A pair of straight lines from an external point F intersects a circle at A and B (FA

Question

A pair of straight lines from an external point F intersects a circle at A and B (FA < FB), and touches the circle at C. O is the centre of the circle. Given that ∠ACF = 50° and ∠AFC = 30°, find ∠AOB.

Accepted Answer

#### 1. **Given:**
- ∠ACF = 50° (angle between chord AC and tangent at C)
- ∠AFC = 30° (angle at F between AF and CF)

#### 2. **Find ∠FCA in triangle AFC:**
$$
\angle FCA = 180^\circ - (\angle ACF + \angle AFC) = 180^\circ - (50^\circ + 30^\circ) = 100^\circ
$$

#### 3. **By the Alternate Segment Theorem:**
- The angle between the tangent and chord (∠ACF) is equal to the angle in the alternate segment, i.e., ∠ABA (angle subtended by chord AB at the circle on the opposite side).
- Similarly, the angle ∠FCA (between tangent and chord at C) is equal to the angle subtended by chord AB at the circumference on the opposite side, i.e., ∠AOB/2.

#### 4. **Angle at the Centre:**
- The angle subtended by chord AB at the centre (O) is twice the angle subtended at the circumference on the same side.
- Here, the angle at the circumference is ∠FCA = 100°.
- Therefore,
$$
\angle AOB = 2 	imes 50^\circ = 100^\circ
$$

#### 5. **Conclusion:**
So, the angle ∠AOB is **100°**.

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**Thus, the correct answer is:**

**3. 100°**

Answer

Answer

Answer