Congruent and similar shapes

Two shapes can match in two ways: congruent (same shape AND size) or similar (same shape, different size). OCR J560 expects fluency on both, including enlargements with fractional and negative scale factors on Higher.

Congruent — identical shapes

Two triangles are congruent if any of these conditions hold:

• SSS — three sides equal.
• SAS — two sides + included angle equal.
• ASA (or AAS) — two angles + a corresponding side equal.
• RHS — right angle, hypotenuse, and one other side equal (right-angled triangles only).

For congruence proofs, state the condition AND list the matching parts.

Similar — same shape, different size

Two shapes are similar if:

• All corresponding angles are equal.
• All corresponding sides are in the same ratio (the scale factor).

For triangles, AA (two angles equal) is enough — the third must match (angles sum to 180°).

Scale factors

Scale factor k = (length on new shape) / (length on original shape).

• k > 1: enlargement.
• 0 < k < 1: reduction.
• k = 1: congruent (no size change).
• k < 0 (Higher): enlargement with rotation 180° about the centre.

Area and volume scale factors

If lengths scale by k:

• Areas scale by k².
• Volumes scale by k³.

Example: a model 1/10 the linear size has 1/100 the area and 1/1000 the volume.

Constructing enlargements

Given a centre of enlargement O and scale factor k:

1. From O, measure the distance to each vertex of the original.
2. Multiply that distance by k.
3. Plot the new vertex at that scaled distance from O.

For negative k, the new vertex sits on the opposite side of O.

OCR mark scheme conventions

• B1 for stating the correct congruence condition (SSS, SAS, etc.).
• M1 for setting up the ratio for similarity.
• A1 for the missing length / area / volume.
• For enlargement constructions: B1 for centre identified, B1 for scale factor used correctly, B1 for accurate plot.