Consider the following statement :
1. There exixts no triangle ABC for which sin A + sin B = sin C.
2. If the angle of a triangle are in the ratio 1 : 2 : 3, then its sides will be in the ratio `1 : sqrt(3) : 2`.
Which of the above statement is/are correct?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Answer

C

Explanation

1. Given, sin A + sin B = sin C
`a+b=c" "(because "By sine law",(sinA)/(a)=(sinB)/(b)=(sinC)/(c)=K)`
Here, the sum of two sides of `Delta`ABC is equal to the third side, but it is not possible (Because by triangle inequality, the sum of the length of two sides of a triangle is always greater than the length of the third side)
`a + b gt c`
2. Ratio of angles of a triangle
`A : B : C = 1 : 2 : 3`
`A + B + C = 180^(@)`
`therefore A = 30^(@)`
`B = 60^(@)`
`C = 90^(@)`
the ratio in sides according to sin rule a : b : c = sin A : sin B : sin C
`=sin 30^(@) : sin 60^(@) : sin 90^(@)`
`=(1)/(2),(sqrt(3))/(2), 1=(1)/(2) : (sqrt(3))/(2) : 1 rArr 1 : sqrt(3) : 2`