Similarity and congruence (Edexcel GCSE Maths): Revision Notes
Similarity and congruence
What are similar shapes?
Similar shapes are shapes that have exactly the same angles but different sizes. When one shape is an enlargement of another, the two shapes are considered similar.
Understanding similarity is crucial in geometry as it helps us recognise patterns and relationships between shapes of different sizes. This concept is widely used in scale drawings, maps, and architectural plans.
Key features of similar shapes:
- All corresponding angles are identical
- The sides are proportional (in the same ratio)
- One shape is an enlargement or reduction of the other
- They have the same shape but different sizes
Scale factors in similar shapes
When shapes are similar, there is a scale factor that connects them. This tells you how many times bigger or smaller one shape is compared to the other.
Worked Example: Finding Scale Factor
If triangle ABC has sides of 3cm, 4cm, and 5cm, and triangle DEF has corresponding sides of 6cm, 8cm, and 10cm, then:
Step 1: Compare corresponding sides
- Side 1: 6 ÷ 3 = 2
- Side 2: 8 ÷ 4 = 2
- Side 3: 10 ÷ 5 = 2
Step 2: Identify the scale factor The scale factor from ABC to DEF is 2
What are congruent shapes?
Congruent shapes are shapes that are exactly identical in both shape and size. They have the same area and the same perimeter.
The word "congruent" comes from Latin meaning "agreeing" or "corresponding". In mathematics, congruent shapes are perfect matches - they could be placed on top of each other with complete overlap.
Key features of congruent shapes:
- Exactly the same shape
- Exactly the same size
- Same area and perimeter
- All corresponding sides and angles are equal
Transformations that create congruent shapes
Certain transformations produce congruent shapes:
- Rotations - turning the shape around a point
- Reflections - flipping the shape over a line
- Translations - sliding the shape to a new position
Critical Point: Enlargements create similar shapes, not congruent shapes, because they change the size. This is a common mistake in exams - remember that congruent shapes must be exactly the same size!
Key differences between similar and congruent shapes
| Similar Shapes | Congruent Shapes |
|---|---|
| Same angles | Same angles |
| Different sizes | Same size |
| Proportional sides | Equal sides |
| Created by enlargements | Created by rotations, reflections, translations |
Working with similar shapes in 3D
When dealing with 3D shapes like cubes, you can calculate how many smaller similar shapes fit inside a larger one. This requires understanding that we must consider all three dimensions.
Worked Example: 3D Similar Shapes
To find how many 2cm cubes fit inside an 8cm cube:
Step 1: Calculate the linear scale factor Scale factor = 8 ÷ 2 = 4
Step 2: Apply the scale factor to all three dimensions
- Length: 4 times
- Width: 4 times
- Height: 4 times
Step 3: Calculate the total number Number of small cubes = small cubes
This works because you need to consider all three dimensions (length, width, and height). The volume scales by the cube of the scale factor.
Exam tips
Essential Exam Strategies:
- Look carefully at the angles when identifying similar shapes - they must be the same
- Remember that congruent shapes are exactly identical, whilst similar shapes are the same shape but different sizes
- When finding how many small 3D shapes fit into larger ones, remember to cube the scale factor
- In grid questions, look for shapes that are the same form but different sizes for similarity
Key Points to Remember:
- Similar shapes have the same angles but different sizes - one is an enlargement of the other
- Congruent shapes are exactly the same shape and size with equal areas and perimeters
- Enlargements create similar shapes, not congruent shapes
- Rotations and reflections create congruent shapes
- In 3D problems, cube the scale factor to find how many smaller shapes fit inside larger ones