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Math Expert V
Joined: 02 Sep 2009
Posts: 58395
Points A and C lie on a circle centered at O, each of BA and BC are ta  [#permalink]

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Difficulty:   75% (hard)

Question Stats: 49% (02:51) correct 51% (02:45) wrong based on 37 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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Senior Manager  G
Joined: 25 Feb 2019
Posts: 336
Re: Points A and C lie on a circle centered at O, each of BA and BC are ta  [#permalink]

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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

award kudos if helpful

Posted from my mobile device
Intern  B
Joined: 14 Mar 2018
Posts: 22
Location: India
Concentration: Finance, Marketing
GPA: 3.55
Re: Points A and C lie on a circle centered at O, each of BA and BC are ta  [#permalink]

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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

award kudos if helpful

Posted from my mobile device

Could you please provide a diagram for the solution?

Thanks!
Director  D
Joined: 19 Oct 2018
Posts: 985
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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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) $$\frac{\sqrt{3}}{2}$$

Attachments a.png [ 8.48 KiB | Viewed 187 times ] Points A and C lie on a circle centered at O, each of BA and BC are ta   [#permalink] 27 May 2019, 09:25
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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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