Congruence and similarity · Year 11
In Year 11 you learn: area and volume in similar shapes.
Congruent shapes are identical in shape and size. Similar shapes have the same shape but may be different sizes: all their lengths are in the same ratio (the scale factor), and their angles are equal.
Key facts
- Triangles are congruent if they match by SSS, SAS, ASA (or AAS) or RHS. AAA is not enough.
- Similar shapes: length scale factor k = new length ÷ old length.
- Area scale factor = k².
- Volume scale factor = k³.
- Equal angles (from parallel lines or vertically opposite angles) often show that triangles are similar.
Worked examples
Example 1
The triangles are similar. Find x.
Answer: x = 10 cm
Example 2
Two similar shapes have lengths in the ratio 2 : 5. The smaller has an area of 12 cm². Find the area of the larger.
Answer: 75 cm²
Example 3
Two similar jugs hold 250 ml and 2,000 ml. The small jug is 10 cm tall. How tall is the large jug?
Answer: 20 cm