Question

Consider the following statements :
1. There exists no triangle ABC for which sin A + sin B = sin C.
2. If the angles 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 statements is/are correct ?

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

Answer 1

The correct option is C Both 1 and 2

According to the Sine Rule

For a triangle (eg. shown in figure)

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R$, where 2 R is the radius of circumcircle of the triangle

Given- $\sin A+\sin B=\sin C$

$\frac{a}{2R}+\frac{b}{2R}=\frac{c}{2R}$

$\Rightarrow a+b=c$

But we know that, for any triangle, sum of 2 sides is always greater than the third side, i.e $a+b>c$

$\therefore$ The statement "There exists no triangle ABC for which sin A + sin B = sin C" is true

$\sin A+\sin B\ne \sin C$

Statement 2-

let the angles be $x,2x,3x$

$\Rightarrow x+2x+3x=6x={180}^o$

$\Rightarrow x={30}^o$

The angles are ${30}^o,{60}^o,and{90}^o$ respectively

Their Sines are $\frac{1}{2},\frac{\sqrt{3}}{2},and1$ respectively

We know that side of a triangle is $\propto$ Sine of Angle opposite to it

$\Rightarrow$ The ratios being $1:\sqrt{3}:2$

Hence both statement 1 and statement 2 are correct.