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In a circle with centre O and radius 1 cm, an arc AB makes an angle 60

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Bunuel
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In a circle with centre O and radius 1 cm, an arc AB makes an angle 60 [#permalink] In a circle with centre O and radius 1 cm, an arc AB makes an angle 60   03 Apr 2020, 05:55
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In a circle with centre O and radius 1 cm, an arc AB makes an angle 60 degrees at O. Let R be the region bounded by the radii OA, OB and the arc AB. If C and D are two points on OA and OB, respectively, such that OC = OD and the area of triangle OCD is half that of R, then what is the length of OC?


A. \(\sqrt{\frac{\pi}{4}}\)

B. \(\sqrt{\frac{\pi}{6}}\)

C. \(\sqrt{\frac{\pi}{3\sqrt{3}}}\)

D. \(\sqrt{\frac{\pi}{4\sqrt{3}}}\)

E. \(\sqrt{\frac{\pi}{5\sqrt{3}}}\)




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yashikaaggarwal
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Re: In a circle with centre O and radius 1 cm, an arc AB makes an angle 60 [#permalink] In a circle with centre O and radius 1 cm, an arc AB makes an angle 60   03 Apr 2020, 06:40
Area of sector =60/360πr^2
Area of Sector=1/6π=R
Area of Triangle OCD is r/2
∆OCD=1/12π
Also, the area of triangle ocd is √3/4x^2(where x is length of OC)
1/12π=√3/4x^2
X=√(π/3√3)
OA is C

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Re: In a circle with centre O and radius 1 cm, an arc AB makes an angle 60 [#permalink] In a circle with centre O and radius 1 cm, an arc AB makes an angle 60   03 Apr 2020, 07:12
Area of region R = \(\pi r^2 * \frac{360}{6}\) = \(\frac{\pi }{6}\) , since r = 1
Now, \(\frac{\pi }{6} * \frac{1}{2}\) = △ OCD
or, △ OCD = \(\frac{\pi }{12}\) .. equation (i)

Now, since OC=OD, so ∠OCD=∠ODC.
Hence, ∠OCD=∠ODC=∠COD=60 deg
Therefore, OC=CD=OD

Now we can equate from equation (i),
△ OCD = \(\frac{\pi }{12}\),
or, \(\frac{\sqrt{3}}{4} * x^2\)= \(\frac{\pi }{12}\)
therefore, x = \(\sqrt{\frac{\pi }{ 3 \sqrt3}}\)
Answer: C
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Re: In a circle with centre O and radius 1 cm, an arc AB makes an angle 60 [#permalink] In a circle with centre O and radius 1 cm, an arc AB makes an angle 60   03 Apr 2020, 07:18
Bunuel wrote:

In a circle with centre O and radius 1 cm, an arc AB makes an angle 60 degrees at O. Let R be the region bounded by the radii OA, OB and the arc AB. If C and D are two points on OA and OB, respectively, such that OC = OD and the area of triangle OCD is half that of R, then what is the length of OC?


A. \(\sqrt{\frac{\pi}{4}}\)

B. \(\sqrt{\frac{\pi}{6}}\)

C. \(\sqrt{\frac{\pi}{3\sqrt{3}}}\)

D. \(\sqrt{\frac{\pi}{4\sqrt{3}}}\)

E. \(\sqrt{\frac{\pi}{5\sqrt{3}}}\)

[/spoiler]


let the OC be a.

R = \( \frac{\pi *(1)^2*60}{360} \) = \( \frac{\pi }{6} \)
Area of triangle OCD = \(\frac{1*(a)^2*sin(60)}{2}\)= \( \frac{\pi }{6*2} \)
\((a)^2\)=\( \frac{\pi }{6*sin(60)} \)=\( \frac{\pi*2 }{6*\sqrt{3}} \)=\( \frac{\pi }{3*\sqrt{3}} \)
\(a= \sqrt{\frac{\pi}{3\sqrt{3}}}\)
Ans: C
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Re: In a circle with centre O and radius 1 cm, an arc AB makes an angle 60 [#permalink]
03 Apr 2020, 07:18
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