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MHT CET202615 April 2026Evening ShiftMathematicsProperties of TrianglesActual

Let P₁ , P₂ , P₃ be the altitudes of a triangle ABC from the vertices A, B, C respectively. If denotes the area of the triangle and s is the semi-perimeter of the triangle, then A P₁ + B P₂ + C P₃ =

Options

  1. AR
  2. B1 R
  3. CR^2
  4. D1 R^2

Correct answer

B. 1 R

Step-by-step solution

The area of the triangle is given by = 1 2 a P₁ = 1 2 b P₂ = 1 2 c P₃ . This gives P₁ = 2 a , P₂ = 2 b , and P₃ = 2 c . Substituting these into the given expression: A P₁ + B P₂ + C P₃ = a A + b B + c C 2 Using the sine rule a = 2R A , b = 2R B , c = 2R C , the numerator becomes: a A + b B + c C = 2R A A + 2R B B + 2R C C = R( 2A + 2B + 2C) In any triangle ABC , the sum of sines of double angles is 2A + 2B + 2C = 4 A B C . Thus, the numerator is 4R A B C . The area of the triangle can be written as = abc 4R = (2R A

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