Congruent shapes
- Congruent shapes are exactly the same shape and size. One may be rotated or reflected.
- Two triangles are congruent if one of these is true: SSS (all three sides match), SAS (two sides and the angle between them), ASA (two angles and the side between them), or RHS (right angle, hypotenuse and one other side).
- Three matching angles (AAA) do not prove congruence: the triangles could be different sizes.
Similar shapes
- Similar shapes have equal angles, and their sides are in the same ratio. One is an enlargement of the other.
- Scale factor = new length ÷ matching old length. If matching sides are 4 cm and 6 cm, the scale factor is 1.5, so a 10 cm side becomes 15 cm.
- Match the corresponding sides carefully. Redrawing the triangles the same way round helps.
Areas and volumes of similar shapes (Higher)
- If the length scale factor is k, the area scale factor is k² and the volume scale factor is k³.
- Two similar cylinders have heights of 3 cm and 6 cm, so k = 2. If the small one has a volume of 50 cm³, the large one has a volume of 50 × 2³ = 400 cm³.
- To go back from an area scale factor to k, take the square root: an area scale factor of 9 gives k = 3.
Key terms
- Congruent
- Exactly the same shape and size.
- Similar
- The same shape, with all lengths multiplied by the same scale factor.
- Scale factor
- The number that lengths are multiplied by to get from one shape to a similar one.
- Corresponding sides
- Sides in matching positions on two shapes.
- SSS
- Side, side, side: all three sides match.
- SAS
- Side, angle, side: two sides and the angle between them match.
- ASA
- Angle, side, angle: two angles and the side between them match.
- RHS
- Right angle, hypotenuse, side.