Properties of Triangles 1 Question 16
16. There exists a $\triangle A B C$ satisfying the conditions
(a) $b \sin A=a, A<\frac{\pi}{2}$
(b) $b \sin A>a, A>\frac{\pi}{2}$
(c) $b \sin A>a, A<\frac{\pi}{2}$
(d) $b \sin A<a, A<\frac{\pi}{2}, b>a$
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Answer:
Correct Answer: 16. (a, d)
Solution:
- The sine formula is
$$ \frac{a}{\sin A}=\frac{b}{\sin B} \Rightarrow a \sin B=b \sin A $$
(a) $b \sin A=a \Rightarrow a \sin B=a$
$\Rightarrow B=\frac{\pi}{2}$
Since, $\angle A<\frac{\pi}{2}$, therefore the triangle is possible.
(b) and (c) $b \sin A>a$
$\Rightarrow a \sin B>a \Rightarrow \sin B>1$
$\therefore \triangle A B C$ is not possible.
(d) $b \sin A<a$
$\Rightarrow a \sin B<a \Rightarrow \sin B<1 \quad \Rightarrow \quad \angle B$ exists.
Now, $b>a \quad \Rightarrow \quad B>A$
Since, $A<\frac{\pi}{2}$
$\therefore$ The triangle is possible.
Hence, (a) and (d) are the correct answers.
