Question

With usual notations, the perimeter of a triangle ABC is 6 times the arithmetic mean of sine of its angles. If \(\mathrm{a}=1\), then \(\angle \mathrm{A}=\)

• A. \(\frac{\pi}{6}\)
• B. \(\frac{\pi}{3}\)
• C. \(\frac{\pi}{2}\)
• D. \(\frac{2\pi}{3}\)

Hint:

Use the sine rule to turn the perimeter condition into a value of the circumradius.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
By the sine rule, \(a=2R\sin A\), \(b=2R\sin B\), \(c=2R\sin C\).

Step 2: Use the Given Condition
Perimeter \(=6\times\dfrac{\sin A+\sin B+\sin C}{3}=2(\sin A+\sin B+\sin C)\).
Also perimeter \(=2R(\sin A+\sin B+\sin C)\), so \(R=1\).

Step 3: Find A
\(a=2R\sin A\Rightarrow1=2\sin A\Rightarrow\sin A=\dfrac{1}{2}\)
So \(A=\dfrac{\pi}{6}\) (or \(\dfrac{5\pi}{6}\), which is not an option).

Step 4: Compare the options
\(A=\dfrac{\pi}{6}\) matches the first option.

Final Answer:
Angle A equals pi by six. \[ \boxed{\text{(A) }\dfrac{\pi}{6}} \]