An equilateral triangle of side 9cm inscribed in a circle The radius of the circle is:
3cm
√3 cm
3√3 cm
3√32 cm
A
√3 cm
B
3√3 cm
C
3cm
D
3√32 cm
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Solution
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△ABC is an equilateral triangle
AB=BC=CA=9cm
O is the circumcentre of △ABC
∴OD id the perpendicular of the side BC
In △OBD and △ODC
OB=OC (Radius of the circle)
BD=DC (D is the mid point of BC)
OD=OD (common)
∴△OBD=△ODC
⇒∠BOD=∠COD
∠BOC=2∠BAC=2×60∘=120∘ (The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle)
∴∠BOD=∠COD=∠BOC2=120∘2=60∘
BD=BC=BC2=92cm
In △BOD
⇒sin∠BOD=sin60∘=BDOB
∴√32=92OB
⇒OB=92×2√3=3√3cm
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