In the following questions, a statement of assertion (Statement-1) is followed by a statement of reason (Statement-2). Mark the correct choice as :
Let $O$ and $O^{'}$ be respectively the circumcentre and the orthocentre of a triangle $A B C$ , and $S$ be any point in the plane of the triangle.
Statement-1 : $\vec{O A} + \vec{O B} + \vec{O C} = \vec{O O}'$
Statement-2 : $\vec{S A} + \vec{S B} + \vec{S C} = 3 \vec{S G},$ where $G$ is the centroid of $△ A B C .$

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Both Statement-1 and Statement-2 are true and Statement-2 is the correct explanation of Statement-1.

Both Statement-1 and Statement-2 are true but Statement-2 is not the correct explanation of Statement-1.

Statement-1 is true but Statement-2 is false.

Statement-1 is true but Statement-2 is true.

answer is A.

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Detailed Solution

$\vec{S A} + \vec{S B} + \vec{S C} = \vec{S A} + (\vec{S B} + \vec{S C})$
$\Rightarrow \vec{S A} + \vec{S B} + \vec{S C} = \vec{S A} + 2 \vec{S D}$
[ $∴$ D is the mid-point of BC]
$\Rightarrow \vec{S A} + \vec{S B} + \vec{S C} = (1 + 2) \vec{S G} = 3 \vec{S G}$
So, statement- $2$ is true. Replacing $S$ by $O$ in statement- $2$ , we get
$\begin{array}{l} \vec{O A} + \vec{O B} + \vec{O C} = 3 \vec{O G} \\ \Rightarrow \vec{O A} + \vec{O B} + \vec{O C} = 2 \vec{O G} + \vec{O G} \\ \Rightarrow \vec{O A} + \vec{O B} + \vec{O C} = \vec{G O^{'}} + \vec{O G} \\ [∵ 2 O G = G O^{'}] \\ \Rightarrow \vec{O A} + \vec{O B} + \vec{O C} = \vec{O G} + \vec{G O^{'}} \\ \Rightarrow \vec{O B} + \vec{O C} = \vec{O O^{'}} \end{array}$
So, statement- $1$ is true and statement- $2$ is a correct explanation for statement- $1$ .