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In Figure the line segment XY is parall...

In Figure the line segment XY is parallel to side AC of `DeltaA B C`and it divides the triangle into two parts of equal areas. Find the ratio `(A X)/(A B)`.

Text Solution

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we have `XY||AC`(GIVEN)
so `angle BXY=angleA` and `angleBYX=angleC`(Corresponding angles)
Therefore,`triangleABC~triangleXBY`(AA similarity criterion)
so
(`AR``(ABC)`)/(`AR``(XBY)`)=`((AB)/(XB))^2`
ALSO,`AR``(triangleABC)`/`AR``(triangleXBY)`=`2/1`
Therefore`((AB)/(XB))^2=2/1`,i.e,`(AB)/(XB)=sqrt2/1`
`(XB)/(AB)`=`1/sqrt2`
or,`1-(XB)/(AB)=1-1/sqrt2`;or,`(AB-XB)/(AB)=(sqrt2-1)/(sqrt2)`,i.e..`(AX)/(AB)`=`(2-sqrt2)/2`
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