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MHT CET202521 Apr 2025Evening ShiftMathematicsProperties of TrianglesActual

With usual notations, the perimeter of a triangle ABC is 6 times the arithmetic mean of sine of its angles. If a=1 , then A =

Options

  1. A6
  2. B3
  3. C2
  4. D2 3

Correct answer

A. 6

Step-by-step solution

Let the sides of the triangle be a , b , c with opposite angles A , B , C respectively, and let R be the circumradius. The perimeter equals six times the arithmetic mean of the sines of the angles: a + b + c = 6 A + B + C 3 = 2( A + B + C) . By the Law of Sines, a = 2R A , b = 2R B , c = 2R C . Substituting gives: 2R( A + B + C) = 2( A + B + C) . Since A + B + C > 0 for any triangle, we cancel these terms to obtain 2R = 2 , hence R = 1 . Given a = 1 , applying the Law of Sines again yields 1 = 2R A = 2 1 A , so A =

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