If the three consecutive vertices of a parallelogram are (-2, -1), (1, 0) and (4, 3), then what are the coordinates of the fourth vertex?

Concept:

Properties of a parallelogram

• The diagonals of a parallelogram bisect each other.
• Opposite sides of a parallelogram are congruent.
• Opposite angles of a parallelogram are congruent.

Calculation:

Diagonals of a parallelogram bisect each other,

Let the point of intersection of diagonals be P(x, y)

So, P is mid-point of AC,

\(\left( {{\rm{x}},{\rm{\;y}}} \right) = \left( {\frac{{ - 2\; + \;4}}{2},\frac{{ - 1\; + \;3}}{2}} \right) = \left( {1,{\rm{\;}}1} \right)\) 

Let the Fourth vertex be D (x_D, y_D)

P is mid-point of BD,

\(\therefore \left( {{\rm{x}},{\rm{\;y}}} \right) = \left( {1,{\rm{\;}}1} \right) = \left( {\frac{{1\; + \;{{\rm{x}}_{\rm{D}}}}}{2},\frac{{0\; + \;{{\rm{y}}_{\rm{D}}}}}{2}} \right)\)  ⇒ (x_D, y_D) = (1, 2)

Fourth vertex of the parallelogram is D = (1, 2)