Three ways to move a shape across the page — without ever changing its size. Set the rule, and watch it glide to a new home.
Start hereThree of them — translation, reflection and rotation — pick a shape up and set it down somewhere new, but they never stretch it or shrink it. Same shape, same size — new spot.
A transformation takes a shape (its starting position is called the object) and produces a new copy (the image) somewhere else on the grid. Sliding, flipping and turning are the three "moves" that keep the shape exactly as big as it started. There's a fourth move — enlargement — and it's the rule-breaker that does change the size. You'll meet all four, and you'll get to drive each one yourself.
Before you can move a shape, you need two names for it. The object is where the shape starts — the original. The image is where it lands after the transformation — the copy in its new position. They are the same shape wearing a different address.
Mathematicians like to mark the image with little dashes, called primes. If a corner of the object is labelled A, the matching corner of the image is A′ (say "A-prime"). So the whole object might be triangle ABC, and its image is triangle A′B′C′. The dashes are just a tidy way of saying "this is the same corner, after it moved."
The shape you start with — the original position. Corners are A, B, C…
The same shape after the move — its new position. Corners are A′, B′, C′…
Here's a habit worth building: a transformation doesn't move "the shape" as one blob — it moves every single point by the same rule, and corner A always maps to corner A′, corner B to B′, and so on. When you check your own work, line up matching corners and ask "did this corner obey the rule?" If every corner did, the whole image is right.
Throughout this page the cyan shape is always the object and the pink shape is always its image. Keep that in your eye, and every move will make sense.
Pick a move. Set its rule with the controls. Then hit Animate and watch the cyan object glide to its pink image on the grid.
Translation by vector (4 right, 2 down): every point of the object slides the same way to its image.
Play for a minute and you'll notice the headline fact of this whole topic: the pink image is always exactly the same size and shape as the cyan object. Wherever you slide, flip or turn it, the shape itself never changes — only where it sits and which way it faces.
A translation slides every point of the shape the same distance in the same direction — like nudging a sticker across a fridge without turning it. The shape doesn't rotate, doesn't flip; it just travels.
How far, and which way? That's captured by a vector: a pair of numbers stacked in a tall bracket, written (3−2). The top number is how far to move across (right is positive, left is negative); the bottom number is how far up or down (up is positive, down is negative). So (3−2) means "3 right and 2 down."
Coordinates make this exact. To translate a point by (3−2) you simply add the vector to the point: a corner at (1, 4) lands at (1 + 3, 4 − 2) = (4, 2). Do that to all the corners and you've drawn the whole image — no protractor, no folding, just adding.
Try it in the playground above: switch to Translate and drag the two sliders. Watch the pink image keep pace with your numbers — left/right with the top slider, up/down with the bottom one.
A reflection flips the shape across a straight line called the mirror line (sometimes the "line of reflection"). Picture folding the paper along that line: the image lands exactly where the object would print through. It's your reflection in a real mirror — same you, but left and right are swapped.
The mirror line follows one beautiful rule that makes reflections easy to predict:
Every point of the image is the same distance from the mirror line as its matching point on the object — just on the opposite side. And the line joining a point to its image always crosses the mirror at a perfect right angle.
You've met reflections your whole life without naming them: the word AMBULANCE printed backwards on the front of the van so it reads correctly in your rear-view mirror, the two halves of a butterfly, the letters in a steamy bathroom mirror. The maths is the same as the glass — distance in, distance out.
In the playground, switch to Reflect and try each mirror: the y-axis (the vertical line x = 0), the x-axis (the horizontal line y = 0), and the slanted line y = x. Each has a tidy coordinate rule — reflecting in the y-axis flips the sign of x so (2, 5) becomes (−2, 5); reflecting in the x-axis flips the sign of y; and reflecting in y = x simply swaps the two coordinates, turning (2, 5) into (5, 2).
A rotation turns the shape around a fixed point called the centre of rotation, by a certain angle, in a chosen direction. Think of a clock hand pinned at the middle: the hand sweeps around, but the pin never moves. The centre is the pin; the shape is the hand.
