Question 4 If from an external point B of a circle with centre O, two tangents BC and BD are drawn such that ∠ DBC = 120∘, prove that BC+BD=BO i.e BO = 2BC.
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Solution
Two tangents BD and BC are drawn from an external point B.
To Prove: BO = 2BC Given: ∠ DBC = 120∘ Join OC, OD and BO. Since, BC and BD are tangents. OC ⊥ BC and OD ⊥ BD We know, OB is an angle bisector of ∠ DBC. ∴∠OBC=∠DBO=60∘ In right angle ∠OBC,cos60∘=BCOB ⇒12=BCOB ⇒ OB = 2BC Also, BC = BD. [Tangent drawn from internal point to circle are equal] ∴OB =BC+BC ⇒OB = BC + BD