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 ________.

Answer 1

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,

$$\begin{gathered} \sin 30°=\frac{\mathrm{AQ}}{\mathrm{AP}} \\ \Rightarrow \frac{1}{2}=\frac{\mathrm{AQ}}{5\mathrm{cm}} \\ \Rightarrow \mathrm{AQ}=\frac{5}{2}\mathrm{cm} \end{gathered}$$

∴ AB = 2AQ = $2\times \frac{5}{2}$ = 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___.