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. Statement - 1 is True, Statement - 2 is True , Statement - 2 is a correct explanation for Statement - 1.
B. Statement -1 is True, Statement - 2 is True, Statement -2 is not a correct explanation for Statement - 1.
C. Statement - 1 is True, Statement - 2 is False.
D. Statement - 1 is False, Statement - 2 is True.

Answer

A

Explanation

Using triangle law of addition in triangles OAC and OBC, we have

`vec(OA)+vec(AC)=vec(OC) and vec(OB)+vec(BC)=vec(OC) `
`rArr vec(OA)+vec(AC) +vec(OB)+vec(BC)=vec(OC)+vec(OC)`
`rArr vec(OA) + vec(OB) + (vec(AC) + vec(BC)) = 2vec(OC) `
`rArr vec (OA)+ vec(OB) = 2 vec(OC) `
` [ because C " is the mid-point of" AB therefore vec(AC) + vec(BC) = vec(0)]`
So, statement -2 is true.
Using statement - 2, we have
`vec (AB) + vec (AD) = 2 vec(AF) and vec(CB) + vec(CD) = 2 vec(CF)`

`rArr vec(AB) + vec(AD) + vec(CB) + vec(CD) = 2 ( vec(AF) + vec(CF))`
`rArr vec(AB) + vec(AD) + vec(CB) + vec(CD) = -2(vec(FA) + vec(FC))`
`rArr vec(AB) + vec(AD) + vec(CB) + vec(CD)= -2 (2vec(FC) " " `[Using statement-2]
`rArr vec(AB) + vec (AD) + vec(CB) + vec(CD) = 4 vec(EF)`
So, statement -1 is true and statement - 2 is a correct explanation for statemetn -1.