Problem 5. Three regular nonagons have a common center, their sides are respectively parallel. The sides of the nonagons are 8 cm and 56 cm. The third nonagon divides the area of the figure enclosed between the first two in the ratio $1: 7$, counting from the smaller nonagon. Find the side of the third nonagon. Answer. $8 \sqrt{7}$.
Solution. Let the side of the third nonagon be $x$. The areas of similar figures are in the ratio of the squares of corresponding sides. Therefore, the areas enclosed between the nonagons are in the ratio $\left(x^{2}-8^{2}\right):\left(56^{2}-x^{2}\right)=1: 7$, from which the answer follows.