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If the angle of triangle $A, B$ and $C$ are in $A P$ and $b: a=\sqrt{3}: 1$, then what is the value of $A$ ?
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Verified Answer
The correct answer is:
$30^{\circ}$
Here, $A, B$ and $C$ are in $A P$
So, $2 B=A+C$
And sum of angles of triangle, $A+B+C=180^{\circ}$
$\Rightarrow 2 B+B=180$
$\begin{aligned} & \Rightarrow 3 B=180 \\ & \Rightarrow B=60^{\circ}\end{aligned}$
Now, by sine rule,
$\frac{a}{\sin A}=\frac{b}{\sin B}$
$\Rightarrow \frac{b}{\mathbf{a}}=\frac{\sin B}{\sin A}$
$\Rightarrow \frac{\sqrt{3}}{1}=\frac{\sin 60^{\circ}}{\sin A}$
$\Rightarrow \sin A=(\sqrt{3} / 2) \times(1 / \sqrt{ } 3)$
$\Rightarrow \sin A=1 / 2$
$\Rightarrow A=30^{\circ}$
Hence, option (1) is correct.
So, $2 B=A+C$
And sum of angles of triangle, $A+B+C=180^{\circ}$
$\Rightarrow 2 B+B=180$
$\begin{aligned} & \Rightarrow 3 B=180 \\ & \Rightarrow B=60^{\circ}\end{aligned}$
Now, by sine rule,
$\frac{a}{\sin A}=\frac{b}{\sin B}$
$\Rightarrow \frac{b}{\mathbf{a}}=\frac{\sin B}{\sin A}$
$\Rightarrow \frac{\sqrt{3}}{1}=\frac{\sin 60^{\circ}}{\sin A}$
$\Rightarrow \sin A=(\sqrt{3} / 2) \times(1 / \sqrt{ } 3)$
$\Rightarrow \sin A=1 / 2$
$\Rightarrow A=30^{\circ}$
Hence, option (1) is correct.
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