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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
2 years 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 EF⊥AB, DG⊥CA
 
Since EF⊥AB,
 
EF is the height of triangles ADE and DBE
 
Area of △ADE=21​× base × height=21​AD×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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