Similarity/Congruency & Ratios of length, area & volume

Notes

Congruency

*3 tests required

• • SSS (3 Sides) - All 3 sides are equal

  • SAS (Side - Angle - Side) - 2 sides are equal, included angle equal

  • AAS (Angle - Angle - Angle) - 2 angles equal, included side equal

  • RHS (Right Angle Hyp. Side) - Longest side of triangle equal, 1 angle equal

Similarity

*2 tests required

• • RRR (3 Sides) - 3 sides equal proportions

  • AAA (3 Angles) - 3 angles equal

  • RAR (Ratio - Angle - Ratio) - 2 sides equal proportion, included angle equal

Area/Volume of similar solids

Area/Surface Area

Volume/Mass

It is important to note the units

• • of length: mm, cm, m

  • of area: mm², cm², m²

  • of volume: mm³, cm³, m³

• If the scale factor for lengths is x, then the scale factor for areas is x², and the scale factor for volumes is x³.

• All the length of the sides of the solid on the right are twice the lengths of corresponding sides on the left.

• If the area of the cuboid on the left is 20 units² and the volume is 50 units³

  • the area of the cuboid on the right is 2² x 20 = 160 units²

  • the volume of the cuboid on the right is 2³ x 50 units³ = 400 units³

• • All the length of the sides on the right are 3 times the lengths of corresponding sides on the left

  • If the area of the cuboid on the right is 180 units² and the volume is 540 units³

    • the area of the cuboid on the left is 180/3² = 20 units²

    • the volume of the cuboid on the left is 540/3³ = 20 units³