The sem of the lenght of one side in each of two regular triangles is 20 cm. Calculate the lenght of the sides of the triangle If the ratio of their areas is 1:4

Let the lengths of the sides of the two regular triangles be x cm and y cm, respectively.

Given that the sum of the length of one side in each triangle is 20 cm, we have:
x + y = 20

The ratio of the areas of the two triangles is 1:4. The ratio of the areas of two similar figures is equal to the square of the ratio of their corresponding sides. So,
\left( \frac{x}{y} \right)^2 = \frac{1}{4}
\frac{x^2}{y^2} = \frac{1}{4}
x^2 = \frac{1}{4}y^2
x^2 = \frac{y^2}{4}
x = \frac{y}{2}

Substitute this into the first equation:
\frac{y}{2} + y = 20
\frac{3y}{2} = 20
3y = 40
y = \frac{40}{3} = 13.\overline{3} \text{ cm}

Therefore, the length of the sides of the triangles are:
x = \frac{13.\overline{3}}{2} = 6.\overline{6} \text{ cm}
y = 13.\overline{3} \text{ cm}

\boxed{x = 6.\overline{6} \text{ cm}, y = 13.\overline{3} \text{ cm}}