3.9 Similarity and congruence
Similarity and congruence
Everything on similarity and congruence that can come up in the WJEC GCSE Mathematics and Numeracy, in Unit 2 and Unit 3. Each skill below has the method, the mistake that costs the most marks, a worked example, and free practice marked the way WJEC marks it.
Practise similarity and congruence free
A worked example
Similar shapes · [3 marks]
Two similar photo frames have areas of 2 cm² and 8 cm². The smaller frame is 14 cm wide. How wide is the larger frame?
- M18 ÷ 2 = 4
- M1√4 = 2
- A114 × 2 = 28 cm
The exam skills (6)
Similar shapes
- Match the corresponding sides (the ones in the same position).
- Scale factor = new length ÷ old length.
- Multiply (or divide) the side you need by the scale factor.
Watch out: Adding the difference instead of multiplying.
Congruence
- SSS: three sides equal. SAS: two sides and the angle between them.
- ASA (or AAS): two angles and a corresponding side.
- RHS: right angle, hypotenuse and one side.
Watch out: Saying AAA. Three equal angles only proves similar, not congruent.
Area scale factor
- If lengths are × k, areas are × k².
- Square the length scale factor.
- Multiply the area by it.
Watch out: Multiplying the area by k instead of k².
Volume scale factor
- If lengths are × k, volumes are × k³.
- Cube the length scale factor.
- Multiply the volume by it.
Watch out: Using k² for volume.
Similar triangles
- The small triangle and the big one are similar because the lines are parallel.
- Use the WHOLE side of the big triangle, not just the extra bit.
- Scale factor = big ÷ small, then multiply.
Watch out: Using DB instead of AB. The big side is AD + DB.
Similar solids
- Area scale factor = k². Square root it to get k.
- Volume scale factor = k³.
- Multiply the volume (or capacity) by k³.
Watch out: Going straight from area to volume without finding k first.
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