Sheet № 58 · Foundation + Higher · AQA · Edexcel · OCR
Enlargement & Scale Factor –
Enlargement is the fourth type of transformation in GCSE Maths and the only one that changes the size of a shape. It is tested at both Foundation and Higher tiers on AQA, Edexcel, and OCR papers. At Foundation you need to enlarge shapes by positive integer and fractional scale factors. At Higher you must also handle negative scale factors
§Key definitions
Question:
Enlarge triangle ABC with vertices A(1, 1), B(3, 1), and C(1, 3) by scale factor 2 from the centre of enlargement (0, 0).
Check:
Original side AB = 2 units. Image side A'B' = 4 units. Scale factor = 4/2 = 2 ✓
Answer:
The image has vertices at A'(2, 2), B'(6, 2), and C'(2, 6).
Question 1:
Enlarge the point (2, 3) by scale factor 3 from the origin.
Question 2:
A shape is enlarged by scale factor 2. The original area is 12 cm². What is the area of the image?
§Formulas to memorise
New length = original length × scale factor
Distance from centre to image point = distance from centre to original point × scale factor
Area scale factor = (linear scale factor)²
Volume scale factor = (linear scale factor)³
Scale factor > 1 — the image is larger than the original (e.g. scale factor 3 triples every length).
Scale factor between 0 and 1 — the image is smaller than the original (e.g. scale factor ½ halves every length). This is sometimes called a reduction.
Negative scale factor — (Higher only) — the image is on the opposite side of the centre of enlargement and inverted. A scale factor of −2 means the image is twice as large and flipped through the centre.
Worked example
Enlarge triangle ABC with vertices A(1, 1), B(3, 1), and C(1, 3) by scale factor 2 from the centre of enlargement (0, 0).
Working:
⚠ Common mistakes
- ✗Forgetting to state the centre of enlargement. The description "enlargement by scale factor 2" is incomplete and will lose marks. You must give the centre.
- ✗Measuring distances from the shape instead of from the centre. All distances must be measured from the centre of enlargement.
- ✗Confusing scale factor with area factor. If the scale factor is 3, lengths are 3 times bigger but the area is 9 times bigger (3²). Do not mix these up.
- ✗Drawing lines that do not pass through the centre. When checking your work, lines through corresponding vertices should all pass through the centre.
- ✗Negative scale factor direction. For a negative scale factor, the image appears on the opposite side of the centre. Forgetting to go through the centre to the other side is a common Higher tier error.
✦ Exam tips
- →For fractional scale factors, the image is smaller. If the scale factor is ⅓, every length is divided by 3. The image will be closer to the centre.
- →Always draw construction lines from the centre through each vertex. This helps you place the image accurately and shows your method.
- →To find the scale factor from a diagram, divide an image length by the corresponding original length. If the answer is less than 1, the shape has been reduced.
- →Area and volume relationships are popular at Higher level. Remember: area factor = (scale factor)² and volume factor = (scale factor)³.
- →Use column vectors to handle distances from the centre — this is the most reliable method and avoids counting errors on the grid.