GCSE · Maths · AQA · Spec 8300 · Foundation

Fractional scale factors for enlargement

Here's an odd one: an enlargement can make a shape smaller. Give it a scale factor like ⅓ and watch it shrink in towards a single point.

Maths · Enlargement

Where does A′ belong?

OA 15.0. OA′ 11.5. AB 6.0. A′B′ 5.0. AC 6.7. A′C′ 7.2. Angle A 63.4°. Angle A′ 3.9°. Classification: Enlargement · scale factor ⅓ · centre O. Relationship: Each image vertex lies on the line from O through its object vertex, ⅓ as far from O. Get that right and every image length is ⅓ of the object length — and every angle stays the same.

OA15.0OA′11.5AB6.0A′B′5.0AC6.7A′C′7.2Angle A63.4°Angle A′3.9°Slide A′ until A′B′C′ is a perfect mini ABC

Enlargement · scale factor ⅓ · centre O

Each image vertex lies on the line from O through its object vertex, ⅓ as far from O. Get that right and every image length is ⅓ of the object length — and every angle stays the same.

Triangle ABC is being enlarged by scale factor ⅓, centre O. B′ and C′ are already in place. Drag A′ along its line from O and watch the measurements change.

Watch out: Measure from the centre O, not from the triangle. A′ sits ⅓ of the way out from O to A, not ⅓ of the way back from A.

What do you think?

Scale factor ½ — what actually happens?

Triangle T is enlarged by scale factor ½ with centre O. O is outside the triangle.

Which is closest to what you think happens?

Every corner ends up half as far from O, so every side halves and T shrinks in towards O.It gets bigger. That's what 'enlargement' means.Every side gets shorter by the same amount, say 2 cm each.Keep one corner of T where it is and halve the sides from that corner.

How sure are you?

I'm sureI think soI'm guessing

Every corner ends up half as far from O, so every side halves and T shrinks in towards O.

This matches the evidence

Spot on. Each corner stays on its own line from O but finishes half as far away. Halve every one of those journeys and every side halves too, while the angles stay exactly as they were. You get a half-size copy of T, pulled in towards O.

It gets bigger. That's what 'enlargement' means.

This is a common misconception

Completely fair guess from the name, but in maths an enlargement means multiplying every distance from the centre by the scale factor. Multiply by ½ and everything is half as big. A scale factor bigger than 1 makes the image bigger; a scale factor between 0 and 1 makes it smaller. Both are enlargements.

Every side gets shorter by the same amount, say 2 cm each.

This is a common misconception

Scale factors multiply — they never subtract. Take 2 cm off a 10 cm side and a 3 cm side and you get 8 cm and 1 cm: the shape gets distorted. Halving gives 5 cm and 1.5 cm, and the shape stays the same.

Keep one corner of T where it is and halve the sides from that corner.

This is a common misconception

That only works if O happens to be that corner. Here the centre is O, so every distance is measured from O. Measure from a corner of T instead and you'd draw a half-size triangle — but in a different place from the real image.

Scale factor ½ halves every distance from the centre — so the image is half the size and closer to the centre.

Worked example

Problem

Enlarge triangle PQR by scale factor ½, centre (1, 1). P is at (5, 7), Q is at (9, 7) and R is at (9, 3).

Your turn

Finish the enlargement

Enlarge triangle DEF by scale factor ¼, centre (2, 1). D is at (6, 9), E is at (10, 9) and F is at (10, 5).

The centre is (2, 1) and the scale factor is ¼. Every journey starts at (2, 1).
missing step

Which line is step 2?

Journeys from the centre: D is 6 right, 9 up; E is 10 right, 9 up; F is 10 right, 5 up.Journeys from the centre: D is 4 right, 8 up; E is 8 right, 8 up; F is 8 right, 4 up.Journeys from the centre: D is 8 right, 10 up; E is 12 right, 10 up; F is 12 right, 6 up.

Same shape — same size?

Congruent or similar?

Is each image congruent to its object, or similar but not congruent?

Still to sort

Congruent (0)

Same shape and same size: every length and every angle matches.

Where the line is: If the angles match but the lengths have changed, it isn't congruent — it's similar.

Similar, not congruent (0)

Same shape, different size: angles match and every length is multiplied by the same scale factor.

Where the line is: Congruent shapes are similar too, so this group is only for images whose size has changed.

6 of 6 still to sort.

Run it backwards

Describe it fully

Triangle P has vertices (4, 7), (10, 7) and (10, 4). Triangle Q has vertices (2, 3), (4, 3) and (4, 2). Two tools for running an enlargement backwards: • Scale factor = an image length ÷ the matching object length. • Centre: draw a line through each vertex of P and its matching vertex on Q. Every image vertex sits on a line from the centre — just like A′ on its line from O — so these lines all meet at the centre.

