Triangles and Similarity
Triangles and Similarity tests proportional sides, similar triangle criteria, the Basic Proportionality Theorem, and Pythagoras. Similar triangles have equal corresponding angles and proportional corresponding sides.
Core rules
- Basic Proportionality Theorem: if a line is parallel to one side of a triangle and intersects the other two sides, it divides them in the same ratio. The converse is also true.
- Similarity criteria: AA, SAS (two sides proportional and included angle equal), and SSS (all three pairs of sides proportional).
- In similar triangles, corresponding sides, medians, altitudes, and perimeters are in the same ratio; areas are in the square of that ratio.
- Pythagoras: in a right triangle, hypotenuse^2 = sum of squares of the other two sides. Its converse identifies a right angle.
Worked example
In triangle ABC, DE is parallel to BC with D on AB and E on AC. If AD=3 cm, DB=5 cm, and AE=4.5 cm, find EC. By BPT, AD/DB = AE/EC, so 3/5 = 4.5/EC. Cross-multiplying gives 3EC=22.5, hence EC=7.5 cm. The parallel condition is essential; without DE||BC, the ratio does not follow.
Traps
Match corresponding vertices in the correct order; triangle ABC similar to triangle DEF is not the same as ABC similar to EDF. Area ratio is the square of the side ratio, not the side ratio. BPT uses segments from the same vertex, not arbitrary side parts. Do not assume triangles are similar just because they look alike.