To describe a rotation completely you need all three pieces of information:
There are tidy coordinate rules for turns about the origin, too. A 90° anticlockwise turn sends (x, y) to (−y, x), a 180° turn sends it to (−x, −y), and a 270° anticlockwise turn sends it to (y, −x). So a corner at (4, 1), turned 90° anticlockwise about (0, 0), lands at (−1, 4). If the centre isn't the origin, you can still do it by hand: count how far the point is across and up from the centre, turn that little arrow, and step it out again.
Notice that a 180° turn looks the same whether you go clockwise or anticlockwise — a half turn ends in the same spot either way. For 90° and 270°, direction really matters. Switch to Rotate above, slide the centre around, and flip between ↺ and ↻ to feel the difference.
Slide, flip or turn — the three moves all share one promise: the image is congruent to the object.
Two shapes are congruent when they are identical in size and shape — you could cut one out and lay it perfectly on top of the other (flipping it over if you need to). Every side length is equal, every angle is equal. Translation, reflection and rotation are called rigid transformations for exactly this reason: nothing bends, stretches or shrinks. They only change position and facing, never measurements. So whenever you only slid, flipped or turned, you can say with total confidence: object and image are congruent.
The cyan object jumped to the pink image. Which single transformation did it? Look at where it landed and which way it faces, then choose.
Which transformation maps the cyan object onto the pink image?
Pick the transformation you think was used.
The giveaway is facing. If the shape points the same way and just moved, it's a translation. If it's been turned to a new angle but still "reads" the same way round, it's a rotation. If it's become a mirror-image — flipped left-to-right or top-to-bottom — it's a reflection.
Do one move, then another on the result. Tap a combo and watch the object travel: cyan object → faint gold halfway shape → pink final image.
Here's the lovely part: a chain of moves can always be replaced by a single move. Two reflections in lines that cross is secretly a rotation about the crossing point. Two reflections in parallel lines is secretly a translation. Slide-then-slide is just one bigger slide. Animators and game designers lean on this all the time — they stack simple moves and let the maths fold them into one.
So far every move kept the shape the same size. Enlargement is the transformation that doesn't. It grows or shrinks the shape from a fixed point (the centre of enlargement) by a number called the scale factor. A scale factor of 2 makes every length twice as long; a scale factor of ½ shrinks it by half.
Because the size changes, the object and image are not congruent. Instead they're similar — same shape, same angles, but scaled. That's the dividing line for this whole topic: the three rigid moves give you congruent images; enlargement gives you a similar one.
These aren't just exam tricks — transformations are everywhere a computer or an artist needs to move something without redrawing it.
The honeycomb on a wall, bathroom tiles, Islamic geometric art — one tile is translated, reflected and rotated again and again to cover a surface with no gaps.
A character walking left is often just the right-facing sprite reflected — one drawing, two directions. Coins spin by rotation; clouds drift by translation.
Cartoons and 3D films move objects frame by frame using translation, rotation and scaling. The computer stores the rule, not a thousand separate drawings.
A butterfly's wings are reflections of each other across its body. Flowers and starfish show rotational symmetry — the same petal turned around a centre.
It's a really common mix-up, and you can see why: a 180° rotation and a reflection can land a symmetric shape in the same place. But for most shapes they give different images, and the difference is one word — handedness.
"Flipping a shape and turning it half-way round do the same thing."
A rotation keeps the shape's handedness — turn the letter R and it's still a normal R, just tilted. A reflection reverses handedness — flip the letter R and you get a backwards Я that no amount of turning can fix. Reflection swaps left and right; rotation never does.
So the test is simple: if the image is a mirror-version of the object (a backwards copy), it was a reflection. If it's the same copy just spun to a new angle, it was a rotation. That single check is what you used in the Spot the move game.
A translation only slides — it never changes size or flips the shape. Object and image are congruent.
Only a reflection reverses handedness — it turns a shape into its mirror-image. Turning and sliding never do that.
Slide along a vector — across and up/down. Same facing.
Flip across a mirror line — same distance, other side. Handedness reversed.
Turn about a centre by an angle and direction. Same handedness.
All three keep the shape congruent — only enlargement changes the size.