If the angles of a triangle are in the ratio | vedclass

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Question

(A) Let the angles of the triangle be $4x, x$,and $x$. Since the sum of angles in a triangle is $180^{\circ}$,we have $4x + x + x = 180^{\circ}$,which

Vedclass · Accepted Answer

$\sqrt{3}:(2+\sqrt{3})$

Vedclass · Answer

(A) Let the angles of the triangle be $4x, x$,and $x$. Since the sum of angles in a triangle is $180^{\circ}$,we have $4x + x + x = 180^{\circ}$,which implies $6x = 180^{\circ}$,so $x = 30^{\circ}$. The angles are $120^{\circ}, 30^{\circ}$,and $30^{\circ}$. Using the Sine Rule,$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = k$,where $a$ is the longest side opposite to $120^{\circ}$. Then $a = k \sin 120^{\circ}$,$b = k \sin 30^{\circ}$,and $c = k \sin 30^{\circ}$. The ratio of the longest side to the perimeter is $\frac{a}{a+b+c} = \frac{\sin 120^{\circ}}{\sin 120^{\circ} + \sin 30^{\circ} + \sin 30^{\circ}}$. Substituting the values: $\frac{\frac{\sqrt{3}}{2}}{\frac{\sqrt{3}}{2} + \frac{1}{2} + \frac{1}{2}} = \frac{\frac{\sqrt{3}}{2}}{\frac{\sqrt{3}+2}{2}} = \frac{\sqrt{3}}{2+\sqrt{3}}$.

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

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