question_answer

Three vertices of a parallelogram taken in order are \[(-1,\,\,-6),\,\,\,(2,\,\,-5)\] and \[(7,\,\,2)\]. The fourth vertex is:

A) \[(1,\,\,4)\]                        

B) \[(4,\,\,1)\]

C) \[(1,\,\,1)\]                                        

D) \[(4,\,\,4)\]

Correct Answer: B

Solution :

Key Idea: The diagonals of a parallelogram bisects each other. Let the coordinate of fourth vertex be\[({{x}_{1}},\,\,{{y}_{1}})\]. \[\therefore \]Mid-point of\[AC\]is\[(3,\,\,-2)\]and mid-point of \[DB\] is \[\left( \frac{{{x}_{1}}+2}{2},\,\,\frac{{{y}_{1}}-5}{2} \right)\]. Since, the diagonals, of a parallelogram bisects each other. \[\therefore \]  \[\frac{{{x}_{1}}+2}{2}=3\]and\[\frac{{{y}_{1}}-5}{2}=-2\] \[\Rightarrow \]               \[{{x}_{1}}=4\]and\[{{y}_{1}}=1\] \[\therefore \]Required coordinates is\[(4,\,\,1)\].