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If the angles of a triangle $\mathrm{ABC}$ are in the ratio $1: 2: 3$. then the corresponding sides are in the ratio
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The correct answer is:
$1: \sqrt{3}: 2$
Given, angles of triangle are in ratio $1: 2: 3$ Consider, $\mathrm{A}=30^{\circ}, \mathrm{B}=60^{\circ}$ and $\mathrm{C}=90^{\circ}$
We know, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$
$\Rightarrow \frac{a}{\sin 30^{\circ}}=\frac{b}{\sin 60^{\circ}}=\frac{c}{\sin 90^{\circ}}$
$\Rightarrow \frac{a}{\frac{1}{2}}=\frac{b}{\frac{\sqrt{3}}{2}}=\frac{c}{1}$
$\Rightarrow a: b: c=\frac{1}{2}: \frac{\sqrt{3}}{2}: 1=1: \sqrt{3}: 2$
We know, $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$
$\Rightarrow \frac{a}{\sin 30^{\circ}}=\frac{b}{\sin 60^{\circ}}=\frac{c}{\sin 90^{\circ}}$
$\Rightarrow \frac{a}{\frac{1}{2}}=\frac{b}{\frac{\sqrt{3}}{2}}=\frac{c}{1}$
$\Rightarrow a: b: c=\frac{1}{2}: \frac{\sqrt{3}}{2}: 1=1: \sqrt{3}: 2$
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