The diagonal of a square is equal to the side of an equilateral triangle. If the area of the square is 15√3 sq cm, what is the area of the equilateral triangle?

Let the side of square = $$s$$ cm and diagonal = $$d$$ cm

=> Area of square = $$(s)^2 = 15\sqrt{3}$$ ----------(i)

In right triangle of the square, => $$(s)^2 + (s)^2 = (d)^2$$

Substituting value of $$s^2$$ from equation (i)

=> $$(d)^2 = 15\sqrt{3} + 15\sqrt{3} = 30\sqrt{3}$$ ----------(ii)

Side of equilateral triangle = Diagonal of square = $$d$$ cm

$$\therefore$$ Area of equilateral triangle = $$\frac{\sqrt{3}}{4} d^2$$

Substituting value of $$d^2$$ from (ii), we get :

= $$\frac{\sqrt{3}}{4} \times 30\sqrt{3} = \frac{90}{4} = \frac{45}{2} cm^2$$

=> Ans - (D)