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write the prove of basic proportionally theorem.prove it ........................................................................................

write the prove of basic proportionally theorem.prove it ........................................................................................
 


 

Grade:10

1 Answers

Armaan Arora
100 Points
one year ago

 

Basic Proportionality Theorem states that, if a line is parallel to a side of a triangle which intersects the other sides into two distinct points,then the line divides those sides of the triangle in proportion.
 
Let ABC be the triangle.
 
The line l parallel to BC intersect AB at D and AC at E.
 
To prove DBAD=ECAE
 
Join BE,CD
 
Draw EFABDGCA
 
Since EFAB,
 
EF is the height of triangles ADE and DBE
 
Area of ADE=21× base × height=21AD×EF
 
Area of DBE=21×DB×EF
areaofΔDBEareaofΔADE=21×DB×EF21×AD×EF=DBAD          ........(1)
Similarly,
areaofΔDCEareaofΔADE=21×EC×DG21×AE×DG=ECAE            ......(2)
 
But ΔDBE and ΔDCE are the same base DE and between the same parallel straight line BC and DE.
 
Area of ΔDBE= area of ΔDCE         ....(3)
 
From (1), (2) and (3), we have
 
DBAD=ECAE
Hence proved.

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