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 \[\left( {x,y} \right) = \left( {\dfrac{{{x_1} + {x_2}}}{2},\dfrac{{{y_1} + {y_2}}}{2}} \right)\].

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 \[AC\]=\[\left( {\dfrac{{2 + \left( { - 2} \right)}}{2},\dfrac{{ - 1 + 3}}{2}} \right)\].
=\[\left( {0,1} \right)\]
Coordinates of midpoint of \[BD\]=\[\left( {\dfrac{{3 + x}}{2},\dfrac{{4 + y}}{2}} \right)\].
Since, midpoint of \[AC\] and \[BD\] is same, we conclude,
\[\left( {\dfrac{{3 + x}}{2},\dfrac{{4 + y}}{2}} \right) = \left( {0,1} \right)\]
\[ \Rightarrow \dfrac{{3 + x}}{2} = 0\]and\[\dfrac{{4 + y}}{2} = 1\]
\[ \Rightarrow x = - 3\] 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.