Maths-
General
Easy

Question

In a ΔABC, let straight angle C equals pi over 2.  If r and R are the inradius and the circumradius respectively of the triangle then 2(r + R) is equal to -

  1. a + b
  2. b + c
  3. c + a
  4. a + b + c

The correct answer is: a + b


    Here, R = OA = OB = OC equals 1 half A B equals c over 2
    r equals straight capital delta over s equals fraction numerator 1 half a b over denominator 1 half left parenthesis a plus b plus c right parenthesis end fraction equals fraction numerator a b over denominator a plus b plus c end fraction

    table attributes columnalign right left right left right left right left right left right left columnspacing 0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em end attributes row cell therefore r plus R equals fraction numerator a b over denominator a plus b plus c end fraction plus c over 2 end cell row cell equals fraction numerator 2 a b plus c left parenthesis a plus b plus c right parenthesis over denominator 2 left parenthesis a plus b plus c right parenthesis end fraction end cell row cell equals fraction numerator 2 a b plus c a plus b c plus a squared plus b squared over denominator 2 left parenthesis a plus b plus c right parenthesis end fraction open parentheses because c squared equals a squared plus b squared close parentheses end cell row cell equals fraction numerator left parenthesis a plus b right parenthesis squared plus c left parenthesis a plus b right parenthesis over denominator 2 left parenthesis a plus b plus c right parenthesis end fraction equals fraction numerator left parenthesis a plus b right parenthesis left parenthesis a plus b plus c right parenthesis over denominator 2 left parenthesis a plus b plus c right parenthesis end fraction end cell row cell therefore 2 left parenthesis r plus R right parenthesis equals a plus b end cell end table

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