Rotation and Translation

Transformations: moving shapes around

In geometry, a transformation is a way of moving or changing a shape. Three of the most important transformations are reflection (flipping), rotation (turning) and translation (sliding). This lesson focuses on the two "moving" transformations — rotation and translation — and you can revisit flipping in our lesson on symmetry and reflection.

All three of these transformations are rigid motions, which means the shape never gets bigger, smaller or squashed. The shape that comes out (called the image) is congruent to the shape you started with (the object): same side lengths, same angles, same area. Only its position changes — and for rotation, also its orientation (the way it faces).

We usually carry out transformations on a coordinate grid, where every point has an (x, y) address. If you need a refresher on plotting points, see our lesson on the coordinate plane.

Translation: sliding a shape

A translation slides every point of a shape by the same distance in the same direction. Nothing turns and nothing flips — it is exactly like pushing a book across a table.

We describe a translation with a vector, written as a pair of numbers in a column or in brackets, like (3, −2):

• The top (first) number tells you the horizontal move: positive means right, negative means left.
• The bottom (second) number tells you the vertical move: positive means up, negative means down.

So the vector (3, −2) means "move 3 right and 2 down". To translate a whole shape, you simply add the vector to every vertex (corner) of the shape, then join the new points up in the same order.

Worked example 1: translating a triangle

A triangle has vertices at A(1, 1), B(4, 1) and C(1, 3). Translate it by the vector (2, 3).

Add 2 to every x-coordinate and 3 to every y-coordinate:

• A(1, 1) → A′(1 + 2, 1 + 3) = (3, 4)
• B(4, 1) → B′(4 + 2, 1 + 3) = (6, 4)
• C(1, 3) → C′(1 + 2, 3 + 3) = (3, 6)

The new triangle A′B′C′ has exactly the same size and shape as the original — it has just slid 2 squares right and 3 squares up. (We use the little dash, called a "prime", so A′ means "the image of A".)

Rotation: turning a shape

A rotation turns a shape around a fixed point called the centre of rotation. Think of the centre as a pin pushed through the paper: the shape spins around that pin like the hand of a clock.

To describe a rotation fully, you must state three things:

1. The centre of rotation (the point you turn around — often the origin, (0, 0)).
2. The angle of turn (commonly 90°, 180° or 270°).
3. The direction: clockwise (the way clock hands move) or anticlockwise (the opposite way).

A 180° rotation is special: clockwise and anticlockwise give the same result, so you do not need to state a direction for it.

A diagram in words

Picture a clock face with its centre at the origin. Place a small flag with its base at the point (4, 0) — that is straight out to the right, where the "3 o'clock" mark is.

Now rotate the flag 90° clockwise about the origin. The flag swings down to point straight down, landing at (0, −4) — the "6 o'clock" position. Rotate another 90° clockwise and it reaches (−4, 0), out to the left (9 o'clock). A final 90° brings it to (0, 4), straight up (12 o'clock). After four turns of 90° it is back where it started — a full 360°.

Worked example 2: rotating a point 90° clockwise

Rotate the point (4, 0) by 90° clockwise about the origin.

Following the clock-face picture above, a point on the positive x-axis swings down onto the negative y-axis. So (4, 0) maps to (0, −4).

A useful rule for rotations about the origin: for a 90° clockwise turn, the point (x, y) becomes (y, −x). Check it: (4, 0) → (0, −4). It works.

Worked example 3: rotating a point 180°

Rotate the point (2, 5) by 180° about the origin.

A 180° rotation about the origin sends (x, y) to (−x, −y) — both signs flip. So (2, 5) maps to (−2, −5). The point ends up diagonally opposite, on the other side of the origin, the same distance away.

How rotation and translation differ

It is easy to mix these up, so here is the key contrast:

Feature	Translation	Rotation
Action	Slides	Turns
Orientation changes?	No (same way up)	Yes (the shape faces a new way)
Described by	A vector (x, y)	Centre, angle and direction
Size changes?	No	No

In both cases the image is congruent to the object. The big difference is that translation keeps the shape facing the same way, while rotation makes it face a new direction.

Why this matters

Rotation and translation are not just classroom exercises — they describe how things actually move in the real world. Every time a video game character walks across the screen, the program translates their sprite. Every time a wheel turns, a robot arm pivots, or a planet spins, that is a rotation about a centre. Animators, engineers and designers describe motion using exactly these vectors and angles. Understanding transformations also deepens your grasp of symmetry: a shape has rotational symmetry if it looks the same after a rotation of less than a full turn (a square looks identical every 90°). Mastering how to move shapes precisely on a grid is the foundation for computer graphics, robotics and the coordinate geometry you will meet again and again.

Activity: transformation challenge

You need squared paper, a pencil, and a piece of tracing paper.

1. Draw and label a small "L-shaped" tromino (three joined squares) on a coordinate grid and label its corners.
2. Translate it by the vector (5, 0), then by (0, −4), and finally by (−5, 4). Where does it end up? You should find the three slides cancel out and it returns close to the start — predict before you check!
3. Rotate the original L-shape 90° clockwise about the origin using the tracing paper: trace the shape, hold the pencil tip on the origin, and physically turn the tracing paper a quarter turn. Mark where the corners land and read off their coordinates.
4. Compare your rotated coordinates with the rule (x, y) → (y, −x). Do they match?
5. Challenge: can you find a single translation that has the same effect as two translations done one after another? What is the connection between its vector and the two original vectors?