Triangles and Similarity
Triangles and Similarity tests the Basic Proportionality Theorem, similarity criteria, ratio of areas, and Pythagoras’ theorem. Proofs and direct computations both appear.
Core rules
- Basic Proportionality Theorem: if DE∥BC in triangle ABC, with D on AB and E on AC, then AD/DB=AE/EC.
- Similarity criteria: AA, SSS (proportional sides), SAS (two sides proportional and included angle equal).
- Corresponding angles are equal; corresponding sides are proportional.
- Ratio of areas of similar triangles = square of ratio of corresponding sides.
- Pythagoras: in a right triangle, hypotenuse²=sum of squares of the other two sides; the converse also holds.
Worked example
In triangle ABC, DE∥BC, AD=3 cm, DB=6 cm, AE=4 cm. Find EC. By BPT, AD/DB=AE/EC. So 3/6=4/EC, giving 3EC=24, EC=8 cm.
Traps
Match corresponding vertices in the correct order; ∆ABC~∆DEF is not the same as ∆ABC~∆EDF. The area ratio is the square of the side ratio, not the side ratio itself. BPT requires the line parallel to one side; without parallel, the proportion is not valid. In Pythagoras, identify the hypotenuse-the side opposite the right angle-before squaring.