Question

Assertion: \[(A):\overline {GA} + \overline {GB} + \overline {GC} = \overline 0 \;\] where $G$ is the centroid of triangle ABC.
Reason \[(R):\,\overline {AB} \, = \,P.V\,of\,B\, - \,P.V\,of\,A\]
A. Both Assertion and Reason are individually true and Reason is the correct explanation of Assertion.
B. Both Assertion and Reason are individually true and Reason is not the correct explanation of Assertion.
C. Assertion is true but Reason is false
D. Assertion is false but Reason is true.

Answer 1

Hint:We know that the centroid of a triangle is the point of intersection of its medians.Here the question is related to the verbal reasoning. The assertion and reason is one kind of verbal reasoning. Here in this question first we check if the reason is true or not.Then we are going to check if the assertion is true or false. While checking the assertion we are going to consider the reason.

Complete step by step answer:
To solve this problem first we have to know about the definition of centroid. The centroid of a triangle is the point of concurrency of the three medians. The centroid is represented as G.The formula for the centroid is given by \[G = \dfrac{{\overline a + \overline b + \overline c }}{3}\], where \[\overline a \], \[\overline b \] and \[\overline c \] are the points of the triangle.
Now we consider the Reason.
\[(R):\,\overline {AB} \, = \,P.V\,of\,B\, - \,P.V\,of\,A\]
P.V means the present value. Thus, the reason is true.

Now we have to check whether the Assertion is true or not. We calculate the \[\overline {GA} \], where $G$ is a centroid and $A$ is the point of the triangle.
\[ \Rightarrow \overline {GA} = P.V\,of\,A\, - P.V\,of\,G\]
On substituting the values
\[ \Rightarrow \overline {GA} = \overline a - \dfrac{{\overline a + \overline b + \overline c }}{3}\]
On simplifying we have
\[ \Rightarrow \overline {GA} = \dfrac{{2\overline a - \overline b - \overline c }}{3}\] -------- (1)
Now we calculate the \[\overline {GB} \]
\[ \Rightarrow \overline {GB} = P.V\,of\,B\, - P.V\,of\,G\]
On substituting the values
\[ \Rightarrow \overline {GB} = \overline b - \dfrac{{\overline a + \overline b + \overline c }}{3}\]
On simplifying we have
\[ \Rightarrow \overline {GB} = \dfrac{{ - \overline a + 2\overline b - \overline c }}{3}\] ------- (2)

Now we calculate the \[\overline {GC} \]
\[ \Rightarrow \overline {GC} = P.V\,of\,C\, - P.V\,of\,G\]
On substituting the values
\[ \Rightarrow \overline {GC} = \overline c - \dfrac{{\overline a + \overline b + \overline c }}{3}\]
On simplifying we have
\[ \Rightarrow \overline {GC} = \dfrac{{ - \overline a - \overline b + 2\overline c }}{3}\] ---------(3)
On adding (1), (2) and (3) we get
\[ \Rightarrow \overline {GA} + \overline {GB} + \overline {GC} = \dfrac{{2\overline a - \overline b - 2\overline c }}{3} + \dfrac{{ - \overline a + 2\overline b - \overline c }}{3} + \dfrac{{ - \overline a - \overline b + 2\overline c }}{3}\]

On simplifying we get
\[ \Rightarrow \overline {GA} + \overline {GB} + \overline {GC} = \dfrac{1}{3}\left( {2\overline a - \overline b - 2\overline c - \overline a + 2\overline b - \overline c - \overline a - \overline b + 2\overline c } \right)\]
On cancelling the like terms we get
\[ \Rightarrow \overline {GA} + \overline {GB} + \overline {GC} = \overline 0 \]
The assertion is also true. Therefore both Assertion and Reason are individually true and Reason is the correct explanation of Assertion.

Hence the option A is the correct one.

Note:In verbal reasoning there are different kinds. The assertion is a simple statement and the reason is the explanation for the assertion. Generally in assertion and reason we have to consider both the statements and we have to check if each one is true or not. Both the statements will be not true, it depends on the assertion and reason. sometimes any one of them is true.