Euclidean Geometry: Triangles and Circles
What the domain tests
This domain tests triangle congruence and similarity, the Basic Proportionality Theorem, Pythagoras’ theorem, and standard circle theorems involving chords, angles, cyclic quadrilaterals and tangents.
Core rules and concepts
- Congruence criteria: SSS, SAS, ASA and RHS. SSA is not a congruence criterion.
- Similarity criteria: AA, SSS (proportional sides) and SAS (two sides proportional with the included angle equal).
- Basic Proportionality Theorem: if a line is parallel to one side of a triangle and intersects the other two sides, it divides them proportionally.
- Pythagoras: in a right triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.
- The angle at the centre is twice the angle at the circumference standing on the same arc. Opposite angles of a cyclic quadrilateral are supplementary.
- A tangent is perpendicular to the radius at the point of contact, and tangents from an external point are equal in length.
Worked example
In triangle ABC, DE is parallel to BC, with D on AB and E on AC. If AD/DB = 2/3 and AE = 4 cm, find EC. By the Basic Proportionality Theorem, AD/DB = AE/EC. So 2/3 = 4/EC, giving 2EC = 12 and EC = 6 cm.
Traps that cost marks
Do not use SSA as a congruence rule. In similarity, name corresponding vertices in the correct order. For circle questions, distinguish between a chord, a secant and a tangent. The perpendicular from the centre bisects a chord, not every line through the centre.