question_answer

ABC is triangle. G is the centroid. D is the mid point of BC. If \[A=(2,3)\] and \[G=(7,5),\] then the point D is

A)  \[\left( \frac{9}{2},4 \right)\]                    

B)  \[\left( \frac{19}{2},6 \right)\]

C)  \[\left( \frac{11}{2},\frac{11}{2} \right)\]                             

D)  \[\left( 8,\frac{13}{2} \right)\]

Correct Answer: B

Solution :

Since, D is the mid point of BC. So, coordinate Of BC are \[\left( \frac{{{x}_{2}}+{{x}_{3}}}{2},\frac{{{y}_{2}}+{{y}_{3}}}{2} \right)\] Given, \[G(7,5)\] is the centroid of \[\Delta ABC\] \[\therefore \]  \[7=\frac{2+{{x}_{2}}+{{x}_{3}}}{3}\] and \[5=\frac{3+{{y}_{2}}+{{y}_{3}}}{3}\]                                 \[\Rightarrow \]               \[{{x}_{2}}+{{x}_{3}}=21-2\] and \[{{y}_{2}}+{{y}_{3}}=15-3\] \[\Rightarrow \]               \[{{x}_{2}}+{{x}_{3}}=19\] and \[{{y}_{2}}+{{y}_{3}}=12\] \[\Rightarrow \]               \[\frac{{{x}_{2}}+{{x}_{3}}}{2}=\frac{19}{2}\]   and \[\frac{{{y}_{2}}+{{y}_{3}}}{2}=6\] \[\therefore \] Coordinate of D are \[\left( \frac{19}{2},6 \right)\]