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 $\triangle A B C$ be $4 x, x$ and $x$.
Then, $4 x+x+x=180^{\circ}$
$\Rightarrow x=30^{\circ}$
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}} $
$=\sqrt{3}: 2+\sqrt{3}$