In the given figure, PA and PB are tangents to a circle centred at O. If \(\angle OAB = 15^\circ\), then \(\angle APB\) equals :
Show Hint
There is a direct geometric relationship for this configuration:
The angle between the tangents (\(\angle APB\)) and the angle formed by the chord with the radius (\(\angle OAB\)) always satisfy the following relation:
\[ \angle APB = 2 \angle OAB \]
Using this direct formula:
\[ \angle APB = 2 \times 15^\circ = 30^\circ \]
This direct shortcut saves significant time during multiple-choice sections!
Step 1: Understanding the Question:
We are given a circle centered at \(O\). From an external point \(P\), two tangents \(PA\) and \(PB\) are drawn to the circle. Inside the circle, the angle \(\angle OAB\) is given as \(15^\circ\). We need to determine the measure of the angle between the two tangents, \(\angle APB\).
Step 2: Key Formula or Approach:
1. The radius of a circle is perpendicular to the tangent at the point of contact: \(\angle OAP = \angle OBP = 90^\circ\).
2. The lengths of tangents from an external point are equal: \(PA = PB\), which means \(\Delta PAB\) is an isosceles triangle.
3. Radii of a circle are equal: \(OA = OB\), making \(\Delta OAB\) an isosceles triangle.
Step 3: Detailed Explanation:
1. Consider the triangle \(\Delta OAB\):
Since \(OA\) and \(OB\) are both radii of the same circle, we have:
\[ OA = OB \]
Therefore, \(\Delta OAB\) is an isosceles triangle, which implies:
\[ \angle OBA = \angle OAB = 15^\circ \]
2. The radius \(OA\) is perpendicular to the tangent \(PA\) at the point of contact \(A\):
\[ \angle OAP = 90^\circ \]
3. Now, we can find the angle \(\angle PAB\):
\[ \angle PAB = \angle OAP - \angle OAB \]
\[ \angle PAB = 90^\circ - 15^\circ = 75^\circ \]
4. Consider the triangle \(\Delta PAB\):
Since the tangents from an external point to a circle are equal in length, we have:
\[ PA = PB \]
Therefore, \(\Delta PAB\) is also an isosceles triangle, which implies:
\[ \angle PBA = \angle PAB = 75^\circ \]
5. Sum of angles in triangle \(\Delta PAB\) is \(180^\circ\):
\[ \angle APB + \angle PAB + \angle PBA = 180^\circ \]
\[ \angle APB + 75^\circ + 75^\circ = 180^\circ \]
\[ \angle APB + 150^\circ = 180^\circ \]
\[ \angle APB = 180^\circ - 150^\circ = 30^\circ \]
The angle \(\angle APB\) is \(30^\circ\).
Step 4: Final Answer:
The value of \(\angle APB\) is \(30^\circ\), which corresponds to option (A).
