43. A, B और C केन्द्र O वाले वृत्त पर स्थित तीन बिन्दु हैं। स्पर्श रेखा C, BA को T तक बढ़ाए जाने पर मिलती है। यदि ∠ATC=30∘\angle \text{ATC} = 30^\circ और ∠ACT=48∘\angle \text{ACT} = 48^\circ हो, तो ∠AOB\angle \text{AOB} का मान क्या है? A, B and C are three points on a circle with center O. Tangent at C meets BA produced to T. If ∠ATC=30∘\angle \text{ATC} = 30^\circ and ∠ACT=48∘\angle \text{ACT} = 48^\circ, then what is the value of ∠AOB\angle \text{AOB}? (SSC CGL Tier - I 2016) (a) 78∘78^\circ (b) 96∘96^\circ (c) 120∘120^\circ (d) 108∘108^\circ

Answer: 108^∘\circ

Step-by-step solution

Step 1: Apply Alternate Segment Theorem

According to the Alternate Segment Theorem, the angle between the tangent CTCT and the chord ACAC through the point of contact equals the angle subtended by the chord in the alternate segment, which is ∠ABC\angle \text{ABC}. Therefore, ∠ABC=48∘\angle \text{ABC} = 48^\circ.

Step 2: Find angle TAC using Exterior Angle Property

In △ATC\triangle \text{ATC}, the side TATA is extended along the line TABTAB. Thus, the exterior angle ∠CAB\angle \text{CAB} (or ∠TAC\angle \text{TAC} viewed as the exterior angle of △ACT\triangle \text{ACT} at vertex AA) equals the sum of the two interior opposite angles: ∠ATC+∠ACT=30∘+48∘=78∘\angle \text{ATC} + \angle \text{ACT} = 30^\circ + 48^\circ = 78^\circ.

Step 3: Find angle ACB in triangle ABC

In △ABC\triangle \text{ABC}, the sum of all three interior angles is 180∘180^\circ. Since ∠CAB=78∘\angle \text{CAB} = 78^\circ and ∠ABC=48∘\angle \text{ABC} = 48^\circ, we calculate ∠ACB=180∘−126∘=54∘\angle \text{ACB} = 180^\circ - 126^\circ = 54^\circ.

Step 4: Calculate the central angle AOB

The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle. Therefore, ∠AOB=2∠ACB=2×54∘=108∘\angle \text{AOB} = 2 \angle \text{ACB} = 2 \times 54^\circ = 108^\circ.

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