Question

The three vertices of a parallelogram taken in order are $\left( -1,0 \right)$, $\left( 3,1 \right)$ and $\left( 2,2 \right)$ respectively. Find the coordinate of the fourth vertex.

Answer 1

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, we have two unknowns and two equations to solve it.
The midpoint formula of two points $\left( a,b \right)$ and $\left( c,d \right)$ is $\left( \dfrac{a+c}{2},\dfrac{b+d}{2} \right)$.
So, we take points $A\equiv \left( -1,0 \right)$ and $C\equiv \left( 2,2 \right)$ which gives us coordinates of O. O is the midpoint of AC diagonal.
$O\equiv \left( \dfrac{-1+2}{2},\dfrac{0+2}{2} \right)\equiv \left( \dfrac{1}{2},1 \right)$.
Similarly, we take points $B\equiv \left( 3,1 \right)$ and $D\equiv \left( x,y \right)$ which gives us coordinates of O. O is the midpoint of BD diagonal.
$O\equiv \left( \dfrac{3+x}{2},\dfrac{1+y}{2} \right)$.
We found two sets of coordinates for point O which have to be the same. This means $O\equiv \left( \dfrac{3+x}{2},\dfrac{1+y}{2} \right)\equiv \left( \dfrac{1}{2},1 \right)$. We equate X and Y coordinates to find the value of x and y.
So, $\dfrac{3+x}{2}=\dfrac{1}{2}\Rightarrow x+3=1\Rightarrow x=-2$ and $\dfrac{1+y}{2}=1\Rightarrow y+1=2\Rightarrow y=1$.
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.