Triangles and Similarity
This section covers Triangles and Similarity. The exam tests similarity criteria, the Basic Proportionality Theorem (BPT), ratios of areas, and Pythagoras’ theorem.
Core rules
- Two triangles are similar if their corresponding angles are equal (AA), corresponding sides are proportional (SSS), or two sides are proportional and the included angle is equal (SAS).
- BPT: if a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides in the same ratio.
- If two triangles are similar, the ratio of their areas equals the square of the ratio of corresponding sides.
- Pythagoras: in a right triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.
Worked example
In triangle ABC, DE is parallel to BC, with D on AB and E on AC. Given AD/DB = 2/3 and AE = 4 cm, find EC. By BPT, AD/DB = AE/EC. So 2/3 = 4/EC, giving 2EC = 12 and EC = 6 cm.
Traps
Match corresponding vertices when writing similarity statements; AB/DE is not necessarily the correct ratio. The area ratio is the square of the side ratio, not the side ratio itself. BPT requires the line to be parallel to a side; equal-looking divisions are not enough. In Pythagoras problems, identify the hypotenuse first-it is opposite the right angle.