Question

If two tangents are drawn to a circle of radius 3 cm such that the angle between them is 60°, then find the length of each tangent.

Answer 1

Let PQ and PR are the tangents on circle from point P. OQ and OR are the radius of circle.

So, PQ ⊥ OQ and PR ⊥ OR

In right angled ∆POQ and ∆POR,

∠OQP = ∠ORP (each 90°)

hypotenuse PO = hypotenuse PO (common side)

and QQ = OR(equal radii of circle)

∴ ∆POQ = ∆POR (by RHS congruence)

⇒ ∠QPO = ∠RPO (CPCT)

⇒ ∠QPO = ∠RPO

= \(\frac { { 60 }^{ \circ } }{ 2 } \) = 30°

In right angled ∆OQP

tan 30° = \(\frac { OQ }{ PQ }\)

⇒ \(\frac { 1 }{ \sqrt { 3 } } \) = \(\frac { 3 }{ PQ }\) ⇒ PQ = 3\(\sqrt { 3 }\)

Since PQ and PQ are the tangents from point P. we know that tangents to a circle from an external points are equal.

So, PR = PQ

= 3\(\sqrt { 3 }\) cm