MISSION · Geometry
Slides, Flips, and Turns
Three ways to move a shape without changing it
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
A translation, a reflection, and a rotation all move a shape to a new place without altering its size or its shape — the copy is congruent to the original, every time.
The three moves
Translation — a slide. Every point moves the same distance in the same direction. Nothing turns, nothing flips. Describe it with two numbers: how far across, how far up.
Reflection — a flip. Every point jumps to the opposite side of a mirror line, landing exactly as far from the line as it started. Points sitting on the mirror line don’t move at all. Describe it by naming the line.
Rotation — a turn. Every point swings around a fixed centre by the same angle. Describe it with three things: the centre, the angle, and the direction. Leave out the centre and the description is useless — a shape turned \(90°\) about its own corner ends up somewhere completely different from the same shape turned \(90°\) about the origin.
What survives, and what doesn’t
All three preserve lengths and angles. The image is congruent to the original: same size, same shape, same everything except position and facing.
What changes is orientation. A reflection reverses it — that’s why your reflection’s left hand is on your right. Slides and turns don’t reverse anything, which is why you can slide or spin a shape into place with your hands, but you can never lay a left glove onto a right one.
There’s a fourth transformation, enlargement, that does change size. It keeps angles and shape but scales every length by the same factor, so the image is similar to the original rather than congruent.
Coordinates make it arithmetic
On a grid, each move becomes a rule you can apply to any point:
| Move | \((x, y)\) becomes |
|---|---|
| Slide \(a\) right, \(b\) up | \((x + a,\; y + b)\) |
| Reflect in the \(y\)-axis | \((-x,\; y)\) |
| Reflect in the \(x\)-axis | \((x,\; -y)\) |
| Rotate \(90°\) anticlockwise about the origin | \((-y,\; x)\) |
| Rotate \(180°\) about the origin | \((-x,\; -y)\) |
Apply the rule to each corner, plot the results, and join them up. You never have to picture the whole shape moving.
Common mix-ups
- Reflecting in the wrong axis. Reflecting in the \(y\)-axis changes the sign of \(x\), not \(y\) — the axis you flip across is the one that stays put.
- Describing a rotation without its centre. Incomplete, and unmarkable.
- Forgetting direction. Anticlockwise is the default in maths, but \(90°\) clockwise is a different image, so say which.
- Thinking a flipped shape is a different shape. It’s congruent. Same shape, mirrored orientation.
Check yourself
Triangle \(ABC\) has a corner at \((3, 2)\). Where does that corner land after a reflection in the \(y\)-axis? After a \(90°\) anticlockwise rotation about the origin?
Reflection: \((-3, 2)\) — the \(x\) flips sign. Rotation: \((-2, 3)\), applying \((x, y) \to (-y, x)\).
LEVEL 3 · REFLECT
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