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# If in any $\Delta ABC$ $A=30^{\large \circ}$ and $B=60^{\large\circ}$ then find $a: b:c$

$\begin {array} {1 1}(a)\;1 : 2 : 3&(b)\;2 : \sqrt 3: 5\\(c)\;1 : \sqrt 3 :2&(d)\;None \;of\;these\end{array}$

$A+B+C=180^{\large\circ}$
$30^{\large\circ}+60^{\large\circ}+C=180^{\large\circ}$
$C=180^{\large\circ}-90^{\large\circ}$
$\;\;\;=90^{\large\circ}$
Now $\large\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$
$a : b: c=\sin A :\sin B :\sin C$
$\qquad\quad\;\;=\sin 30^{\large\circ} :\sin 60^{\large\circ} :\sin 90^{\large\circ}$
$\qquad\quad\;\;=\large\frac{1}{2} :\frac{\sqrt 3}{2}$$: 1$
$\qquad\quad\;\;=1:\sqrt 3 : 2$