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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: 61385
Points A and C lie on a circle centered at O, each of BA and BC are ta  [#permalink]

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28 Mar 2019, 23:35
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Difficulty:

75% (hard)

Question Stats:

50% (02:48) correct 50% (02:45) wrong based on 38 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: 333
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, 04:47
2
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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Intern
Joined: 13 Mar 2018
Posts: 22
Location: India
Concentration: Finance, Marketing
GMAT 1: 760 Q50 V42
GPA: 3.48
Re: Points A and C lie on a circle centered at O, each of BA and BC are ta  [#permalink]

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27 May 2019, 03:49
m1033512 wrote:
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

Posted from my mobile device

Could you please provide a diagram for the solution?

Thanks!
VP
Joined: 19 Oct 2018
Posts: 1303
Location: India
Points A and C lie on a circle centered at O, each of BA and BC are ta  [#permalink]

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27 May 2019, 08:25
Let AB=BC=CA=a
and OA=OC=OD=r

a/sin150=r/sin30
$$a=\sqrt{3}r$$

$$AC^2=BD*BE$$
$$a^2= BD*(BD+2r)$$
$$3r^2=BD^2+2r*BD$$
$$BD^2+2r*BD-3r^2=0$$
BD=r or -3r
BD can't be -ve, Hence BD=r

BD/BO=r/(r+r)=1/2

Bunuel wrote:
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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Points A and C lie on a circle centered at O, each of BA and BC are ta   [#permalink] 27 May 2019, 08:25
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