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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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V
Joined: 02 Sep 2009
Posts: 54371
Points A and C lie on a circle centered at O, each of BA and BC are ta  [#permalink]

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New post 29 Mar 2019, 00:35
00:00
A
B
C
D
E

Difficulty:

  65% (hard)

Question Stats:

44% (02:42) correct 56% (02:02) wrong based on 16 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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Re: Points A and C lie on a circle centered at O, each of BA and BC are ta  [#permalink]

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New post 18 Apr 2019, 02:50
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Joined: 25 Feb 2019
Posts: 191
Re: Points A and C lie on a circle centered at O, each of BA and BC are ta  [#permalink]

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New post 18 Apr 2019, 05:47
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


award kudos if helpful

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

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