UPSC 2015 CDS I Math Paper

Let the angles be 4k, k and k, respectively.
Now,4k+k+k=180°
⇒6k=180°
⇒k=30°
Therefore, the angles of the triangle are 120°, 30° and 30°.
Let the sides opposite to theseangles be a, b and c. So, a is the largest side of the triangle.
∴

a

sin‌120°

=

b

sin‌30°

=

c

sin‌30°

⇒

a

( √3 2 )

=

b

( 1 2 )

=

c

( 1 2 )

(sin‌120°=sin‌60°=

√3

2

)
⇒

a

√3

=

b

1

=

c

1

=

a+b+c

√3+1+1

⇒

a

a+b+c

=

√3

2+√3