Question

In the given figure, two tangents RQ, and RP and RP are drawn from an external point R to the circle with centre O. If ∠PRQ =120° , then prove that OR = PR + RQ.

Answer 1

Construction Join PO and OQ
In ΔPOR and ΔQOR
OP = OQ(Radii)
RP = RQ(Tangents from the external point are congruent)
OR = OR (Common)
By SSS congruency, ΔPOR ≅ ΔQOR
∠PRO = ∠QRO(C.P.C.T)

Now,∠PRO+ ∠QRO= ∠PRQ
⇒ 2 ∠PRO = 120°
⇒ ∠PRO = 60°
Now. In ΔPOR
cos 60° `=(PR)/(OR)`

⇒ `1/2 =(PR)/(OR)`
⇒ OR =  2PR
⇒  OR = PR + PR
⇒  OR = PR +RQ