KS3 · Maths

Enlarging shapes (positive integer scale factor)

Zoom in on a photo and everything grows, but nothing bends out of shape. That's enlargement: every length multiplied, every angle untouched, and one point deciding where it all lands.

Maths · Geometry

Where does A′ belong?

O to A 5.0. O to A′ 8.0. side AB 3.0. side A′B′ 7.4. angle at A 90°. angle at A′ 65°. Classification: enlargement, scale factor 2. Relationship: Scale factor 2 from O: O to A′ = 2 × 5 = 10, and only there is A′B′ = 2 × 3 = 6 and the angle at A′ = 90°, the same as at A.

O to A5.0O to A′8.0side AB3.0side A′B′7.4angle at A90°angle at A′65°slide A′ along its line from O

enlargement, scale factor 2

Scale factor 2 from O: O to A′ = 2 × 5 = 10, and only there is A′B′ = 2 × 3 = 6 and the angle at A′ = 90°, the same as at A.

Triangle A′B′C′ is meant to be triangle ABC enlarged by scale factor 2 from the centre O. B′ and C′ are already in the right places, each on a faint line from O. A′ is not. Drag A′ along its line from O and watch the readings: find the one spot where the image becomes a true enlargement.

Watch out: The angle at A′ matches the 90° at A only at that one spot. Angles are never multiplied by the scale factor — they stay the same.

Maths · Geometry

What does an enlargement actually do?

Be honest — which of these is closest to what you thought before this lesson?

'Enlarge' means make bigger, so an enlargement can never make a shape smaller.Everything gets multiplied by the scale factor — the side lengths and the angles.You enlarge a shape by adding the same amount onto every side.It changes the size, multiplies every length by the scale factor, and leaves every angle the same.

How sure are you?

I'm sureI think soI'm guessing

'Enlarge' means make bigger, so an enlargement can never make a shape smaller.

This is a common misconception

It's a very natural reading of the word, but in maths an enlargement is any change of size: the shape can be stretched or shrunk. What makes it an enlargement is that the shape itself doesn't change. In this lesson every scale factor is a whole number, so our images come out bigger — but keep in mind that the word covers shrinking too.

Everything gets multiplied by the scale factor — the side lengths and the angles.

This is a common misconception

Try it. A triangle with angles 40°, 60° and 80° enlarged by scale factor 2 would get 80°, 120° and 160°. Those add up to 360° — that isn't a triangle any more! The angles don't change at all: they are invariant. Only lengths, and distances from the centre, get multiplied. That's exactly why the image is the same shape.

You enlarge a shape by adding the same amount onto every side.

This is a common misconception

Adding usually changes the shape. A 1 cm by 4 cm rectangle with 2 cm added to each side becomes 3 cm by 6 cm: the short side has tripled but the long side hasn't even doubled, so the shape has changed. An enlargement MULTIPLIES every length by the same number, the scale factor.

It changes the size, multiplies every length by the scale factor, and leaves every angle the same.

This matches the evidence

Spot on. Size changes, shape doesn't. Every length — and every distance from the centre — is multiplied by the scale factor, and every angle in the image equals the matching angle in the object. That's why we say the image is similar to the object.

Enlargement = a change of size. Lengths get multiplied; angles don't budge.

How to enlarge a shape on a grid

Problem

Enlarge triangle PQR by scale factor 2 from the centre X. On the grid, X is at (1, 8), P is at (2, 7), Q is at (4, 7) and R is at (2, 5).

Maths · Geometry

Is one rectangle an enlargement of the other?

Sort each pair: similar (one is an enlargement of the other) or not similar?

Still to sort

Similar — an enlargement (0)

Every side multiplied by the same number.

Where the line is: One multiplier for BOTH pairs of sides. If the height and width use different multipliers, it's not an enlargement.

Not similar (0)

The sides changed by different multipliers, or by adding.

Where the line is: Both rectangles getting bigger isn't enough — the multipliers have to match.

6 of 6 still to sort.

Check the multipliers. If every side of the small rectangle is multiplied by the same number to give the big one, they're similar.

Your turn

Find the scale factor, then a missing length

A triangle has sides of 3 cm, 5 cm and 6 cm. It is enlarged, and the side that was 3 cm becomes 12 cm. How long is the image of the 5 cm side?

Find a pair of corresponding lengths where we know both: the 3 cm side and its image, 12 cm.
missing step

Which line is step 2?

