Equilateral triangles are drawn on the hypotenuse and one of the perpendicular sides of a right-angled isosceles triangles. Their areas are H and A respectively. $$\frac{A}{H}$$ is equal to:

Let the length of the two equal sides of right-angled isosceles triangle be x cm then hypotenuse be $$x\sqrt{2}  cm$$

Area of equilateral triangle drawn on hypotenuse (H) = $$\frac{\sqrt{3}}{4}\times a^{2}=\frac{\sqrt{3}}{4}\times (x\sqrt{2})^2$$

Area of equilateral triangle drawn on side (A) = $$\frac{\sqrt{3}}{4}\times a^{2}=\frac{\sqrt{3}}{4}\times x^2$$

$$\frac{A}{H}=\left(\frac{\sqrt{3}}{4}\times x^{2}\right)\times\frac{4}{\sqrt{3}}\times\frac{1}{(x\sqrt{2})^2} = \frac{1}{2}$$