Similarity

4.4 Similarity – Cheat Sheet

1. Key Idea of Similarity

• Similar shapes: Same shape, proportional sides, equal corresponding angles.
• Scale factor (k) relates sizes of similar figures.

2. Showing Triangles Are Similar

• Use any of these geometric reasons:
  1. AAA – all angles equal
  2. SAS – one equal angle, and sides around it in the same ratio
  3. SSS – all sides in same ratio

3. Length, Area, Volume Ratios in Similar Figures

4. Calculating Lengths in Similar Shapes

Formula:
Length in shape B = (Length in shape A) × k

Example:
Length in A = 5 cm, k = 1.4 → Length in B = 5 × 1.4 = 7 cm

5. Areas of Similar Shapes

Formula:
Area in shape B = (Area in shape A) × k²

Example:
Area in A = 20 cm², k = 1.5 → Area in B = 20 × (1.5)² = 45 cm²

6. Volumes of Similar Solids

Formula:
Volume in shape B = (Volume in shape A) × k³

Example:
Volume in A = 40 cm³, k = 2 → Volume in B = 40 × 2³ = 320 cm³

7. Reverse Problems

• If given areas, k = √(Area ratio)
• If given volumes, k = ³√(Volume ratio)

Example:
Area ratio = 49 : 25 → k = √(49/25) = 7/5

8. Quick Reference Table