The angles of a triangle are in the ratio 3: | vedclass

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(D) Let the angles of the triangle be $3x, 5x,$ and $10x$. Since the sum of angles in a triangle is $180^{\circ}$,we have $3x + 5x + 10x = 180^{\circ}

Vedclass · Accepted Answer

$1: 2 \cos 10^{\circ}$

Vedclass · Answer

(D) Let the angles of the triangle be $3x, 5x,$ and $10x$. Since the sum of angles in a triangle is $180^{\circ}$,we have $3x + 5x + 10x = 180^{\circ}$. $18x = 180^{\circ} \Rightarrow x = 10^{\circ}$. The angles are $30^{\circ}, 50^{\circ},$ and $100^{\circ}$. By the Sine Rule,the sides are proportional to the sines of their opposite angles: $a : b : c = \sin A : \sin B : \sin C$. The smallest side corresponds to the smallest angle $(30^{\circ})$ and the greatest side corresponds to the greatest angle $(100^{\circ})$. Ratio $= \sin 30^{\circ} : \sin 100^{\circ}$. Since $\sin 100^{\circ} = \sin(180^{\circ} - 80^{\circ}) = \sin 80^{\circ} = \cos 10^{\circ}$. Ratio $= \frac{1}{2} : \cos 10^{\circ} = 1 : 2 \cos 10^{\circ}$.

The angles of a triangle are in the ratio $3: 5: 10$. Then the ratio of the smallest side to the greatest side is:

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