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Question

# If PA and PB are two tangents drawn from an external point P to a circle such that PA = 5 cm and ∠APB = 60°, then the length of chord AB is ________.

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Solution

## PA and PB are tangents drawn from P to circle with centre O. Let OP and AB intersect at Q. ∠APQ = ∠BPQ = $\frac{60°}{2}$ = 30º (Tangents drawn from an external point to a circle are equally inclined to the segment joining the centre to that point) In ∆PAQ and ∆PBQ, AP = BP (Length of tangents drawn from an external point to a circle are equal) ∠APQ = ∠BPQ (30º each) PQ = PQ (Common) ∴ ∆PAQ $\cong$ ∆PBQ (SAS congruence axiom) ⇒ ∠AQP = ∠BQP and AQ = BQ (CPCT) Also, ∠AQP + ∠BQP = 180º (Linear pair of angles) ⇒ 2∠AQP = 180º (∠AQP = ∠BQP) ⇒ ∠AQP = 90º In right ∆APQ, $\mathrm{sin}30°=\frac{\mathrm{AQ}}{\mathrm{AP}}\phantom{\rule{0ex}{0ex}}⇒\frac{1}{2}=\frac{\mathrm{AQ}}{5\mathrm{cm}}\phantom{\rule{0ex}{0ex}}⇒\mathrm{AQ}=\frac{5}{2}\mathrm{cm}$ ∴ AB = 2AQ = $2×\frac{5}{2}$ = 5 cm (AQ = BQ) Thus, the length of the chord AB is 5 cm. If PA and PB are two tangents drawn from an external point P to a circle such that PA = 5 cm and ∠APB = 60°, then the length of chord AB is ___5 cm___.

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