2026 · Q86
For the next three (03) items that follow : A circle with centre O passes through the vertex A of an equilateral triangle ABC and touches BC at its midpoint M. The circle cuts AB at D and AC at E. What is the ratio of the area of the triangle ADO to the area of the quadrilateral BDOM?
Elementary MathematicsGeometry: Lines, Angles and Triangles
- A3 : 5
- B4 : 5
- C3 : 3
- D2 : 3
Explanation
Place M = (0, 0), A = (0, s√3/2), O = (0, s√3/4) and D = (−3s/8, s√3/8) (dividing AB in ratio AD : DB = 3 : 1). Triangle ADO has base AO = s√3/4 and horizontal distance 3s/8 of D, so area = ½·(s√3/4)(3s/8) = 3√3s²/64. Quadrilateral BDOM with vertices B(−s/2, 0), D, O, M has shoelace area 5√3s²/64. The ratio is 3 : 5, option A.