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(x_{1} , y_{1}), C(x_{2} , y_{2}).

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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