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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
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