In the given figure, AB is a tangent to the circle with center O. If angle BOC = 21 degrees, find angle BAC.

Solution:

Given that AB is a tangent to the circle with center O and angle BOC = 21 degrees.

We know that the angle subtended by an arc at the center is double the angle subtended by the same arc at any point on the remaining part of the circle.

In the given figure, arc BC subtends angle BOC at the center and angle BAC at the circumference.

Therefore, we have the relationship:

\( \angle BOC = 2 \times \angle BAC \)

We are given \( \angle BOC = 21^{\circ} \).

Substituting the value of \( \angle BOC \) into the formula:

\( 21^{\circ} = 2 \times \angle BAC \)

To find \( \angle BAC \), we divide both sides by 2:

\( \angle BAC = \frac{21^{\circ}}{2} \)

\( \angle BAC = 10.5^{\circ} \)

Answer: \( 10.5^{\circ} \)