If o is the circumcenter, G is the centroid and O' is orthocenter or triangle ABC then prove that:

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

We have,
`vec(SA) + vec (SB) + vec(SC)=vec(SA) + (vec (SB) + vec(SC))`
`rArr vec(SA) + vec (SB) + vec(SC)=vec(SA) + 2 vec(SD) " " [ because D " is the mid-point of " BC]`
`rArr vec(SA) + vec (SB) + vec(SC)= (1+2)vec(SG)=3 vec(SG)`
So, statement- 2 is true.
Replacing S by O in statement -2, we get
`vec(OA) + vec (OB) + vec(OC)=3 vec(OG)`
`rArr vec(OA) + vec (OB) + vec(OC)=2 vec(OG)+vec(OG)`
`rArr vec(OA) + vec (OB) + vec(OC)=vec(GO')+ vec(OG) " " [ because 2 OG = GO']`

`rArr vec(OA) + vec (OB) + vec(OC)=vec(OG) + vec(GO)' `
`rArr vec(OA) + vec (OB) + vec(OC)= vec(OO)' `
So, statement - 1 is true and statement -2 is a correct explanation for statement -1.