Explanation
$$ {\textbf{Step -1: Form equation}} $$
$$ {\text{Let A }= (3,4), \text{C } = (6,5) \text{as they are end points of the diagonal and B }= (2,1)} $$
$$ {\text{Let the fourth vertex D }= (x,y)} $$
$$ {\text{We know that the diagonals of a parallelogram bisect each other}}{\text{. }} $$
$$ {\text{So, the midpoint of AC is same as the mid point of BD}}{\text{.}} $$
$$ {\text{Mid point of two points }}({x_1},{y_1}) {\text{and }}({x_2},{y_2}) $$
$$ {\text{which}}\;{\text{ is calculated by the formula}};(\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}) $$
$$ {\textbf{Step -2: Find fourth point of vertex}} $$
$$ {\text{So, midpoint of AC = Mid point of BD}} $$
$$ \Rightarrow \left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right) = \left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right) $$
$$ \Rightarrow \left( {\dfrac{{3 + 6}}{2},\dfrac{{4 + 5}}{2}} \right) = \left( {\dfrac{{2 + x}}{2},\dfrac{{1 + y}}{2}} \right) $$
$$ {\text{So }}\dfrac{{3 + 6}}{2} = \dfrac{{2 + x}}{2}\;{\text{and }}\dfrac{{4 + 5}}{2} = \dfrac{{1 + y}}{2} $$
$$ {\text{So, }}\dfrac{9}{2} = \dfrac{{2 + x}}{2} $$
$$ \Rightarrow 9 = 2 + x $$
$$ \Rightarrow x = 7 $$
$$ {\text{and }}\dfrac{9}{2} = \dfrac{{1 + y}}{2} $$
$$ \Rightarrow 9 = 1 + y $$
$$ \Rightarrow y = 8 $$
$$ {\text{So fourth vertex is (7,8)}} $$
$$ {\textbf{Hence the correct answer is option D}}{\text{.}} $$
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