KS3 · Maths

Translating shapes

Drag a photo across your screen. It doesn't flip, spin or stretch — it just ends up somewhere else. That's a translation, and two numbers say exactly where.

Finish the slide

Where does C′ have to go?
5.02.03.23.63.65.852°25°ABCA′B′C′

B to B′: across 5.0. C to C′: across 2.0. AC 3.2. A′C′ 3.6. BC 3.6. B′C′ 5.8. angle C 52°. angle C′ 25°. Classification: Translation. Relationship: It's only a translation when every corner makes the same move — here, right 5 and up 2. Then every length and every angle matches the object. Only the position has changed.

B to B′: across5.0C to C′: across2.0AC3.2A′C′3.6BC3.6B′C′5.8angle C52°angle C′25°slide C′ along its row

Translation

It's only a translation when every corner makes the same move — here, right 5 and up 2. Then every length and every angle matches the object. Only the position has changed.

Triangle ABC is the object. A′B′C′ is its image — say 'A prime' for A′: it's the corner that A moved to. B has slid 5 squares right and 2 squares up to reach B′, and A has made exactly the same move to A′. C′ is already 2 squares up, on the right row, but in the wrong place. Drag C′ along its row until the image is a perfect copy of the object.

Watch out: Close isn't good enough. Put C′ one square too far and angle C′ can look right while A′C′ and B′C′ are both wrong. Only one spot makes every reading match.

Translation or not?

Sort the pairs

Is each image a translation of its object?

Still to sort

A translation (0)

Same shape and size, facing the same way, just somewhere else.

Not a translation: flipped or turned (0)

Same shape and size, but facing a different way.

Where the line is: Same size and shape is not enough. If you'd have to flip it over or twist it round to make it fit, it wasn't a slide.

Not a translation: not congruent (0)

A length or angle has changed.

Where the line is: Check the lengths before anything else. A slide never stretches or shrinks a shape.

7 of 7 still to sort.

Each pair is an object and an image. Decide whether a translation took place — and if not, what gives it away.

Predict, then check

Meet the column vector: two numbers stacked in tall brackets, with no line between them. The top number is the move across (positive means right, negative means left). The bottom number is the move up or down (positive means up, negative means down). We'll write it here as 'top 2, bottom 4'.

Kai translates a triangle by the column vector top 2, bottom 4. Mia translates the same triangle by top 4, bottom 2. How do their two moves compare?

Say it properly

Describe a translation in full

Shape P has corners at (1, 5), (3, 5) and (1, 8). Shape Q has corners at (4, 3), (6, 3) and (4, 6). Q is the image of P. Describe the translation fully in words. Then write it using a column vector. [4 marks]

0 words · your answer stays on this page and is not sent anywhere.

Your turn to slide

Translate a triangle by a vector

Triangle ABC has A(3, 2), B(5, 2) and C(5, 5). Translate it by the column vector top −2, bottom 4.

  1. The vector is top −2, bottom 4.Top number: across. Bottom number: up or down.
  2. missing step
Which line is step 2?

Spot the slip

Which corners did they count between?

Triangle FGH has F(1, 1), G(4, 1) and H(1, 3). After a translation, its image is F′(6, 2), G′(9, 2) and H′(6, 4). Describe the translation.

A student's answer — which line goes wrong?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Slide a shape without flipping, turning or stretching it. Everything about it stays the same — except where it is.

What you need to know

  • A translation slides a shape. Every point moves the same distance in the same direction.
  • The object is the original shape; the image is where it ends up. A′ (A prime) is the image of corner A.
  • Lengths, angles and orientation are invariant (they don't change). Only the position changes, so the image is congruent and faces the same way.
  • Congruent isn't enough: if the image has been flipped or turned, it isn't a translation.
  • Describe it fully: translated, left or right and how far, up or down and how far. Leave out a direction with no move.
  • A column vector gives both moves: top = across (negative is left), bottom = up or down (negative is down). Write 0 where there's no move.
  • Count the move between matching corners (A to A′), never between the two closest corners.
  • Displacement is the distance from the starting point in a straight line, together with its direction. A translation's displacement is given as a move across and a move up or down.

The big picture

A translation slides a shape: every point moves the same distance in the same direction. The original is the object; the moved shape is its image, and A′ marks where corner A went. Lengths, angles and orientation stay the same — only the position changes. Describe it fully: translated, the move left or right, and the move up or down, each with a distance. A column vector says the same in two numbers: across on top, up or down underneath, with negatives for left and down.

Key points

1A translation is a slide: nothing flips, twists or stretches.
2Invariant means doesn't change. Under a translation, lengths, angles and orientation are invariant; only position changes.
3With coordinates: the same amount is added to (or taken from) every x, and the same amount to every y.
4An increase in x is a move right, a decrease is left. An increase in y is up, a decrease is down.
5'Moved across and down' is incomplete — no direction for 'across' and no distances.
6In a column vector the order of the numbers matters (across on top). The order you do the two moves in doesn't.
7Why the same move for every point? The gap between x = 2 and x = 5 is 3. Add 5 to both and it's still 3; add different amounts and the gap — and so a length — changes.
8On a grid the grid's units are assumed; with no grid, give a unit such as cm.

Worked example

Problem

A translation moves the point (−3, 0) to (2, −2). Describe the translation in words and as a column vector. Then find where the point (−1, 3) goes.

⚠ Watch out

Swapping the numbers in a column vector. The top number is always the move across (right is positive, left is negative) and the bottom is always the move up or down (up is positive, down is negative) — the same order as coordinates.

🧠

Memory hook

Slide, don't spin. Across on top, up-or-down underneath — the same order as (x, y).

✓

Check yourself

Translate (6, 1) by the column vector top −4, bottom 3. Where does it land? Describe the move in words. (Answers: (2, 4); translated left by 4 squares and up by 3 squares.)

Flashcards

(15)
What is a translation?
A transformation that slides a shape: every point moves the same distance in the same direction.
What do 'object', 'image' and A′ mean?
The object is the original shape and the image is where it ends up. A′ (A prime) is the corner of the image that A moved to.
What does 'invariant' mean?
Doesn't change.
Under a translation, which properties are invariant — and which one changes?
Lengths, angles and orientation stay the same. Only the position changes.
Two shapes are congruent. Is one definitely a translation of the other?
No. It's only a translation if you can slide one onto the other with no flipping and no twisting.
How can coordinates show that a translation has happened?
The same amount has been added to (or taken from) every x coordinate, and the same amount to every y coordinate.
x goes up, x goes down, y goes up, y goes down — which moves are these?
x up: right. x down: left. y up: up. y down: down.
What must a full description of a translation in words include?
That it's translated, plus the move left or right and the move up or down — each with how far.
Why is 'moved across and down' not a full description?
'Across' doesn't say left or right, and neither move has a distance.
In a column vector, what do the top and bottom numbers mean?
Top: the move across (positive right, negative left). Bottom: the move up or down (positive up, negative down).
A shape moves 2 left and doesn't move up or down. How do you write that in words, and as a vector?
Words: translated left by 2 squares (no need to mention up or down). Vector: top −2, bottom 0 — the 0 must be written.
Does top 2, bottom 4 give the same image as top 4, bottom 2?
No. The first is right 2, up 4; the second is right 4, up 2. The order of the numbers matters.
Does it matter whether you move across first or up/down first?
No — you land in the same place either way. Doing across first is just a good habit, like coordinates.
Between which corners do you count the move?
A corner and its own image (A to A′), not the two corners that happen to be closest.
When do you need to give units like cm in a description?
When there's no grid. On a grid, the grid's units are assumed.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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