Question

if from an external point p of a circle with centre o two tangents pq and pr are drawn such that angle qpr = 120 prove that 2pq=po

Answer 1

in ∆POR and ∆POQ
1) PO = PO (common side)
2) PR = PQ (tangents from same external point)
3) OR = OQ (radii of same. circle)
so ∆POR congruent to ∆POQ (SSS test)
and <OPR = <OQR (CPCT)
<QPR = <OPR + <OQR = 120°
so <OPR = <OQR = 60°
<ORP = <OQP = 90° (radius perpendicular to tangent)
therefore ∆POQ and ∆POR are 90-60-30 triangles
in ∆POQ,
as cos 60° = ½
therefore cos P = PQ/PO = 1/2
therefore 2PQ = PO

Answer 2

Sorry for these cuttings