Triangles and Similarity
This domain tests the basic proportionality theorem, the AA, SSS and SAS similarity criteria, the ratio of areas of similar triangles, and Pythagoras' theorem with its converse.
Core rules
- Basic Proportionality Theorem (Thales): a line parallel to one side of a triangle divides the other two sides in the same ratio. If DE ∥ BC in △ABC, then AD/DB = AE/EC.
- Similarity criteria: AA (two angles equal), SSS (all three side pairs proportional), SAS (two side pairs proportional with the included angle equal).
- Corresponding angles are equal and corresponding sides are proportional in similar triangles.
- Ratio of areas of similar triangles = square of the ratio of corresponding sides.
- Pythagoras: in a right triangle, hypotenuse² = sum of squares of the other two sides; the converse proves a triangle right-angled.
Worked example
In △ABC, DE ∥ BC with AD = 3 cm, DB = 5 cm and AE = 4.5 cm. By the Basic Proportionality Theorem, AD/DB = AE/EC, so 3/5 = 4.5/EC. Cross-multiplying gives 3EC = 22.5, hence EC = 7.5 cm.
Traps
Using the ratio of areas as the ratio of sides instead of its square; naming vertices in the wrong correspondence order; treating SAS similarity as SAS congruence; applying the proportionality theorem when the line is not parallel to a side; forgetting the included angle in SAS.