Similarity, congruence and scale factors
Congruent shapes
Two shapes are congruent if one can be mapped onto the other by translation, reflection or rotation — they are identical in size and shape.
Triangle congruence criteria (must memorise)
| Code | Means |
|---|---|
| SSS | All three sides equal |
| SAS | Two sides + included angle equal |
| ASA (or AAS) | Two angles + a corresponding side equal |
| RHS | Right angle + hypotenuse + one other side equal |
SSA is NOT a valid criterion (ambiguous case).
Similar shapes
Same shape, different size. Corresponding angles equal; corresponding sides in the same ratio.
Linear scale factor (k)
Length ratio between two similar shapes: k = corresponding side of new ÷ corresponding side of original.
If k = 2, all lengths are doubled.
Length, area and volume scale factors
| Quantity | Scale factor |
|---|---|
| Length | k |
| Area | k² |
| Volume | k³ |
Example: Two cones are similar. Heights are 6 cm and 9 cm. Length scale factor k = 9/6 = 1.5. Area scale factor = 2.25; volume scale factor = 3.375.
If small cone has volume 80 cm³, large cone has volume 80 × 3.375 = 270 cm³.
Proving similar triangles
Show one of:
- All three pairs of angles equal (AAA — but two pairs are enough since the third follows).
- All three pairs of corresponding sides in the same ratio.
- Two pairs of sides in the same ratio with the included angle equal.
Finding missing lengths in similar shapes
Set up a ratio: small : large = a : b. Use proportion to solve.
Common CCEA exam tip
For a 3-mark "show two triangles are similar" question, you usually need:
- State the similarity statement (e.g. △ABC similar to △XYZ).
- Identify two pairs of equal angles with reasons.
- Conclude "AAA" (or AA).
Skipping the reason for any equal angle costs the M1.
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