Describe fully the single transformation that maps triangle P onto triangle Q. [3 marks]

Your answer

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

The mark scheme

Names the transformation: an enlargement. (1 mark)
Scale factor ⅓ — image length ÷ object length, for example 2 ÷ 6. (1 mark)
Centre (1, 1) — the point where lines through matching vertices meet. (1 mark)

Model answer

An enlargement, scale factor ⅓, centre (1, 1). The top side of P is 6 long and the matching side of Q is 2 long, and 2 ÷ 6 = ⅓. The line through (4, 7) and (2, 3) and the line through (10, 7) and (4, 3) both pass through (1, 1).

A full description has three parts: the word 'enlargement', the scale factor and the centre. Leave one out and the description isn't full.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

When an enlargement makes a shape smaller — and the one rule that works either way.

What you need to know

To enlarge a shape you need two things: the centre of enlargement and the scale factor.
Each vertex's distance from the centre is multiplied by the scale factor. The image vertex lies on the line from the centre through the object vertex.
Every length in the shape is multiplied by the scale factor, and the angles stay the same — so the image is similar to the object.
A scale factor between 0 and 1, such as ½ or ⅓, makes the image smaller and pulls it in towards the centre.
To describe an enlargement fully, give the word 'enlargement', the scale factor and the centre.

The big picture

To enlarge a shape you need a centre of enlargement and a scale factor. Every vertex's distance from the centre is multiplied by the scale factor, and so is every length in the shape; the angles don't change. A fractional scale factor between 0 and 1, such as ½ or ⅓, gives an image that is smaller than the object — the same shape (similar), just smaller and pulled in towards the centre.

Key points

1Method: count the journey from the centre to each vertex, multiply it by the scale factor, then step out from the centre again.

2Only multiply the coordinates themselves when the centre is (0, 0).

3Scale factor = image length ÷ object length.

4To find the centre, draw lines through each object vertex and its image vertex; they meet at the centre.

5Rotations, reflections and translations give congruent images. An enlargement changes the size whenever the scale factor, ignoring any minus sign, isn't 1 — and then the image is similar but not congruent. In this lesson that means a scale factor between 0 and 1: the image is smaller, so it's similar but not congruent.

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Worked example

Problem

Enlarge the square with vertices (2, 1), (8, 1), (8, 7) and (2, 7) by scale factor ⅓, centre (5, 4).

⚠ Watch out

Getting the scale factor upside down when describing an enlargement. It is image length ÷ object length: a side that goes from 6 to 2 gives 2 ÷ 6 = ⅓, not 6 ÷ 2 = 3. A scale factor of 3 would make the shape bigger, not smaller.

🧠

Memory hook

Start at the centre, multiply the journey. ×½ lands you halfway out; ×⅓ lands you a third of the way out.

✓

Check yourself

A 9 cm side is enlarged by scale factor ⅓ from a centre C. How long is the image side, and does the image end up closer to C or further away? (3 cm; closer.)

Flashcards

(13)

What two things do you need to perform an enlargement?

A centre of enlargement and a scale factor.

In an enlargement, what happens to a vertex's distance from the centre?

It is multiplied by the scale factor. With scale factor ⅓, a vertex 12 units from the centre ends up 4 units from it.

Where does an image vertex lie, compared with the centre and its object vertex?

On the straight line from the centre through the object vertex.

What happens to the lengths of a shape when it is enlarged?

Every length is multiplied by the scale factor.

Do the angles change in an enlargement?

No. The angles stay the same, which is why the image is the same shape.

What does a scale factor between 0 and 1 do to the image?

Makes it smaller, pulled in towards the centre of enlargement.

Is it still called an enlargement if the shape gets smaller?

Yes. In maths, an enlargement multiplies distances from a centre by a scale factor — whether the image grows or shrinks.

Similar vs congruent — what's the difference?

Similar shapes have the same angles but can be different sizes. Congruent shapes match exactly: same angles and same size.

Which transformations always give a congruent image?

Rotation, reflection and translation — they move a shape without changing its size.

How do you find the scale factor from an object and its image?

Divide an image length by the matching object length.

How do you find the centre of an enlargement from a drawing?

Draw lines through each object vertex and its matching image vertex. The point where they meet is the centre.

'Describe fully' an enlargement — what must your answer include?

The word 'enlargement', the scale factor, and the coordinates of the centre.

When can you just multiply the coordinates by the scale factor?

Only when the centre of enlargement is the origin, (0, 0). Otherwise count the journey from the centre.

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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Scale factor ½ — what actually happens?

Finish the enlargement

Congruent or similar?

Describe it fully

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