Q. 34.0( 5 Votes )

Prove analytically that the line segment joining the middle points of two sides of a triangle is equal to half of the third side.

Answer :

Let ∆ABC be any triangle such that O is the origin.


Let coordinates be A(0, 0), B(x1 , y1), C(x2 , y2).


Let D and E are the mid-points of the sides AB and AC respectively.


We have to prove that line joining the mid-point of any two sides of a triangle is equal to half of the third side which means,


DE = BC


By midpoint formula,


x = , y =


For midpoint D on AB,


x =, y =


x = and y =


Coordinate of D is (, )


For midpoint E on AC,


x =, y =


x = and y =


Coordinate of E is ( , )


By distance formula,


XY =


For BC,


BC =


For DE,


DE =


= ( )


= BC


DE = BC


Hence, we proved that line joining the mid-point of any two sides of a triangle is equal to half of the third side.


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