The perimeter of a \Delta ABC is 6 times the arithmetic mean of the sines of ...

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Question Id : 11504 | Context :NIMCET 2023

The perimeter of a

$\Delta ABC$ is 6 times the arithmetic mean of the sines of its angles. If the side a is 1, then the angle A is

🎥 Video solution / Text Solution of this question is given below:

Given: Perimeter = 6 × Arithmetic Mean of sin A, sin B, sin C

Using Law of Sines: $\frac{a}{\sin A} = 2R$ , and a = 1 ⇒ $\sin A = \frac{1}{2R}$

Assume $A = 30^\circ \Rightarrow \sin A = \frac{1}{2} \Rightarrow 2R = 2$

⇒ b = 2 sin B, c = 2 sin C

Perimeter = $1 + b + c = 1 + 2 \sin B + 2 \sin C$

Mean = $\frac{\sin A + \sin B + \sin C}{3}$

Check: $1 + 2\sin B + 2\sin C = 6 \cdot \frac{1/2 + \sin B + \sin C}{3}$ ✅

✅ Final Answer: 30°

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