Question

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 1

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 = \(\frac{1}2\) BC

By midpoint formula,

x = \(\frac{x_1+x_2}2\), y = \(\frac{y_1+y_2}2\)

or midpoint D on AB,

 x = \(\frac{x_1+0}2\), y = \(\frac{y_1+0}2\)

∴ x = \(\frac{x_1}2\) and y = \(\frac{y_1}2\)

∴ Coordinate of D is (\(\frac{x_1}2\),\(\frac{y_1}2\))

For midpoint E on AC,

  x = \(\frac{x_2+0}2\), y = \(\frac{y_2+0}2\)

 ∴ x = \(\frac{x_2}2\) and y = \(\frac{y_2}2\)

∴ Coordinate of E is ( \(\frac{x_2}2\), \(\frac{y_2}2\)) 

By distance formula,

Hence, 

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