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AP EAMCET201921 Apr 2019Morning ShiftMathematicsProperties of TrianglesActual

In a A B C , if 2 r₂ r₃ r₂-r₁ =r₃-r₁ , then r₁ (r₂+r₃ ) r₁ r₂+r₂ r₃+r₃ r₁ =

Options

  1. Aa^2+b^2+c^2 ^2
  2. Bb-c
  3. C1 2 R
  4. D2R

Correct answer

D. 2R

Step-by-step solution

Given, aligned & 2 r₂ r₃ r₂-r₁ =r₃-r₁ & 2 r₂ r₃= (r₂-r₁ ) (r₃-r₁ ) & 2 (s-b) (s-c) = ( s-b - s-a ) ( s-c - s-a ) & 2 ^2 (s-b)(s-c) = ^2 s-a-s+b (s-b)(s-a) s-a-s+c (s-c)(s-a) & 2 (s-b)(s-c) = (b-a) (s-b)(s-a) (c-a) (s-c)(s-a) & 2(s-a)^2=(b-a)(c-a) & 2(b+c-a)^2 4 =(b-a)(c-a) & b^2+c^2+a^2+2 b c-2 c a-2 a b=2 & b^2+c^2+a^2+2 b c-2 c a-2 a b & b^2+c^2+a^2=2 a^2 a^2=b^2+c^2 & Now r₁ (r₂+r₃ ) r₁ r₂+r₂ r₃+r₃ r₁ aligned aligned & = s-a 1 s-b + 1 s-c s & = ^2(2 s-b-c) s(s-a)(s-a)(s-c) & = ^2(a+b+c-b-c) ^2 =a=2 R aligned

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