GCSE · Maths · AQA · Spec 8300 · Foundation

Transformations — rotation, reflection, translation, enlargement

Slide a shape, turn it, flip it or resize it. Three of those keep its size. The fourth is called an enlargement, and it can shrink a shape.

Geometry · Enlargement

Can an enlargement make a shape smaller?
12.58.04.04.097°97°97°OABCA′B′C′A″B″C″P

OP 12.5. AB 8.0. A′B′ 4.0. A″B″ 4.0. Angle C 97°. Angle C′ 97°. Angle C″ 97°. Classification: Enlargement, centre O. Relationship: OP = scale factor × OA, and OA = 10. With P on A′, OP = 5.0, so the scale factor is ½. Every length halves (AB = 8.0, A′B′ = 4.0), but angle C′ still equals angle C.

OP12.5AB8.0A′B′4.0A″B″4.0Angle C97°Angle C′97°Angle C″97°Drag P along the ray

Enlargement, centre O

OP = scale factor × OA, and OA = 10. With P on A′, OP = 5.0, so the scale factor is ½. Every length halves (AB = 8.0, A′B′ = 4.0), but angle C′ still equals angle C.

Triangle ABC is the object and O is the centre of enlargement. Triangle A′B′C′ is its image with scale factor ½. P is where corner A would land for any scale factor you like: drag it onto A, then onto A′, then out beyond A, and watch OP. Going further: drag P all the way through O to A″. Triangle A″B″C″ is the image with scale factor −½.

Watch out: Don't subtract to find a scale factor. A′ is half as far from O as A is, so the scale factor is ½, not 10 take away 5.
UK note

Higher tier: negative scale factors, like the upside-down image A″B″C″, are Higher-only content.

Now you choose the steps

Describe an enlargement that shrinks

Triangle A has corners (8, 4), (14, 4) and (8, 7). Triangle B has corners (4, 2), (6, 2) and (4, 3). Describe fully the single transformation that maps triangle A onto triangle B.

  1. Triangle B is smaller than triangle A but faces the same way, with the same angles. A change of size means an enlargement.
  2. Read off matching lengths. The bottom of A runs from x = 8 to x = 14, so it is 6 long. The bottom of B runs from x = 4 to x = 6, so it is 2 long.
  3. missing step
Which line is step 3?

All four side by side

What does each transformation keep?

Quick reminder: a translation slides a shape, a rotation turns it about a centre, a reflection flips it over a mirror line, and an enlargement changes its size from a centre. For each row, tick every property the image has, then check the grid.

Translation
Rotation (90° or 180°)
Reflection
Enlargement, scale factor 2
Enlargement, scale factor ½
Enlargement, scale factor −2
UK note

Higher tier: the negative scale factor row is Higher-only content.

Check the answer

Where does this answer go wrong?

Shape A has corners (1, 1), (3, 1) and (1, 2). Shape B has corners (−1, 1), (−1, 3) and (−2, 1). Describe fully the single transformation that maps shape A onto shape B.

A student's answer — which line goes wrong?
Higher

Predict, then check

Two moves, one after the other.

A shape is reflected in the x-axis. Then the image is reflected in the y-axis. Which single transformation takes the original shape straight to the final one?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Before asking what changed, ask what stayed the same. Same size means the shape slid, turned or flipped. A different size means an enlargement, even when the image is smaller.

What you need to know

  • The shape you start with is the object. The shape you end up with is the image.
  • Translation slides, rotation turns about a centre, reflection flips in a mirror line, and enlargement changes size from a centre.
  • Rotation, reflection and translation keep every length and every angle, so the image is congruent to the object: identical in shape and size.
  • Enlargement keeps every angle but multiplies every length by the scale factor, so the image is similar: same shape, different size.
  • A full description gives everything that pins the transformation down. Rotation: centre, angle and direction. Reflection: the mirror line's equation. Translation: a column vector. Enlargement: scale factor and centre.

The big picture

A transformation moves a shape (the object) to a new position (the image). Translation slides it, rotation turns it, reflection flips it and enlargement changes its size. Rotation, reflection and translation keep every length and angle, so the image is congruent. Enlargement keeps every angle but multiplies every length by the scale factor, so the image is similar. A scale factor between 0 and 1 makes the image smaller. To describe a transformation fully, give everything that pins it down.

Key points

1Scale factor = image length ÷ object length. Divide, never subtract.
2A scale factor between 0 and 1 makes the image smaller. It's still called an enlargement.
3The centre of enlargement is where the lines through matching corners of the object and image meet.
4A mirror line is halfway between each point and its image. The line x = 3 is vertical, and y = 3 is horizontal.
5A column vector has two numbers, one above the other. The top number is the move right (negative means left) and the bottom number is the move up (negative means down). On this page it's written as (top over bottom): (3 over −2) means 3 right and 2 down.
6Higher: a negative scale factor puts the image on the other side of the centre, upside down.
7Higher: to combine transformations, follow a point through each one. Two translations combine by adding their column vectors, and changing the order of two different transformations can change the result.

Worked example

Problem

Shape A has corners (1, 3), (1, 5) and (2, 5). Shape B has corners (3, 1), (5, 1) and (5, 2). Describe fully the single transformation that maps shape A onto shape B.

⚠ Watch out

Mixing up vertical and horizontal mirror lines. Every point on the line x = 3 has an x-coordinate of 3, so x = 3 is vertical and y = 3 is horizontal. Check which coordinate stays the same all along the line.

🧠

Memory hook

What stayed the same? Same size: it slid, turned or flipped. New size: enlargement, even if it shrank.

✓

Check yourself

Triangle T has corners (2, 1), (4, 1) and (2, 4). It is translated 5 left and 2 up. Write the column vector, then find the corners of the image.

Flashcards

(10)
Which transformations always give a congruent image?
Rotation, reflection and translation. Every length and every angle stays the same.
What does an enlargement keep, and what does it change?
It keeps every angle and multiplies every length by the scale factor, so the image is similar, not congruent.
What makes a description of a rotation full?
The centre, the angle, and the direction of turn (clockwise or anticlockwise).
What makes a description of an enlargement full?
The scale factor and the centre of enlargement.
How do you describe a reflection fully?
Give the equation of the mirror line, for example y = x or x = 3.
How do you describe a translation fully?
With a column vector: the top number is the move right (negative = left), and the bottom number is the move up (negative = down).
How do you find the scale factor of an enlargement?
Divide a length on the image by the matching length on the object.
What does a scale factor between 0 and 1 do?
It makes the image smaller: every point ends up nearer the centre. It's still an enlargement.
Higher: what does a negative scale factor do?
It puts the image on the opposite side of the centre, upside down. The number without its minus sign sets the size.
What is an invariant point?
A point that stays in the same place after a transformation, such as the centre of a rotation or any point on a mirror line.

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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