In the given figure, O is the center of the circle. AB is a tangent to the circle at point B. If \(\angle\) OAB = 30^{\(\circ\)} and OA = 16, find the length of the radius OB.

Solution:

We are given that AB is a tangent to the circle at point B, and O is the center of the circle. This means that the radius OB is perpendicular to the tangent AB at the point of tangency B. Therefore, \( \angle OBA = 90^{\circ} \).

We are given that \( \angle OAB = 30^{\circ} \) and \( OA = 16 \).

We can consider the right-angled triangle OBA. In this triangle, we can use trigonometric ratios to find the length of OB.

Using the sine function:

\[ \sin(\angle OAB) = \frac{\text{Opposite side}}{\text{Hypotenuse}} = \frac{OB}{OA} \]\[ \sin(30^{\circ}) = \frac{OB}{16} \]\[ \frac{1}{2} = \frac{OB}{16} \]\[ OB = 16 \times \frac{1}{2} \]\[ OB = 8 \]

Geometric Construction:

Ответ: OB = 8.