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In the given figure, line the segment XY...

In the given figure, line the segment XY is parallel to side AC of `Delta ABC` and it divides the triangles into two parts of equal area. Prove that `AX:AB=(2-sqrt(2)):2`

Text Solution

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Since `XY||AC`, we have,
`angle A= angle BXY and angle C= angle BYX` [ coress `angle`]
`:. Delta ABC~ Delta XBY`
`rArr (ar (Delta ABC))/(ar (Delta XBY))=(AB^(2))/(XB^(2))" ....(i)`
But, `ar (Delta ABC)~ 2xx ar (Delta XBY)` [ given]
`rArr ar (Delta ABC)/(ar (Delta XYB))=2 " "....(ii)`
From (i) and (ii), we get
`(AB^(2))/(XB^(2))=2 rArr ((AB)/(XB))^(2)=2`
`rArr (AB)/(XB)=sqrt(2) AB= sqrt(2)(XB)`
`rArr AB=sqrt(2)(AB-AX)`
`rArr (AX)/(AB)=sqrt(2(-1))/(sqrt(2))xx(sqrt(2))/(sqrt(2))=((2-sqrt(2)))/(2)`
Hence, `AX:AB(2-sqrt(2)):2`
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