Properties of Triangles Questions with Solutions

Properties of triangles connect the sides a, b, c and the opposite angles A, B, C of any triangle, not just right-angled ones. This topic covers the sine rule (a/sin A = b/sin B = c/sin C = 2R), the cosine rule (a^2 = b^2 + c^2 - 2bc cos A), the projection formulas, half-angle formulas, and formulas for the area of a triangle, including (1/2)ab sin C and Hero's formula. It also covers the circumradius R, the inradius r and their relationships with the area and semi-perimeter, such as r = area/s. Exam questions test applying these rules to find an unknown side, angle or area, and identifying the type of triangle from a given condition. Choosing between the sine rule and the cosine rule is usually the key decision. Practise the properties of triangles questions below, each with a step-by-step solution.