Mensuration and calculation: apply the concepts of congruence and similarity, including the relationships between lengths in similar figures

The idea

Congruent figures are identical in shape and size, meaning corresponding lengths and angles are equal. Similar figures have equal corresponding angles and corresponding lengths in a constant ratio called the scale factor. To apply these concepts, identify corresponding vertices and sides, then set up a proportion. For example, if triangle ABC is similar to DEF with AB = 6 cm and DE = 9 cm, the scale factor from ABC to DEF is 1.5, so every length in DEF is 1.5 times the matching length in ABC. Congruence is the special case where the scale factor is exactly 1. Match corresponding parts correctly before calculating an unknown length, and use similarity to find missing dimensions in geometric figures.

What earns the marks

• States that congruent figures have equal corresponding lengths and angles, while similar figures have equal corresponding angles and proportional corresponding lengths.
• Identifies the scale factor as the ratio of a length in the image to the corresponding length in the original figure.
• Sets up and solves a proportion to find an unknown length in a pair of similar figures.
• Recognises that congruence is the special case of similarity with a scale factor of 1.
• Applies the relationship between lengths in similar figures to calculate missing dimensions.

Common mistakes and how to put them right

• Matching non-corresponding sides when forming a ratio.

  Put it right by tracking the order of vertices in the similarity statement, for example ABC ~ DEF means AB corresponds to DE.

• Assuming that equal angles alone prove congruence.

  Put it right by remembering that similarity needs equal angles and proportional sides, while congruence also requires equal lengths.

• Using the scale factor in the wrong direction.

  Put it right by checking whether the unknown length belongs to the original or the image and multiplying or dividing accordingly.

Exam tips

• Write the similarity statement with vertices in corresponding order before doing any calculation.
• Label the scale factor clearly and state whether you are enlarging or reducing the figure.
• Check your answer by substituting back into the proportion to confirm both ratios are equal.