rotation about the origin follows the rule (x, y) → (−y, x) — which makes sense geometrically: turning a quarter-turn swaps which axis a point's distance is measured along, and reverses one direction, since a point that was to the right of center is now above it, and a point that was above center is now to the left.

Worked Example — Rotating a Point 90°

Rotate the point (5, 2) counterclockwise 90° about the origin. Apply the rule (x, y) → (−y, x): (−2, 5).

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

Check a rigid motion's coordinate rule by verifying it preserves distance on a simple test case, like the distance from the origin to (1, 0) — a genuine isometry rule will always keep that distance exactly 1 no matter which rule you're testing.

Common Mistakes

Reflecting over the wrong axis, such as negating the x-coordinate when reflecting over the x-axis instead of the y-axis.
Reflecting over the x-axis flips vertically, changing the sign of y; reflecting over the y-axis flips horizontally, changing the sign of x — match the axis name to the coordinate it changes.
Assuming a rotation changes a figure's size, since it can look visually different after turning.
Rotation is an isometry — it preserves every distance and angle in the figure. Only its orientation around the center point changes, never its size or shape.

Key Takeaways

Translations, reflections, and rotations are all isometries — rigid motions that preserve every distance and angle.
Each has a specific coordinate rule: translation shifts by a fixed vector, reflection flips over a line, rotation turns around a fixed point.
An isometry's image is always exactly congruent to its preimage.

Summary

Each individual rigid motion preserves distance and angle on its own. The next lesson asks what happens when several transformations are applied one after another in sequence.

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