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

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(C) Let the angles be $x, 2x,$ and $7x.$ Since the sum of angles in a triangle is $180^{\circ},$ we have $x + 2x + 7x = 180^{\circ}$ $\Rightarrow 10x

Vedclass · Accepted Answer

$(\sqrt{5} + 1):(\sqrt{5} - 1)$

Vedclass · Answer

(C) Let the angles be $x, 2x,$ and $7x.$ Since the sum of angles in a triangle is $180^{\circ},$ we have $x + 2x + 7x = 180^{\circ}$ $\Rightarrow 10x = 180^{\circ}$ $\Rightarrow x = 18^{\circ}.$ Thus,the angles are $18^{\circ}, 36^{\circ},$ and $126^{\circ}.$ By the Sine Rule,the sides are proportional to the sines of the opposite angles. Let the sides be $a, b, c$ opposite to $18^{\circ}, 36^{\circ}, 126^{\circ}$ respectively. The greatest side is $c$ and the least side is $a.$ The ratio is $\frac{c}{a} = \frac{\sin(126^{\circ})}{\sin(18^{\circ})}.$ Since $\sin(126^{\circ}) = \sin(180^{\circ} - 54^{\circ}) = \sin(54^{\circ}) = \cos(36^{\circ}) = \frac{\sqrt{5} + 1}{4}$ and $\sin(18^{\circ}) = \frac{\sqrt{5} - 1}{4}.$ Therefore,the ratio is $\frac{(\sqrt{5} + 1)/4}{(\sqrt{5} - 1)/4} = \frac{\sqrt{5} + 1}{\sqrt{5} - 1}.$

If the angles of a triangle are in the ratio $1 : 2 : 7,$ then the ratio of its greatest side to the least side is

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