Question

Three vertices of a rhombus taken in order are $$\left(2,-1\right),\left(3,4\right)$$ and $$\left(-2,3\right)$$. Find the fourth vertex.

Answer 1

Hint: Here, in the given question, the three vertices of a rhombus are given and we are asked to find the fourth one. As we know that, rhombus has a special property that the diagonals of rhombus bisect each other. We will use this property and midpoint formula to get the required vertex of a rhombus.
Formula used:
Midpoint formula: If $$\left(x_1,y_1\right)$$ and $$\left(x_2,y_2\right)$$ are the two points joining a line segment, then the midpoint, say $$\left(x,y\right)$$, of this line segment can be calculated by $$\left(x,y\right)=\left({\displaystyle \frac{x_1+x_2}{2}},{\displaystyle \frac{y_1+y_2}{2}}\right)$$.

Complete step-by-step solution:
Given, three vertices of rhombus $$A\left(2,-1\right)$$, $$B\left(3,4\right)$$ and $$C\left(-2,3\right)$$.
Let the fourth vertex be $$D\left(x,y\right)$$

 Now, we have, $$ABCD$$ is a rhombus, where $$AB=BC=CD=AD$$.
We know that the rhombus has a property that the diagonals of rhombus bisect each other perpendicularly. In the given rhombus,$$AC$$ and $$BD$$ are the diagonals.
If diagonal bisects each other, then the midpoint of the diagonals should be the same.
Using Midpoint formula,
Coordinates of midpoint of $$AC$$=$$\left({\displaystyle \frac{2+\left(-2\right)}{2}},{\displaystyle \frac{-1+3}{2}}\right)$$.
=$$\left(0,1\right)$$
Coordinates of midpoint of $$BD$$=$$\left({\displaystyle \frac{3+x}{2}},{\displaystyle \frac{4+y}{2}}\right)$$.
Since, midpoint of $$AC$$ and $$BD$$ is same, we conclude,
$$\left({\displaystyle \frac{3+x}{2}},{\displaystyle \frac{4+y}{2}}\right)=\left(0,1\right)$$
$$\Rightarrow {\displaystyle \frac{3+x}{2}}=0$$and$${\displaystyle \frac{4+y}{2}}=1$$
$$\Rightarrow x=-3$$ and $$y=-2$$
Hence, the fourth vertex of the given rhombus is $$D\left(-3,-2\right)$$.

Note: Alternatively, this question can also be solved using distance formula between two points.
Some properties of rhombus are:
> All the four sides of a rhombus are equal to one another.
> Opposite sides of a rhombus are parallel to each other.
> Opposite angles of rhombus are equal.
> Diagonals of a rhombus bisect each other perpendicularly.
> The adjacent angles of rhombus are supplementary.
using this property we can solve the problem easily.