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# Points A and C lie on a circle centered at O, each of BA and BC are ta

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Math Expert
Joined: 02 Sep 2009
Posts: 55188
Points A and C lie on a circle centered at O, each of BA and BC are ta  [#permalink]

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29 Mar 2019, 00:35
00:00

Difficulty:

75% (hard)

Question Stats:

40% (02:22) correct 60% (02:33) wrong based on 20 sessions

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Points A and C lie on a circle centered at O, each of BA and BC are tangent to the circle, and triangle ABC is equilateral. The circle intersects BO at D. What is BD/BO ?

(A) $$\frac{\sqrt{2}}{3}$$

(B) $$\frac{1}{2}$$

(C) $$\frac{\sqrt{3}}{3}$$

(D) $$\frac{\sqrt{2}}{2}$$

(E) $$\frac{\sqrt{3}}{2}$$

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Joined: 25 Feb 2019
Posts: 277
Re: Points A and C lie on a circle centered at O, each of BA and BC are ta  [#permalink]

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18 Apr 2019, 05:47
1
IMO B

extend OB so that it cut circle at M

now by circle property

BD*BM = AB^2

BD*(BO+OM) = AB^2

let the raidus B r

then
BD*(BO+OM) = AB^2

find the value of BO and AB in terms of r

in right angle triangle AOB , AB = √3r

and OB = 2r

we get BD = r

so BD/BO = r/2r = 1/2

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Re: Points A and C lie on a circle centered at O, each of BA and BC are ta   [#permalink] 18 Apr 2019, 05:47
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