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



 None of these
The correct answer is:
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)
So, the ratio of the longest side and perimeter of triangle is
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