Enlargement
Section: Position and Transformation | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
What is an Enlargement?
Unlike translation, reflection, and rotation, an enlargement changes the size of a shape. The image is similar to the object (same shape, same angles) but not congruent, unless the scale factor is 1.
- Scale factor (k): how many times bigger every length becomes.
- Centre of enlargement: the fixed point the enlargement is measured from.
Enlarging a Shape from a Centre
For each vertex: find the vector from the centre to that vertex, multiply it by the scale factor, then add it back to the centre. Image point = Centre + k × (Object point − Centre).
An enlargement with scale factor 2, from a centre outside the shape
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Question: Enlarge point A(4, 3) by scale factor 2, centre of enlargement (0, 0).
- Step 1: Vector from centre to A = (4, 3)
- Step 2: Multiply by 2: (8, 6)
- Step 3: Add to the centre: (0 + 8, 0 + 6)
- Answer: A′ = (8, 6)
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Question: Enlarge point B(3, 5) by scale factor 3, centre of enlargement (1, 1).
- Step 1: Vector from centre to B = (3 − 1, 5 − 1) = (2, 4)
- Step 2: Multiply by 3: (6, 12)
- Step 3: Add to the centre: (1 + 6, 1 + 12)
- Answer: B′ = (7, 13)
Centre of Enlargement: Outside, On, or Inside the Shape
The centre of enlargement can be positioned outside the shape, on the shape (a vertex or edge), or inside it - the same method applies in every case. If a vertex happens to BE the centre, that vertex doesn't move, since its distance from the centre is zero.
Identifying an Enlargement: Centre and Scale Factor
To find the scale factor, divide a length (or coordinate distance from the centre) on the image by the corresponding length on the object.
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Question: A point at (3, 4) maps to (9, 12) under an enlargement centred at the origin. Find the scale factor.
- Step 1: Since the centre is the origin, compare coordinates directly: 9 ÷ 3 = 3, and 12 ÷ 4 = 3
- Answer: scale factor = 3
Effect of Enlargement on Perimeter
Since perimeter is a length, it scales by the same factor as the sides: new perimeter = original perimeter × k.
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Question: A rectangle has perimeter 20 cm. It is enlarged by scale factor 3. Find the new perimeter.
- Step 1: New perimeter = 20 × 3
- Answer: 60 cm
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Question: A square has a perimeter of 8 cm. It is enlarged so that the image has a perimeter of 88 cm. Find the scale factor.
- Step 1: Rearrange the perimeter rule to find k: scale factor = image perimeter ÷ original perimeter
- Step 2: scale factor = 88 ÷ 8
- Answer: 11
Effect of Enlargement on Area
Since area is a squared measure, it scales by the scale factor squared: new area = original area × k². This is a common point of confusion - area does NOT scale by the same factor as length.
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Question: A rectangle is 4 cm by 6 cm (area 24 cm²). It is enlarged by scale factor 2. Find the new area.
- Step 1: New rectangle: 8 cm by 12 cm
- Step 2: New area = 8 × 12 = 96 cm² (check: 24 × 2² = 24 × 4 = 96 cm² ✓)
- Answer: 96 cm²
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Question: A square has area 9 cm². It is enlarged by scale factor 5. Find the new area.
- Step 1: New area = 9 × 5²
- Answer: 225 cm²
Real-World Applications
Enlargement is used in many practical situations:
- Photography and printing: enlarging photos to different sizes.
- Architecture: scale models and enlarging blueprints to full size.
- Map making: enlarging a small section of a map for detail.
- Projectors: enlarging an image onto a screen.
- Manufacturing: scaling up a prototype design.
Exam Tips
Common Mistakes
MistakeAssuming area scales by the same factor as length, e.g. expecting area to double when lengths double
Fixarea scales by the scale factor SQUARED - doubling lengths (factor 2) quadruples area (2² = 4)
MistakeAssuming the centre of enlargement must be outside the shape
Fixthe centre can be outside, on, or inside the shape
MistakeAssuming an enlarged shape is congruent to the original
Fixan enlargement changes size, so the image is similar but NOT congruent (unless k = 1)
MistakeApplying the scale factor to only some vertices or sides of a shape
Fixevery length in the shape is multiplied by the SAME scale factor
MistakeConfusing perimeter scaling (× k) with area scaling (× k²)
Fixperimeter (a length) scales by k; area (a squared measure) scales by k²
MistakeDividing the original perimeter by the image perimeter (the wrong way round) when finding a scale factor
Fixscale factor = image (new) measurement ÷ object (original) measurement
For Exams
- Learn the formula: Image point = Centre + k × (Object point − Centre).
- Perimeter scales by k; area scales by k².
- The centre can be outside, on, or inside the shape - the method is the same.
- To find the scale factor: divide an image length by the corresponding object length.
- Check your answer: the image should be similar to (same shape as) the object.
Interactive revision notes, videos and practice questions load below.