Transformations
Enlargement
In this note
- 7.1Draw an enlargement from a centre with a whole-number or fractional scale factor.
- 7.1Use a negative scale factor .
- 7.1Describe an enlargement: find the scale factor and the centre.
Before you start: Rotation. Next: Combined transformations.
Formula bank
Drawing an enlargement
An enlargement changes the size but keeps the shape. Every distance from the centre is multiplied by the scale factor k, so every side is multiplied by k too. Angles stay the same.
General methodHow to draw an enlargement
- Mark the centre.
- For each corner, count how far it is from the centre (across, up).
- Multiply both counts by k and count that far from the centre. Mark the new corner.
- Join the corners. Check: each side is k times as long.
Example Enlarge the triangle \((3,1),(3,2),(1,2)\) by scale factor 2, centre \((1,1)\).
From the centre, double the steps.
- \((3,1)\): 2 across \(\to\) 4 across: \((5,1)\)
- \((3,2)\): (2, 1) \(\to\) (4, 2): \((5,3)\)
- \((1,2)\): (0, 1) \(\to\) (0, 2): \((1,3)\)
Answer \((5,1),(5,3),(1,3)\)
A scale factor between 0 and 1 makes the shape smaller: with \(k=\tfrac12\), centre O, \((4,6)\to(2,3)\).
Try it yourself
Show answer
Answer (6, 3)
Show answer
Answer (4, 3)
Negative scale factors
With a negative scale factor, go THROUGH the centre to the other side. \(k=-2\) means twice as far, on the opposite side; the image comes out upside down. Centre O, \(k=-2\): \((2,1)\to(-4,-2)\).
Try it yourself
Show answer
Answer (−1, 0)
Describing an enlargement
General methodHow to describe an enlargement fully
- Scale factor: image side ÷ object side (negative if the image is upside down on the other side).
- Centre: join each corner to its image with a straight line and extend; the lines meet at the centre.
- Answer: 'Enlargement, scale factor k, centre (a, b)'.
Example A: \((1,1),(2,1),(1,2)\). B: \((3,3),(5,3),(3,5)\). Describe the transformation.
Bottom side 1 → 2, so k = 2. Lines through matching corners meet at the centre.
- \((1,1)\to(3,3)\) and \((2,1)\to(5,3)\) both extend back to \((-1,-1)\)
- check: \((-1,-1)\) + 2 × (2, 2) = \((3,3)\) ✓
Answer Enlargement, scale factor 2, centre (−1, −1)
Ways they test it
Type 1 Draw an enlargement
"Enlarge shape A by scale factor 3, centre (1, 2)." (2-3 marks)
Spot it a centre and a scale factor.
- Count from the centre to each corner.
- Multiply by k.
Type 2 Fraction or negative k
"…scale factor \(-\tfrac12\), centre (2, 0)." (2 marks)
Spot it a fraction or a minus sign.
- Fraction: closer to the centre.
- Negative: through the centre to the other side.
Type 3 Describe
"Describe fully the single transformation." (3 marks)
Spot it the image is a different size.
- Enlargement + k + centre.
- Rays to find the centre.
For example \((1,1)\to(3,3)\), \((2,1)\to(5,3)\)
- k = 2, rays meet at (−1, −1)
Answer Enlargement, k = 2, centre (−1, −1)
Type 4 Area after enlarging
"The area of A is 4 cm². Find the area of the image." (2 marks)
Spot it area with a scale factor.
- Multiply by \(k^{2}\).
Mistakes to avoid
- ✗ Multiplying the coordinates when the centre is not O✓ multiply the distances FROM THE CENTRE
- ✗ Calling a scale factor ½ a 'reduction'✓ it is still an 'enlargement, scale factor ½'Use the exam word.
- ✗ Leaving out the centre✓ enlargement + scale factor + centre
- ✗ Area × k✓ area × \(k^{2}\)
Summary
- Enlargement = centre + scale factor.
- Distances from the centre × k; k < 1 shrinks, negative k flips through the centre.
- Find the centre where the corner-to-image lines meet.
- Area scales by \(k^{2}\).
If you see …
- a different sizeenlargement.
- upside down AND across the centrenegative scale factor.
Practice
Practice: easy, medium, hard
Easy
Show answer
Answer (2, 3)
Show answer
Answer 3
Show answer
Answer 36
Medium
Guide me
What is it asking? a point
What do we know? from the centre: (3, 2)
Plan × 3: (9, 6); add (2, 1)
Show working
- (11, 7)
Answer (11, 7)
Guide me
What is it asking? a point
What do we know? halve the distances from O
Plan \((-1,2)\)
Show working
- (−1, 2)
Answer (−1, 2)
Guide me
What is it asking? a point
What do we know? × −2 from O
Plan other side
Show working
- (−4, −2)
Answer (−4, −2)
Hard
Guide me
What is it asking? a point
What do we know? from the centre: (2, 1)
Plan × −2: (−4, −2); add (1, 0)
Show working
- (−3, −2)
Answer (−3, −2)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| (−3, −2) | B2 | ||
| Total: 2 marks | |||
Guide me
What is it asking? a point
What do we know? k = 2
Plan centre = 2 × object − image
Show working
- (−1, −1)
Answer (−1, −1)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| k = 2 | B1 | ||
| (−1, −1) | B1 | ||
| Total: 2 marks | |||
Guide me
What is it asking? a point
What do we know? from the centre: (2, 1)
Plan reverse: (−2, −1); add (1, 1)
Show working
- (−1, 0)
Answer (−1, 0)
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| (−1, 0) | B1 | ||
| Total: 1 mark | |||
Can I do this?
- I can draw an enlargement with whole and fraction scale factors.
- I can use a negative scale factor.
- I can find the scale factor and centre of an enlargement.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
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