Congruence and similarity; areas and volumes of similar figures
This topic combines G5 (congruence proof) and G7 (similarity by enlargement) with the powerful scale factor rules for areas and volumes (also covered in R12).
Congruence — recap
Two shapes are congruent if they have the same shape AND size. Use SSS, SAS, ASA, RHS to prove.
Similarity — recap
Two shapes are similar if corresponding angles are equal and corresponding sides are in proportion. The constant ratio is the linear scale factor (LSF).
Three scale factors
For similar shapes:
- LSF = k (linear).
- ASF = k² (area).
- VSF = k³ (volume).
✦Worked example— Worked examples
Example 1. Two similar triangles, sides in ratio 3 : 5. Areas in ratio?
- ASF = 9 : 25.
Example 2. Two similar cones with surface areas 64 cm² and 100 cm². Find volume ratio.
- ASF = 64/100 → LSF = 8/10 = 4/5 → VSF = 64/125.
- Volume ratio = 64 : 125.
Example 3. Two similar bottles. Smaller has volume 250 ml; larger 1080 ml. Find linear scale factor.
- VSF = 250/1080 = 25/108. Hmm — actually 1080/250 = 4.32. ∛4.32 ≈ 1.629.
- More commonly used numbers: 1080/250 = 27/6.25... not clean. Try VSF = 8/27 ratios in cleaner exam Qs.
Similar triangles — finding sides
When given similar triangles, set up the ratio of corresponding sides and solve.
Worked example: △ABC ~ △PQR. AB = 6, BC = 8, AC = 10, PQ = 9. Find QR and PR.
- LSF = 9/6 = 1.5.
- QR = 8 × 1.5 = 12; PR = 10 × 1.5 = 15.
Similar triangles in parallel-line problems
Lines parallel to one side of a triangle create similar triangles (the small one similar to the big one). Use this to find missing sides quickly.
⚠Common mistakes
- Wrong direction of ratio — small : big vs big : small consistency.
- Linear ratio used as area ratio without squaring.
- Cube vs square confusion when dealing with volume vs area.
- Treating congruent shapes as merely similar — congruence is stronger.
- Mixing ratios of different things (length and area) in one calculation.
➜Try this— Quick check
Two similar prisms have heights 4 cm and 6 cm. Smaller has volume 32 cm³. Find volume of larger.
- LSF = 6/4 = 3/2; VSF = 27/8.
- 32 × 27/8 = 108 cm³.
AI-generated · claude-opus-4-7 · v3-deep-geometry