Enlargements (Edexcel GCSE Maths): Revision Notes
Enlargements
What is an enlargement?
An enlargement is a transformation that changes the size of a shape whilst keeping it similar to the original. When you enlarge a shape, you create a new image that is either bigger or smaller than the original object.
To fully describe an enlargement, you must specify two key pieces of information:
- The scale factor
- The centre of enlargement
These two elements work together to determine exactly how the shape will be transformed and where the new image will be positioned.
Scale factor
The scale factor tells you how many times larger or smaller the new shape will be compared to the original. You can calculate it using this formula:
Understanding scale factors:
- Scale factor > 1: The image will be larger than the original object
- Scale factor = 1: The image will be the same size as the original object
- Scale factor between 0 and 1: The image will be smaller than the original object
When working with fractional scale factors (like or ), remember that these create images smaller than the original shape.
Worked Example: Calculating Scale Factor
If a line segment measures 6 cm in the original shape and 18 cm in the enlarged image:
Step 1: Apply the formula Scale factor =
Step 2: Substitute the values Scale factor =
This means the enlargement has a scale factor of 3, making the image three times larger than the original.
Centre of enlargement
The centre of enlargement is a fixed point from which the enlargement takes place. This point determines where the enlarged shape will be positioned.
Key Property to Remember:
When you draw straight lines connecting corresponding points on the original shape and its enlarged image, these lines will all meet at the centre of enlargement. This is a crucial property that helps you locate the centre of enlargement when it's not given.
Properties of enlargements
Understanding what changes and what stays the same during an enlargement is fundamental to mastering this transformation.
When a shape is enlarged:
- Angles remain unchanged - all angles in the enlarged shape are identical to those in the original
- Side lengths change - they are multiplied by the scale factor
- The shape remains similar - it has the same form but different size
This is why enlarged shapes are called "similar" to their originals - they have the same angles but proportionally different side lengths.
How to enlarge a shape
Learning the systematic approach to enlarging shapes ensures accuracy and consistency in your work.
Worked Example: Step-by-Step Enlargement Process
Step 1: Draw construction lines Draw straight lines from the centre of enlargement through each vertex (corner) of the original shape.
Step 2: Measure and multiply distances For each vertex, measure the horizontal and vertical distances from the centre of enlargement to that vertex. Then multiply both distances by the scale factor.
Step 3: Plot new vertices Use your calculated distances to plot the new vertices of the enlarged shape.
Step 4: Connect and check Join up the new vertices with straight lines to complete the enlarged shape. Check that each side length is the scale factor times the corresponding original length.
Working with fractional scale factors
When the scale factor is a fraction (such as ), the resulting image will be smaller than the original object. This is still called an enlargement in mathematics, even though the shape gets smaller.
Worked Example: Fractional Scale Factor
With a scale factor of :
Step 1: Apply to horizontal distance A horizontal distance of 4 units becomes units
Step 2: Apply to vertical distance
A vertical distance of 8 units becomes units
The resulting image is exactly half the size of the original in both dimensions.
Exam tips
Success in enlargement questions requires attention to detail and systematic working.
Essential Exam Strategies:
- Always state both the scale factor and centre of enlargement when describing an enlargement
- Use a ruler to measure distances accurately
- Check your work by verifying that corresponding sides are in the correct ratio
- Remember that fractional scale factors create smaller images
- Show all your working clearly, especially when calculating distances and scale factors
Key Points to Remember:
- Scale factor determines how much bigger or smaller the image will be
- Centre of enlargement is the fixed point from which the enlargement occurs
- Angles never change during enlargements - only side lengths change
- Fractional scale factors (between 0 and 1) create smaller images
- Always check your work by measuring corresponding lengths
- Construction lines from the centre through vertices are essential for accurate enlargements