Properties of Triangles 3 Question 4
4. Which of the following pieces of data does not uniquely determine an acute angled $\triangle A B C$ ( $R$ being the radius of the circumcircle)?
(2002, 1M)
(a) $a, \sin A, \sin B$
(b) $a, b, c$
(c) $a, \sin B, R$
(d) $a, \sin A, R$
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Answer:
Correct Answer: 4. (d)
Solution:
- First solve each option separately.
(a) If $a, \sin A, \sin B$ are given, then we can determine $b=\frac{a}{\sin A} \sin B, c=\frac{a}{\sin A} \sin C$. So, all the three sides are unique.
So, option (a) is incorrect.
(b) The three sides can uniquely make an acute angled triangle. So, option (b) is incorrect.
(c) If $a, \sin B, R$ are given, then we can determine $b=2 R \sin B, \sin A=\frac{a \sin B}{b}$. So, $\sin C$ can be determined.
Hence, side $c$ can also be uniquely determined.
(d) If $a, \sin A, R$ are given, then
$$ \frac{b}{\sin B}=\frac{c}{\sin C}=2 R $$
But this could not determine the exact values of $b$ and $c$.