Scale factor = 12 ÷ 3 = 4Scale factor = 12 − 3 = 9Scale factor = 12 ÷ 5 = 2.4Scale factor = 3 ÷ 12 = 0.25

Find the centre

Commit to a point before you reveal.

Triangle ABC has vertices A(3, 2), B(5, 2) and C(3, 3). After an enlargement, its image has vertices A′(5, 4), B′(11, 4) and C′(5, 7). If you drew straight lines through A and A′, B and B′, and C and C′, where would they meet?

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Same shape, new size — and one special point decides where the new shape lands.

What you need to know

An enlargement changes a shape's size. Every length is multiplied by the scale factor, but every angle stays exactly the same.
The centre of enlargement decides where the image goes: each image vertex sits on the line from the centre through its object vertex, scale factor times as far from the centre.
Scale factor = length on the image ÷ the corresponding length on the object.
To find the centre, draw straight lines through matching vertices of the object and the image. They cross at the centre.
A full description of an enlargement gives the scale factor AND the centre of enlargement.

The big picture

An enlargement changes the size of a shape without changing its shape. Every side length, and every distance from the centre of enlargement, is multiplied by the scale factor, while every angle stays exactly the same, so the image is similar to the object. You find a scale factor by dividing an image length by its corresponding object length, find the centre by drawing lines through matching vertices, and describe an enlargement fully by giving both the scale factor and the centre.

Key points

1Enlargement = same shape, new size. The image is similar to the object.

2Lengths and distances from the centre are multiplied by the scale factor; angles don't change.

3Only divide lengths that correspond when you work out a scale factor.

4A point at the centre of enlargement stays where it is.

5With positive scale factors like these, the object and image face the same way.

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Worked example

Problem

Rectangle ABCD is 2 squares wide and 1 square tall. Its bottom-left corner A is 1 square right and 2 squares up from the centre O. Enlarge ABCD by scale factor 3 from O, using the shortcut: find one image vertex, then build the rest from the side lengths.

⚠ Watch out

Counting from the object instead of from the centre. The image vertex is scale factor times as far from the CENTRE as the object vertex — so every count starts at the centre, not at the shape.

🧠

Memory hook

Centre says WHERE, scale factor says HOW BIG — and the angles never budge.

✓

Check yourself

Scale factor 4, centre K: what does a kite's 110° angle become? A vertex 3 squares right and 1 up from K — where is its image? (Still 110°; 12 right, 4 up from K.)

Flashcards

(14)

What does an enlargement change, and what does it keep the same?

It changes the size (every length is multiplied by the scale factor). It keeps the shape: every angle stays the same.

Can an enlargement make a shape smaller?

Yes. In maths, enlargement means a change of size — a stretch or a shrink. The whole-number scale factors in this lesson all make the image bigger.

What is a scale factor?

The multiplier from the object to the image: the number every length is multiplied by.

How do you work out a scale factor from two shapes?

Length on the image ÷ the corresponding length on the object.

What does it mean for two shapes to be similar?

The only difference between them is their size: same angles, and side lengths in the same proportions. An enlarged image is similar to its object.

How can you test whether two rectangles are similar?

Check the multipliers: the same number must take each side of one to the matching side of the other. If the multipliers differ, they're not similar.

What is the centre of enlargement?

The point a shape is enlarged from. Where it is decides where the image lands.

A vertex is 4 squares from the centre of enlargement. The scale factor is 3. How far from the centre is its image vertex?

12 squares — along the same line from the centre.

How do you check an image is in the right place?

Draw lines from the centre through each object vertex. Every image vertex should sit on one of those lines.

What happens to a point that is exactly at the centre of enlargement?

It stays where it is: it's 0 from the centre, and 0 × any scale factor is still 0.

How do you find the centre of enlargement from an object and its image?

Draw straight lines through pairs of corresponding vertices. The centre is where they cross.

What must a full description of an enlargement include?

The scale factor AND the centre of enlargement (as coordinates).

On a grid, how do you place an image vertex?

Count across and up or down from the centre to the object vertex, multiply both counts by the scale factor, then count that from the centre.

Do the object and its image face the same way?

Yes, for positive scale factors: an enlargement doesn't turn or flip the shape.

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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What does an enlargement actually do?

Is one rectangle an enlargement of the other?

Find the scale factor, then a missing length

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