Question

In Fig. 2, 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 = \[90^o\]

\[\therefore ∆ PTO \cong ∆ PSO\]                   (By SAS congruency)

Thus, 
∠TPO = ∠SPO =\[\frac{120^o}{2} = 60^o\]

Now, in \[∆ PSO,\]

\[\cos60^o = \frac{PS}{PO}\]
\[ \Rightarrow \frac{1}{2} = \frac{PS}{PO}\]
\[ \Rightarrow PO = 2PS\]

Hence proved.