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A line divides an equilateral triangle A...

A line divides an equilateral triangle `ABC` of side 1 unit into two parts with same perimeter and different are `S_(1)` and `S_(2)` sq. units. If maximum value of `(S_(1))/(S_(2))=m/n` (where `m` and `n` coprime), then `|m-n|` is _____

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Verified by Experts

The correct Answer is:
2
`x+y=1-x+1-y1`
`x+y=3/2`
`(S_(1))/(S_(3))=(1/2xy sin 60^(@))/((sqrt(3))/4)=(xysqrt(3))/(((sqrt(3))/4))=xy=x(3/2-x)`
`:' S_(2)=S_(3)-S_(1)`
`(S_(2))/(S_(1))=1-(S_(1))/(S_(3))=1-xy=1-x(3/2-x)`
`(S_(1))/(S_(3))("max")=3/16` as `(S_(2))/(S_(3))=16/9-1=(7/9)`
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