Geometry and measures - Mensuration and calculation: apply the concepts of congruence and similarity, including the relationships between lengths, in similar figures (Foundation tier)

The idea

Congruent shapes are identical in shape and size; one can be transformed onto the other by rotation, reflection or translation. To prove triangles are congruent, you must show one of four conditions is met: SSS, SAS, ASA, or RHS, stating the condition and listing the three pairs of corresponding equal parts. Similar shapes have the same shape but can be different sizes. Their corresponding angles are equal, and the ratio of corresponding side lengths is constant. This ratio is the length scale factor. To find it, divide a side length on the new shape by the corresponding side length on the old shape. For example, if a 4 cm side corresponds to a 10 cm side, the scale factor is 10 ÷ 4 = 2.5. Use this to find other lengths by multiplying or dividing.

What earns the marks

• Correctly identifying a pair of corresponding sides or angles.
• Calculating the length scale factor correctly from a pair of corresponding sides.
• Using the scale factor to find a missing length by multiplying or dividing appropriately.
• For a congruence proof, stating one of the four conditions (SSS, SAS, ASA, RHS) and listing the three supporting facts with reasons.

Common mistakes and how to put them right

• Mixing up multiplication and division with the scale factor.

  Put it right: Check if the shape is getting bigger (enlarging) or smaller (reducing) to decide whether to multiply or divide.

• Incorrectly pairing corresponding sides, especially when shapes are rotated.

  Put it right: Redraw the shapes in the same orientation and match them based on equal angles.

• Adding or subtracting a fixed length instead of multiplying by a scale factor.

  Put it right: Similarity involves a multiplicative relationship (a ratio), not an additive one.

Exam tips

• Redraw similar shapes separately and in the same orientation to make corresponding sides obvious before calculating.
• For congruence proofs, structure your answer clearly: state the three pairs of equal sides/angles with reasons, then state the congruence condition as your conclusion.