3. The double area of a regular hexagon is equal to the triple area of an equilateral triangle. Determine the ratio of the perimeters of the hexagon and the triangle.
Solution. If $a$ is the length of the side of the hexagon, and $b$ is the length of the side of the triangle, then $2 \cdot 6 \frac{a^{2} \sqrt{3}}{4}=3 \frac{b^{2} \sqrt{3}}{4}$, from which we get $4 a^{2}=b^{2}$. Since $a, b>0$, from the last equality it follows that $b=2 a$, i.e., $6 a=3 b$. Therefore, the perimeters of the given figures are equal, so their ratio is 1:1.