Question

In the given figure, PA and PB are tangents to the given circle such that PA = 5 cm and ∠APB = 60° . The length of chord AB is

(a) 5 √2 cm (b) 5 cm (c) 5 √3 cm (d) 7.5 cm

Answer 1

Correct option is (b) 5 cm

Given: PA and PB are tangents of a circle, PA = 5 cm and ∠APB = 60°

Let O be the center of the given circle and C be the point of intersection of OP and AB

In ΔPAC and ΔPBC

PA = PB (\(\because\) Tangents from an external point are equal)

∠APC = ∠BPC (\(\because\) Tangents from an external point are equally inclined to the segment joining center to that point)

PC = PC (\(\because\) Common)

Thus ΔPACΔPBC (By SAS congruency rule) ..........(1)

∴ AC = BC

Also ∠APB = ∠APC + ∠BPC

∠APC = \(\frac 12\)∠APB     (∠APC = ∠BPC)

\(= \frac 12 \times 60°\)

\(= 30°\)

∠ACP + ∠BCP = 180°

∠ACP = \(\frac 12\) x 180°  (∠ACP + ∠BPC)

= 90°

Now in right triangle ACP

sin30° = \(\frac{AC}{AP}\)

\(\frac 12 = \frac{AC}{5}\)

\(AC =\frac 52 \)

AB = AC + BC 

= AC + AC     (\(\because\) AC = BC)

⇒ AB = \(\frac 52 + \frac 52 \)

= 5 cm

Answer 2

(b) 5 cm

The lengths of tangents drawn from a point to a circle are equal

Each angle of ΔPAB is 60° and therefore, it is an equilateral triangle.

AB = PA = PB = 5 cm 

The length of the chord AB is 5 cm.