[ The inscribed angle is half the central angle ] [ Central angle. Arc length and circumference ]

Chord $AB$ divides the circle into two arcs, the smaller of which is $130^{\circ}$, and the larger is divided by chord $AC$ in the ratio $31: 15$, counting from point $A$. Find the angle $BAC$.
An inscribed angle is measured by half the angular magnitude of the corresponding arc.

## Solution

The angular magnitude of the larger arc is $360^{\circ}-130^{\circ}=230^{\circ}$. Angle BAC subtends arc $BC$, the angular magnitude of which is $\frac{15}{46} \cdot 230^{\circ}=75^{\circ}$. Therefore,

$$
\angle B A C=\frac{1}{2} \cup B C=37^{\circ} 30^{\prime}
$$

![](https://cdn.mathpix.com/cropped/2024_05_06_34fd2032e75750f56a92g-29.jpg?height=383&width=391&top_left_y=1984&top_left_x=845)

## Answer

$37^{\circ} 30^{\prime}$.