Properties of Triangles 1 Question 6
6. In a $\triangle A B C$, among the following which one is true?
(a) $(b+c) \cos \frac{A}{2}=a \sin \frac{B+C}{2}$
(2005, 1M)
(b) $(b+c) \cos \frac{B+C}{2}=a \sin \frac{A}{2}$
(c) $(b-c) \cos \frac{B-C}{2}=a \cos \frac{A}{2}$
(d) $(b-c) \cos \frac{A}{2}=a \sin \frac{B-C}{2}$
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Answer:
Correct Answer: 6. (d)
Solution:
- Let $a, b, c$ are the sides of $\triangle A B C$.
Now, $\quad \frac{b+c}{a}=\frac{k(\sin B+\sin C)}{k \sin A} \quad$ [by sine rule]
$$ =\frac{2 \sin \frac{B+C}{2} \cos \frac{B-C}{2}}{2 \sin \frac{A}{2} \cos \frac{A}{2}} \Rightarrow \frac{b+c}{a}=\frac{\cos \frac{B-C}{2}}{\sin \frac{A}{2}} $$
Also,
$$ \frac{b-c}{a}=\frac{\sin \frac{B-C}{2}}{\cos \frac{A}{2}} $$
