Someone divided the quarter circle into two sectors with areas in the ratio of $2: 3$ by dividing the chord of the quarter circle into five equal segments, then connecting the second division point to the center of the circle, and claimed that this line divides the quarter circle in the desired ratio. Is this construction accurate, and if not, what is the magnitude of the error?
It would be good if the division were such that $\alpha=90^{\circ} \frac{3}{5}=54^{\circ}$. Let's calculate what it is here.

![](https://cdn.mathpix.com/cropped/2024_05_02_a62c54e935739a348b41g-1.jpg?height=388&width=372&top_left_y=245&top_left_x=863)

According to the labels in the figure,

$$
\operatorname{tg} \alpha=\frac{N N^{\prime}}{O N^{\prime}}
$$

Since $B A O \varangle=45^{\circ}$, it follows that $N N^{\prime}=N^{\prime} A$.

Consequently,

$$
\operatorname{tg} \alpha=\frac{N N^{\prime}}{O N^{\prime}}=\frac{N^{\prime} A}{O N^{\prime}}=\frac{N A}{B N}=\frac{3}{2}
$$

From this, $\alpha \sim 56.3^{\circ}$, which is 2.3 degrees more than it should be. The discrepancy is

$$
\varepsilon=r^{2} \pi \frac{2.3}{360}=0.0064 r^{2} \pi
$$