O-Level Equations and inequalities: worked solution

Question

For that triangle (196=x²+y²+xy) with perimeter 30 cm, find its exact area.

The given relation 196 = x² + y² + xy is the cosine rule c² = x² + y² − 2xy cos C with cos C = −1/2, i.e. the included angle between the sides x and y is 120°, and the third side is c = √196 = 14 cm. Use the perimeter: x + y + 14 = 30, so x + y = 16. Use the identity x² + y² + xy = (x + y)² − xy. Substitute: 196 = 16² − xy = 256 − xy, so xy = 256 − 196 = 60. With sum 16 and product 60, x and y are the roots of t² − 16t + 60 = 0. Factorise: (t − 6)(t − 10) = 0, so {x, y} = {6, 10}. Area = (1/2) × x × y × sin 120° = (1/2) × 6 × 10 × (√3/2). Area = 30 × (√3/2) = 15√3 cm² (exact).