Using vectors , prove that the line joining the midpoints of two sides of a triangle, is parallel to the base and half its length.

Explanation

In Fig. 2.12., from the triangle law of vector addition in `Delta ABC`
`vec(AB)+vec(BC)=vec(AC)`
or, `1/2 vec(AB) +1/2 vec(BC)=1/2 vec(AC).......(1)`

As D and E are the mid-points of sides AB and AC,
`vec(AD)=1/2vec(AB) and vec(AE)=1/2vec(AC)`
From the triangle law of vector addition in `Delta ADE`
`vec(AD)+vec(DE)=vec(AE)`
`or, 1/2vec(AB)+vec(DE)=1/2vec(AC).........(2)`
From (1) and (2) ,`vec(DE)=1/2vec(BC)`
Hence , `vec(DE) and vec(BC)` are parallel and `DE=1/2BC.`