UPSC 2017 CDS II Math Paper

Let O be the centre of thecircumcircle of ΔABC.
Remember: In a circle, the angle subtended by the arc at the centre is twice the angle subtended by it at any point on the remaining circle.
In ΔABC,∠A=30°
∴ ∠BOC=2∠A=60°
In ΔBOC,OB = OC = 10 cm(Circum-radius of the circle)
⇒ ∠OBC = ∠OCB
Remember: Equal sides have equal angles opposite to them.
∴ ∠BOC = ∠OBC = ∠OCB = 60°
∴ ΔBOC is an equilateral triangle
⇒ BC = OB = OC = 10 cm
Trick :
By sine rule:

a

SinA

=

b

SinB

=

c

SinC

=2R
where a is the side opposite to angle A and R is the circumradius
∴

a

Sin‌30°

=2×10⇒a=10cm