A circle is divided into two arcs by points A and B, and the ratio of their lengths is 5:7. We need to calculate the peripheral angles that correspond to these arcs.

1. Problem Description

A circle is divided into two arcs by points A and B, and the ratio of their lengths is 5:

7. We need to calculate the peripheral angles that correspond to these arcs.

2. Solution Steps

The ratio of the lengths of the arcs is 5:

7. This means that the corresponding central angles also have the same ratio.

Let the central angles be $5x$ and $7x$. The sum of the central angles in a circle is $360^{\circ}$.

So, $5x + 7x = 360^{\circ}$.

$12x = 360^{\circ}$

$x = \frac{360^{\circ}}{12} = 30^{\circ}$

The central angles are:

$5x = 5 \cdot 30^{\circ} = 150^{\circ}$

$7x = 7 \cdot 30^{\circ} = 210^{\circ}$

The peripheral angle is half of the central angle that subtends the same arc.

Therefore, the peripheral angles are:

$\frac{150^{\circ}}{2} = 75^{\circ}$

$\frac{210^{\circ}}{2} = 105^{\circ}$

3. Final Answer

The peripheral angles are $75^{\circ}$ and $105^{\circ}$.