Question

In a circular museum hall of radius 14 m, some statues are displayed. Statues are kept inside the inner concentric circle of radius 7 m. One such statue lying in sector OAB, is fenced along line segments OA, AP, PB and BO where P is a point on outer circle. Based on above information, answer the following questions :

(i) Find m\(\angle\)AOP.

Hint:

Whenever the adjacent side of a right triangle is exactly half of the hypotenuse, the angle between them is always \(60^\circ\) because \(\cos 60^\circ = 0.5\).

Answer 1

Correct Answer: 60

Step 1: Understanding the Question:
This is a Case Study question from Circles and Coordinate Geometry.
The problem describes two concentric circles of radii \(r = 7\text{ m}\) and \(R = 14\text{ m}\) with common centre \(O\).
The segment \(AP\) is a tangent to the inner circle from point \(P\) on the outer circle.
We need to find the measure of the angle \(\angle AOP\).

Step 2: Key Formula or Approach:
1. The radius \(OA\) is perpendicular to the tangent line \(AP\) at the point of contact \(A\). Thus, \(\angle OAP = 90^\circ\).
2. In the right-angled triangle \(\triangle OAP\), the hypotenuse is \(OP = 14\text{ m}\) (radius of the outer circle) and the adjacent side is \(OA = 7\text{ m}\) (radius of the inner circle).
3. Use the cosine trigonometric ratio: \[ \cos \angle AOP = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{OA}{OP} \]

Step 3: Detailed Explanation:
1. Identify the lengths from the given concentric circles: - Radius of inner circle, \(OA = 7\text{ m}\)
- Radius of outer circle, \(OP = 14\text{ m}\)
2. Since \(AP\) is tangent to the inner circle at \(A\): \[ \angle OAP = 90^\circ \] 3. In right-angled triangle \(\triangle OAP\), write the formula for cosine of \(\angle AOP\): \[ \cos \angle AOP = \frac{OA}{OP} \] 4. Substitute the given values: \[ \cos \angle AOP = \frac{7}{14} = \frac{1}{2} \] 5. Identify the angle whose cosine is \(\frac{1}{2}\): \[ \angle AOP = 60^\circ \]

Step 4: Final Answer:
The measure of \(\angle AOP\) is \(60^\circ\).