Question

If the angles of a triangle are in the ratio $4:1:1$, then the ratio perimeter is the longest side and perimeter is

• A. $\sqrt{3}:2+\sqrt{3}$
• B. $1:6$
• C. $1:2+\sqrt{3}$
• D. $2:6$

Answer 1

Answer: (A)

Let the angle of $△ABC$ be $4x,x$ and $x$.
Then, $4x+x+x=180^{\circ }$
$\Rightarrow x=30^{\circ }$
So, the angles are $A=120^{\circ },B=30^{\circ }$ and $C=30^{\circ }$
Now, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=k$
$\Rightarrow a=\frac{\sqrt{3}}{2}k,b=\frac{1}{2}k\text{ and }c=\frac{1}{2}k$
$\therefore$ Required ratio $=\frac{a}{a+b+c}=\frac{\frac{\sqrt{3}}{2}k}{\frac{\sqrt{3}}{2}k+\frac{1}{2}k+\frac{1}{2}k}$
$=\frac{\sqrt{3}}{2+\sqrt{3}}$
$=\sqrt{3}:2+\sqrt{3}$