B2. In a circle with radius $r$, we inscribe a deltoid, one side of which is twice as long as the other. Calculate the ratio of the areas of the inscribed deltoid and the circle.
B2. Let the side lengths of the inscribed kite be denoted by $a$ and $2a$. Due to symmetry, the longer diagonal of the kite passes through the center of the circle. By the theorem of the angle in a semicircle, the angle between the sides of lengths $a$ and $2a$ is a right angle. By the Pythagorean theorem, the length of the longer diagonal is $\sqrt{a^2 + (2a)^2} = \sqrt{5}a$, which means $r = \frac{\sqrt{5}a}{2}$. The area of the kite is $2 \cdot \frac{a \cdot 2a}{2} = 2a^2$, and the area of the circle is $\pi r^2 = \frac{5\pi a^2}{4}$. The sought ratio of the areas is $\frac{8}{5\pi}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_a0d1506e8e2028d6d427g-24.jpg?height=416&width=420&top_left_y=226&top_left_x=1435)

A clearly drawn and labeled diagram (including the observation that the longer diagonal of the kite is the diameter of the circle) 1 point  
A justified observation that the angle between the sides of lengths $a$ and $2a$ is a right angle 1 point  
Calculation of $r = \frac{\sqrt{5}a}{2}$ or consideration of this calculation in the area of the circle 2 points  
Calculation of the area of the kite 1 point  
Calculated ratio 1 point