Question
→− is tangent to circle O at A.− If =64° , what is the ? (1 point) Responses 58° 58degree 180° 180degree 22° 22degree 26°
Answers
GPT-4o mini
To solve for the angle given that line \( \overline{BC} \) is tangent to circle \( O \) at point \( A \), and the angle \( \angle BAC = 64^\circ \), we can use the principle that a tangent line at a point on a circle is perpendicular to the radius drawn to that point.
1. The radius \( OA \) is perpendicular to the tangent \( BC \) at point \( A \). Therefore, \( \angle OAB = 90^\circ \).
2. We have triangle \( OAB \) formed, where \( \angle OAB = 90^\circ \) and \( \angle BAC = 64^\circ \).
3. We can find angle \( \angle OBA \) using the fact that the sum of the angles in a triangle equals \( 180^\circ \):
\[
\angle OAB + \angle BAC + \angle OBA = 180^\circ
\]
Plugging in the known values:
\[
90^\circ + 64^\circ + \angle OBA = 180^\circ
\]
\[
154^\circ + \angle OBA = 180^\circ
\]
\[
\angle OBA = 180^\circ - 154^\circ = 26^\circ
\]
So the angle \( \angle OBA \) is \( 26^\circ \).
Therefore, the answer is \( 26^\circ \).
1. The radius \( OA \) is perpendicular to the tangent \( BC \) at point \( A \). Therefore, \( \angle OAB = 90^\circ \).
2. We have triangle \( OAB \) formed, where \( \angle OAB = 90^\circ \) and \( \angle BAC = 64^\circ \).
3. We can find angle \( \angle OBA \) using the fact that the sum of the angles in a triangle equals \( 180^\circ \):
\[
\angle OAB + \angle BAC + \angle OBA = 180^\circ
\]
Plugging in the known values:
\[
90^\circ + 64^\circ + \angle OBA = 180^\circ
\]
\[
154^\circ + \angle OBA = 180^\circ
\]
\[
\angle OBA = 180^\circ - 154^\circ = 26^\circ
\]
So the angle \( \angle OBA \) is \( 26^\circ \).
Therefore, the answer is \( 26^\circ \).