InterviewSolution
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InterviewSolution
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In Fig. if tangents PA and PB are drawn to a circle such that ∠APS = 30° and chord AC is drawn parallel to the tangent PB, then ∠ABC =A. 60° B. 90° C. 30° D. None of these |
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Answer» Answer is C. 30° Given: APB = 30° Property 1: If two tangents are drawn to a circle from one external point, then their tangent segments (lines joining the external point and the points of tangency on circle) are equal. Property 2: Sum of all angles of a triangle = 180° By property 1, PA = PB (tangent from P) And, ∠PAB = ∠PBA [∵PA = PB] By property 2, ∠PAB + ∠PBA + ∠APB = 180° ⇒ ∠PAB + ∠PBA + 30° = 180° ⇒ ∠PAB + ∠PBA = 180° - 30° ⇒ ∠ PAB + ∠ PBA = 150° ⇒ ∠ PBA + ∠ PBA = 150° [∵∠PAB = ∠PBA] ⇒ 2∠PBA = 150° ⇒ ∠PBA = \(\frac{150^°}{2}\) ⇒ ∠PBA = 75° Now, ∠PBA = ∠CAB = 75° [Alternate angles] ∠PBA = ∠ACB = 75° [Alternate segment theorem] Again by property 2, ∠CAB + ∠ACB + ∠CBA = 180° ⇒ 75° + 75° + ∠CBA = 180° ⇒ 150° + ∠CBA = 180° ⇒ ∠CBA = 180° - 150° ⇒ ∠CBA = 30° Hence, ∠CBA = 30° |
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