Congruent and similar shapes; scale factors
Edexcel 1MA1 examines this on every Paper 1 reasoning section, with congruence proofs on Higher and similar-triangle calculations on both tiers.
Congruent shapes
Two shapes are congruent if one can be mapped onto the other by a combination of translations, rotations and reflections — they have the same shape AND the same size.
Properties preserved: side lengths, angles, area.
Conditions for congruent triangles
- SSS — three pairs of equal sides.
- SAS — two pairs of equal sides + the included angle equal.
- ASA (or AAS) — two pairs of equal angles + a corresponding side.
- RHS — right angle + hypotenuse + one other side equal.
Similar shapes
Two shapes are similar if one is an enlargement of the other — same shape, possibly different size. All corresponding angles are equal and all corresponding sides are in the same ratio (the scale factor).
Scale factors
If two shapes are similar with scale factor k from shape A to shape B:
- Lengths scale by k.
- Areas scale by k².
- Volumes scale by k³.
Fractional scale factors (enlargements that shrink)
A scale factor of 1/2 reduces all lengths by half. The "enlargement" terminology is retained even though the image is smaller.
Negative scale factors (Higher only)
A scale factor of −1 maps each point through the centre to the diametrically opposite side. Visually, it is a 180° rotation about the centre. A scale factor of −2 enlarges by 2 and inverts through the centre.
✦Worked example
Triangles ABC and PQR are similar. AB = 4 cm, PQ = 6 cm. Triangle ABC has area 12 cm².
- Scale factor (from A to P) = 6/4 = 3/2.
- Area scale factor = (3/2)² = 9/4.
- Area of PQR = 12 × 9/4 = 27 cm².
Common Edexcel exam tip
When proving congruence (Higher 4-mark question), end with the named condition ("△ABC ≡ △PQR by SAS"). Mark schemes specify B1 for the conclusion with condition stated.
⚠Common mistakes— Common errors
- Using only AAA (three equal angles) for congruence — this proves similarity, not congruence (size matters).
- Forgetting to square the scale factor for areas (using k instead of k²).
- Mixing up "scale factor from A to B" vs "from B to A" — they are reciprocals.
- Confusing congruent (≡) with similar (∼) symbols.
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