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(D) In $\Delta ABC$,the Sine Rule states $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$.For option $(a)$: Given $a, \sin A, \sin B$. Fr

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$a, \sin A, R$

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(D) In $\Delta ABC$,the Sine Rule states $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$.For option $(a)$: Given $a, \sin A, \sin B$. From $\frac{a}{\sin A} = 2R$,we find $R$. Then $b = 2R \sin B$ and $c = 2R \sin C = 2R \sin(180^{\circ} - (A+B)) = 2R \sin(A+B)$. Thus,the triangle is uniquely determined.For option $(b)$: Given $a, b, c$. By the $SSS$ congruence criterion,the triangle is uniquely determined.For option $(c)$: Given $a, \sin B, R$. We have $b = 2R \sin B$. Now we have two sides $a, b$ and the circum-radius $R$. Since $\sin A = \frac{a}{2R}$,angle $A$ is determined. Thus,the triangle is uniquely determined.For option $(d)$: Given $a, \sin A, R$. We have $\frac{a}{\sin A} = 2R$. This is an identity that holds for any triangle with side $a$ and angle $A$ having circum-radius $R$. It does not provide enough information to fix the other sides or angles. Therefore,the triangle is not uniquely determined.

Which of the following sets of data does not uniquely determine an acute-angled $\Delta ABC$ ($R$ = circum-radius)?

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