The perimeter of a triangle and an equilateral triangle are same. Also, one of the sides of the rectangle is equal to the side of the triangle. The ratio of the areas of the rectangle and the triangle is

Given that the perimeters of rectangle and triangle are equal

Let the length and breadth of rectangle be 'l' and 'b' respectively

Let the side of triangle be 'a'

$$\Rightarrow$$ 2(l+b)=3a

Given that one side of rectangle of rectangle = side of triangle

Let l=a

$$\Rightarrow$$ 2(a+b)=3a
$$\Rightarrow$$ 2a+2b=3a
$$\Rightarrow$$ a=2b

 Area of rectangle : Area of triangle = ab : $$\frac{\sqrt{3}}{4}a^{2}$$

Substituting a=2b in above equation

$$\Rightarrow$$ $$2b^{2} : \frac{\sqrt{3}}{4}\times4b^{2}$$

$$\Rightarrow 2:\sqrt{3}$$