If `A B C D` is quadrilateral and `E and F` are the mid-points of `AC and BD` respectively, prove that ` vec A B+ vec A D` +` vec C B` +` vec C D` =4 ` vec E Fdot`

Options

A. `3 vec(EF)`
B. `4vec(EF)`
C. `4 vec(FE)`
D. `3 vec(FE)`

Answer

B

Explanation

Since F is the middle point of BD. Therefore,
`A vec(B)+vec(AD)=2 vec(AF) " " ` …(i)
Similarly, we have
`vec(CB)+vec(AD)+2vec(CF) " " ` …(ii)
Adding (i) and (ii), we get
`A vec(B)+vec(AD)+C vec(B)+vec(CD)=2(vec(AF)+vec(CF))= -2(Fvec(A)+Fvec(C)) `
`rArr A vec(B) +vec(AD)+C vec(B)+vec(CD)= -2(2vec(FE))`
`rArr A vec(B)+vec(AD) +C vec B +vec(CD)=4vec(EF) " " [ because E " is the mid-point of " AC]`