Question

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

• A. $ \sqrt 3 : (2 + \sqrt 3)$
• B. 1:03:02
• C. 1 : 2 + $ \sqrt 3 $
• D. 2:03

Answer 1

The Correct Option is A

Given, ratio of angles are 4 : 1: 1.
$\Rightarrow 4 x + x + x = 180^\circ$
$\Rightarrow x = 30^\circ $
$\therefore \angle A = 120^\circ, \, \angle B = \angle C = 30^\circ $
Thus, ratio of longest side to perimeter = $ \frac{a}{ a + b + c}$
Let b = c = x
$\Rightarrow a^2 = b^2 + c^2 - 2bc \, cos A $
$\Rightarrow a^2 = 2x^2 - 2x^2 \, cos \, A = 2x^2 (1 - cos \, A)$
$\Rightarrow a^2 = 4x^2 \, sin^2 \, A/2$
$\Rightarrow a = 2x sin A/2$
$\Rightarrow a = 2x sin 60^\circ = \sqrt 3 x $
Thus, required ratio
= $ \frac{a}{ a + b + c} = \frac{ \sqrt 3 x}{ x + x + \sqrt 3 x } = \frac{ \sqrt 3 }{ 2 + \sqrt 3} = \sqrt 3 : 2 + \sqrt 3 $