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AP EAMCET202123 Aug 2021Evening ShiftMathematicsProperties of TrianglesActual

In A B C, 16 R s A 2 B 2 C 2 s-c is equal to

Options

  1. Ar₁ r₂ r₃
  2. Br₁ r₂ r₃
  3. Cr₁+r₂+r₃
  4. D4 r₁ r₂ r₃

Correct answer

D. 4 r₁ r₂ r₃

Step-by-step solution

In A B C, A+B+C= ...(i) When A, B, C are the angles of A B C . Let R be the circumradius and s be semiperimeter and be the area of A B C 16 R s A 2 B 2 C 2 s-c = (4 R )(4 s) s-c (s-b)(s-c) b c (s-a)(s-c) a c s(s-c) a b A 2 = (s-b)(s-c) b c , B 2 = (s-a)(s-c) a c C 2 = s(s-c) a b r₁= s-a , r₂= s-b , r₃= s-c aligned & and = a b c 4 R & or 4 R =a b c aligned aligned & = (a b c)(4 s) s-c (s-c) s(s-a)(s-b)(s-c) a b c & =4 s aligned aligned & = 4 s ^3 ^2 = 4 s ^3 s(s-a)(s-b)(s-c) & =4 s-a s-b s-c aligned =4 r₁ r₂ r₃

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