In traiangle ABC, F and E are the points on sides AB and AC, respectively, such that FE || BC and FE divides the triangle in parts of equal area. If AD perpendicular to BC ....; AD intersects FE at G, then GD:AG ?
Answers
Answer:
Ratio GD : AG is 1 : ( √2 + 1 ).
Step-by-step explanation:
Given,
FE divides the triangle in parts of equal area.
It means :
= > Area of ∆AFE = Area of remaining part
= > Area of ∆AFE = Area of whole figure - Area of ∆AFE
= > Area of ∆AFE = Area of ∆ABC - Area of ∆AFE
= > Area of ∆AFE + Area of∆AFE = Area of ∆ABC
= > 2 x Area of ∆AFE = Area of ∆ABC
From the properties of triangles,
- 1 / 2 x base x height = Area of triangle
= > 2 x 1 / 2 x AG x FE = 1 / 2 x AD x BC
= > AG x FE = 1 / 2 x AD x BC
= > 2 x FE / BC = AD / AG
= > 2 x FE / BC = ( AG + GD ) / AG { AD = AG + GD }
= > 2 x FE / BC = AG / AF + GD / AG
= > 2 x FE / BC = 1 + GD / AG
= > [ 2 x FE / BC ] - 1 = GD / AG ... ( 1 )
As we can observe, quadrilateral FECB is a trapezium. Therefore :
= > Area of ∆AFE = Area of trapezium FECB
= > 1 / 2 x AG x FE = 1 / 2 x ( FE + BC ) x GD { Area of trapezium = 1 / 2 x sum of || sides x height }
= > AG x FE = ( FE + BC ) x GD
= > FE / ( FE + BC ) = GD / AG ... ( 2 )
Comparing the value of GD / AG from ( 2 ) :
= > 2 x [ FE / BC ] - 1 = FE / ( FE + BC )
= > ( 2 FE - BC ) / BC = FE / ( FE + BC )
= > ( 2FE - BC ) ( FE + BC ) = FE.BC
= > 2FE^2 + 2BC.FE - BC.FE - BC^2 = FE.BC
= > 2FE^2 + BC.FE - BC^2 = FE.BC
= > 2EF^2 = BC^2
= > √2 EF = BC
Therefore,
= > GD / AG = FE / ( FE + BC )
= > GD / AG = FE / ( FE + √2 FE )
= > GD / AG = 1 / ( √2 + 1 )
Hence the required ratio GD : AG is 1 : ( √2 + 1 ).
Answer:
In traiangle ABC, F and E are the points on sides AB and AC, respectively, such that FE || BC and FE divides the triangle in parts of equal area. If AD perpendicular to BC ....; AD intersects FE at G, then GD:AG ?
Step-by-step explanation:
here