2026 · Q75
A square of maximum area is inscribed in an equilateral triangle. If the side of the triangle is equal to (6 + 4√3) cm, then what is the area of the square?
Elementary MathematicsGeometry: Lines, Angles and Triangles
- A36 cm²
- B27 cm²
- C21 cm²
- D18 cm²
Explanation
Let the square of side x rest on one side of the equilateral triangle of side s. By similar triangles, (s√3/2 − x)/(s√3/2) = x/s, giving x = s(2√3 − 3). With s = 6 + 4√3 = 2(3 + 2√3), x = 2(3 + 2√3)(2√3 − 3) = 2·3 = 6 cm. The square's area is 36 cm², option A.