Correct Answer: $36\sqrt{5}:{100}$

Solution : Given: In an isosceles triangle, the length of each equal side is twice the length of the third side.
the third side.
Let the sides of the isosceles triangle be $x, 2x, 2x$, and the sides of the equilateral triangle be $y$ each.
Perimeter of isosceles triangle = Perimeter of equilateral triangle
⇒ $5x = 3y$
⇒ $y = \frac{5}{3}x$
Semi perimeter(s) $=\frac{5x}{2}$
Using Heron's formula, the area of the isosceles triangle is given as,
Area of isosceles triangle
$=\sqrt{[s(s – a)(s – b)(s – c)]}$
$=\sqrt{[\frac{5x}{2}(\frac{5x}{2} – 2x)(\frac{5x}{2} – 2x)(\frac{5x}{2} – x)]}$
$=\sqrt{[\frac{5x}{2}(\frac{x}{2})(\frac{x}{2})(\frac{3x}{2})]}$
$=\frac{\sqrt{15}x^{2}}{4}$
Area of equilateral triangle $=\frac{\sqrt{3}}{4}y^{2}$
Now, the ratio of the area of the isosceles triangle to the area of the equilateral triangle is given as,
$\frac{\frac{\sqrt{15}x^{2}}{4}}{\frac{\sqrt{3}}{4}y^{2}} = \frac{\sqrt{5}x^{2}}{y^{2}} = \frac{9\sqrt{5}}{25}=\frac{9\sqrt{5}}{25} × 4 = \frac{36\sqrt{5}}{100}$
Hence, the correct answer is $36\sqrt{5}:{100}$.