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Question

In triangle $$ABC, $$ if $$\sin A \sin B=\dfrac{ab}{c^{2}}$$, then the triangle is 


A
Equilateral
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B
Isosceles
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C
Right angled
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D
Obtuse angled
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Solution

The correct option is A Equilateral
$$ \begin{aligned} &\begin{array}{l} \sin A \cdot \sin B=\frac{a b}{c^{2}} \\ \begin{aligned} \Rightarrow \frac{a b}{\sin A \cdot \sin B}=c^{2} &=\left(\frac{a}{\sin A}\right)\left(\frac{b}{\sin B}\right) \\ &=\left(\frac{c}{\sin c}\right)\left(\frac{c}{\sin c}\right) \end{aligned} \Rightarrow \sin ^{2} C=1 \end{array}\\ &\Rightarrow \quad C=90^{\circ} \quad \Rightarrow \quad \triangle A B C \text { is might - angled triangle } \end{aligned} $$

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