2026 · Q85
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. If x is the area of the circle and y is the area of the triangle such that z = (x/y)², then which one of the following is correct?
Elementary MathematicsGeometry: Lines, Angles and Triangles
- A0.5 < z < 1
- B1 < z < 1.5
- C1.5 < z < 2
- Dz > 2
Explanation
The centre O lies on AM (perpendicular to the tangent BC at M); with OA = OM = r and O on segment AM, 2r = AM = s√3/2 gives r = s√3/4. Then x = πr² = 3πs²/16 and y = s²√3/4, so x/y = 3π/(4√3) = π√3/4 ≈ 1.360. Thus z = (x/y)² ≈ 1.85, which lies in (1.5, 2), option C.