Three scale factors: length, area, volume
If the length scale factor is , the area scale factor is and the volume scale factor is .
Go backwards with roots
From an area ratio, take the square root to get the length ratio; from a volume ratio, take the cube root.
Congruence vs similarity
Congruent = identical (same size and shape): SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), RHS (Right angle-Hypotenuse-Side). Similar = same shape, enlarged (equal angles, sides in proportion).
Drawn from real examiner reports.
Length SF used for area or volume
Multiplying an area by instead of (or a volume by instead of ).
Matching the wrong corresponding sides
Pairing sides that are not in the same position in the two similar shapes.
Identifying corresponding sides in similar triangles was central to Nov 2024 Paper 1H (Q10).
Square-root instead of cube-root
Going from a volume ratio back to lengths with instead of . Length ratio ; the square root is only for an area ratio.
(general exam technique)
Adding the difference, not scaling
Scaling a side by adding the size difference (e.g. ) instead of multiplying by the length scale factor . Similar shapes multiply, so a cm side becomes .
(general exam technique)
Equal angles prove similarity only
Treating AAA (all angles equal) as a congruence condition. Equal angles prove SIMILARITY only; congruence needs SSS, SAS, ASA or RHS with matching sides.
(general exam technique)
Decide which scale factor first
Write k, then k² for area or k³ for volume, before substituting.
k the right way up
Form the length scale factor as new ÷ old. An enlargement should give and a reduction ; check it matches whether the shape grows or shrinks.
Name the congruence condition
When proving two triangles congruent, quote SSS, SAS, ASA or RHS explicitly with the matching sides and angles — the named condition earns the reasoning mark.
If corresponding lengths are in the ratio (the length scale factor):
Find corresponding sides by matching equal angles / positions in the two shapes.
Difference between congruent and similar?
Two triangles are similar. A side of cm on the small triangle corresponds to cm on the large one. Another side of the small triangle is cm. Find the corresponding side on the large triangle.
