Question

If the angles of a triangle are in the ratio $1:1:4$, then the ratio of the perimeter of the triangle to its largest side is

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

Answer 1

The correct option is A $\sqrt{3}+2:\sqrt{3}$

Let $A,B,C$ be the three angles of the triangle

$\frac{A}{1}=\frac{B}{1}=\frac{C}{4}$ ...[Given]

$\therefore B=A$ and $C=4A$ ...[1]

$A+B+C={180}^0$

$A+A+4A={180}^0$ ...(Using [1] )

$⟹A={30}^0$

$\therefore B=A={30}^0$ and $C=4A={120}^0$

Using sine rule,

$\frac{a}{sinA}=\frac{b}{sinB}=\frac{c}{sinC}$

$\therefore \frac{a}{sin{30}^0}=\frac{b}{sin{30}^0}=\frac{c}{sin{120}^0}$

$\therefore \frac{a}{\frac{1}{2}}=\frac{b}{\frac{1}{2}}=\frac{c}{\frac{\sqrt{3}}{2}}$

$\therefore b=a$ and $c=\sqrt{3}a$

$Permeter=a+b+c=a+a+\sqrt{3}a=(2+\sqrt{3})a$

$LargestSide=c=\sqrt{3}a$

$\frac{Perimeter}{LargestSide}=\frac{(2+\sqrt{3})a}{\sqrt{3}a}$

$=\frac{2+\sqrt{3}}{\sqrt{3}}$

So, the answer is option (A)