question_answer

Three vertices of a rhombus are \[(2,-1),\]\[(3,4)\]and \[(-\,\,2,3).\]Find the fourth vertex.

A) \[(1,2)\]             

B) \[(3,2)\]

C) \[(-\,\,2,-\,\,3)\]               

D) \[(-\,\,3,\,\,-2)\]

E) \[(2,3)\]

Correct Answer: D

Solution :

Let coordinate of D be \[(x,y)\]and M be the intersection point of diagonals AC and BD. Since, M is the mid-point of AC. \[\therefore \]Coordinate of \[M=\left( \frac{-\,2+2}{2},\frac{-1+3}{2} \right)=(0,1)\] Also, M is the mid-point of BD. \[\therefore \]Coordinate of \[M=\left( \frac{x+3}{2},\frac{y+4}{2} \right)\] \[\therefore \]                  \[\frac{x+3}{2}=0\]\[\Rightarrow \]\[x=-\,\,3\] and                   \[\frac{y+4}{2}=1\]\[\Rightarrow \]\[y=-\,\,2\]