Question

a circle with points A, B, and C, and line AC tangent to the circle at point O, with angle BY measuring 64 degrees. What is the measure of angle BAC

Answer 1

To find the measure of angle \( \angle BAC \), we can use some properties of circles and tangents.

1. Since line \( AC \) is tangent to the circle at point \( O \), it implies that the radius \( AO \) is perpendicular to the tangent line at point \( O \). Therefore, we have:
\[
\angle AOB = 90^\circ
\]

2. Given that \( \angle BY = 64^\circ \) and assuming \( Y \) lies on the tangent line \( AC \), we will also need to observe the relationships in the triangle formed by \( A \), \( B \), and \( C \).

3. The angle \( \angle BOA \) relates to \( \angle BAC \) because:
\[
\angle BAC + \angle BOA + \angle AOB = 180^\circ \text{ (since they form a straight line)}
\]
Since we know \( \angle AOB = 90^\circ \):
\[
\angle BAC + \angle BOA + 90^\circ = 180^\circ
\]
This simplifies to:
\[
\angle BAC + \angle BOA = 90^\circ
\]

4. Moreover, the angle at point \( B \) (external angle) created by points \( A \), \( B \), and line segment \( AC \) gives us the relation:
\[
\angle BOA = \angle BY
\]
Thus, we have that:
\[
\angle BOA = 64^\circ
\]

5. Substituting \( \angle BOA \) in our previous equation:
\[
\angle BAC + 64^\circ = 90^\circ
\]

6. Therefore, solving for \( \angle BAC \):
\[
\angle BAC = 90^\circ - 64^\circ = 26^\circ
\]

Thus, the measure of angle \( \angle BAC \) is \( \boxed{26^\circ} \).