Question

The perimeter of a triangle PQR is six times the arithmetic mean of the sines of its angles. If a = 1, then $\angle A$ = .

• A. $60°$
• B. $90°$
• C. $30°$
• D. $45°$

Answer 1

The correct option is C $30°$

$a+b+c=6\times \frac{\mathrm{Sin}A+\mathrm{Sin}B+\mathrm{Sin}C}{3}\mathrm{or}2R(\mathrm{Sin}A+\mathrm{Sin}B+\mathrm{Sin}C)=2(\mathrm{Sin}A+\mathrm{Sin}B+\mathrm{Sin}C)\Rightarrow R=1\mathrm{Now},\frac{a}{\mathrm{Sin}A}=2R\Rightarrow \mathrm{Sin}A=\frac{1}{2}[\because a=1]\Rightarrow a=30°$