Answer
Hint: We use the theorem of the midpoint of two points for finding the solution. We know the diagonals of a parallelogram bisects each other which helps us to find the midpoint of the diagonals. We get two sets of equations, which on solving gives us the vertices of the fourth.
Complete step-by-step solution:
The given three vertices of a parallelogram taken in order are $\left( -1,0 \right)$, $\left( 3,1 \right)$ and $\left( 2,2 \right)$ respectively. Let’s take them in anti-clockwise.So, $A\equiv \left( -1,0 \right)$, $B\equiv \left( 3,1 \right)$ and $C\equiv \left( 2,2 \right)$.Let’s assume the other vertices as $D\equiv \left( x,y \right)$.Now they are in order and they are vertices of a parallelogram.We know the diagonals of a parallelogram bisect each other which means AC and BD bisect each other at a fixed point. Let the fixed point be O.
So, the coordinate of D is $D\equiv \left( -2,1 \right)$.
Note: We can use the distance formula for finding the variables but that will increase the variables in their quadratic form which will be tough to solve. Using the midpoint theorem helps in that case. We also need to remember that we are equating two coordinates of the same point, that’s why the individual coordinates are equal.