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Easy

Question

If the angles of a triangle are in ratio 4 : 1 : 1 then the ratio of the longest side and perimeter of triangle is -

  1. fraction numerator 1 over denominator 2 plus square root of 3 end fraction    
  2. fraction numerator 2 over denominator square root of 3 minus 2 end fraction    
  3. fraction numerator square root of 3 over denominator 2 plus square root of 3 end fraction    
  4. None of these    

The correct answer is: fraction numerator square root of 3 over denominator 2 plus square root of 3 end fraction


    Given that the angles of a triangle are in ratio 4 : 1 : 1 then we have to find the ratio of the longest side and perimeter of triangle.
    Given: Ratio of angles of a triangle =4:1:1
    Let the angles of a triangle are 4x,x and x respectively
    4x+x+x=180
    x=30

    A=4x=4×30=120

    B=x=30

    and C=x=30

    By sine Law, we have
    a/sinA=b/sinB=c/sinC=k (assume)
    a=ksinA=k×sin120=k×sin(90+30)=kcos30=(3/2)k
    b=ksinB=ksin30=k×1/2=k/2
    c=ksinC=ksin30=k×1/2=k/2
    A is the biggest angle.
    So, the largest side will be BC. [Side opposite to biggest angle is longest.]
    Ratio of longest side to the perimeter =a/(a+b+b)
    fraction numerator begin display style fraction numerator square root of 3 k over denominator 2 end fraction end style over denominator fraction numerator square root of 3 k over denominator 2 end fraction plus begin display style k over 2 end style plus k over 2 end fraction
fraction numerator fraction numerator square root of 3 over denominator 2 end fraction over denominator fraction numerator square root of 3 over denominator 2 end fraction plus begin display style 2 over 2 end style end fraction
fraction numerator square root of 3 over denominator square root of 3 plus 2 end fraction

    So, the ratio of the longest side and perimeter of triangle isfraction numerator square root of 3 over denominator square root of 3 plus 2 end fraction

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