Question

In the given figure, from a point P, two tangents PT and. PS are drawn to a circle with centre O such that ∠SPT = 120° Prove that OP = 2PS.

Answer 1

It is given that PS and PT are tangents to the circle with centre O. Also, ∠SPT = 120°.

To prove: OP = 2PS

Proof: 

In △PTO and △PSO,

PT = PS    (Tangents drawn from an external point to a circle are equal in length.)

TO = SO   (Radii of the circle)

∠PTO = ∠PSO = 90°

\(\therefore \triangle PTO \cong\triangle PSO\)     (By SAS congruency)

Thus, 

∠TPO = ∠SPO = \(\frac{120°}2 = 60° \)

Now,

In ∆PSO,

\(\cos 60° = \frac{PS}{OP}\)

⇒ \(\frac 12 = \frac{PS}{OP}\)

⇒ \(OP = 2PS\)

Hence proved.

Answer 2

Given that  ∠SPT = 120°