Triangles and Similarity: BPT, Criteria, Area Ratio and Pythagoras
Triangles and Similarity carries the Basic Proportionality Theorem, the similarity criteria, the area ratio theorem and Pythagoras - often as a 5-mark proof.
Basic Proportionality Theorem (Thales)
A line drawn parallel to one side of a triangle, cutting the other two sides in distinct points, divides those sides in the same ratio: DE ∥ BC in △ABC gives AD/DB = AE/EC. The converse also holds and is the standard way to prove two lines parallel.
Two triangles are similar if AA (two angles equal), SSS (all three pairs of corresponding sides in the same ratio) or SAS (two sides in the same ratio with the included angle equal). Similar triangles have equal corresponding angles and proportional corresponding sides.
Area ratio and Pythagoras
The ratio of the areas of two similar triangles equals the ratio of the squares of any pair of corresponding sides. The ratio of perimeters, medians, altitudes and angle bisectors equals the ratio of corresponding sides. In a right triangle the square on the hypotenuse equals the sum of the squares on the other two sides; the converse identifies right triangles.
Worked example
In △ABC, DE ∥ BC with AD = 3 cm, DB = 5 cm and AE = 4.5 cm. Find EC. By BPT, AD/DB = AE/EC → 3/5 = 4.5/EC → 3·EC = 22.5 → EC = 7.5 cm. Check: AD/AB = 3/8 and AE/AC = 4.5/12 = 3/8 ✓.
Traps that cost marks
- Write similarity with vertices in corresponding order (△ABC ~ △DEF); a wrong order makes every ratio wrong.
- The area ratio is the square of the side ratio, not the side ratio itself.
- AD/DB uses the two parts of a side; AD/AB uses the whole side - pick one convention and stay with it.
- In Pythagoras, identify the hypotenuse first: it lies opposite the right angle.
- Proofs need a labelled figure and a stated reason for each step.