question_answer

If \[(3,\,\,3)\] is a vertex of a triangle and \[(-3,\,\,6)\] and \[(9,\,\,6)\] are the mid points of the two sides through this vertex, then the centroid of the triangle is

A)  \[(3,\,\,7)\]          

B)  \[(1,\,\,7)\]

C)  \[(-3,\,\,7)\]        

D)  \[(-1,\,\,7)\]

Correct Answer: A

Solution :

Given,  \[A=(3,3),E=(-3,6)\] and \[F=(9,6)\] Let  \[B=({{x}_{1}},{{y}_{1}})\] and \[C=({{x}_{2}},{{y}_{2}})\] Then,   \[\frac{{{x}_{1}}+3}{2}=-3,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{{y}_{1}}+3}{2}=6\] \[\Rightarrow \] \[{{x}_{1}}=-9,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{y}_{1}}=9\] and \[\frac{{{x}_{2}}+3}{2}=9,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{{{y}_{2}}+3}{2}=6\] \[\Rightarrow \] \[{{x}_{2}}=15,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{y}_{2}}=9\] Now, centroid \[=\left( \frac{-9+15+3}{3},\frac{9+9+3}{3} \right)\] \[=(3,7)\]