Question

The vertices of a quadrilateral are 0,\[3 + 4i\],\[7 + 7i\],\[4 + 3i\]. Then this quadrilateral is a ________
A) Square
B) Rectangle
C) Parallelogram
D) Rhombus

Answer 1

Hint: The length of sides and diagonals is used to determine the type of quadrilateral.
In square, all the sides are equal and diagonals are equal.
In rectangle and parallelogram, opposite sides are equal.
In rhombus, all the sides are equal and the length of diagonals are not equal.
If two points \[P\] and \[Q\], which are represented by complex numbers \[{z_1} = {a_1} + i{b_1}\]and \[{z_2} = {a_2} + i{b_2}\], then the distance between P and Q is \[PQ = \left| {{z_2} - {z_1}} \right|\]
                                                                                          \[ = \left| {\left( {{a_2} - {a_1}} \right) + i\left( {{b_2} - {b_1}} \right)} \right|\]
                                                                                          \[ = \sqrt {{{\left( {{a_2} - {a_1}} \right)}^2} + {{\left( {{b_2} - {b_1}} \right)}^2}} \]

Complete step-by-step solution:
Let \[A,B,C,D\] represent the vertices of a quadrilateral having coordinates \[0\],\[3 + 4i\],\[7 + 7i\]and \[4 + 3i\]respectively.
Given, \[0\],\[3 + 4i\],\[7 + 7i\],\[4 + 3i\] are vertices of a quadrilateral
Therefore we find length of sides and diagonals of the quadrilateral using the formula for line joining two points \[P\] and \[Q\] represented by \[{z_1} = {a_1} + i{b_1}\] and \[{z_2} = {a_2} + i{b_2}\] respectively.
i.e. \[PQ = \left| {{z_2} - {z_1}} \right|\]
            \[ = \left| {\left( {{a_2} - {a_1}} \right) + i\left( {{b_2} - {b_1}} \right)} \right|\]
            \[ = \sqrt {{{\left( {{a_2} - {a_1}} \right)}^2} + {{\left( {{b_2} - {b_1}} \right)}^2}} \]
Length of side \[AB = \left| {(3 + 4i) - (0)} \right|\]
\[
  AB = \sqrt {{{\left( {3 - 0} \right)}^2} + {{\left( {4 - 0} \right)}^2}} \\
   = \sqrt {{{(3)}^2} + {{(4)}^2}} \\
   = \sqrt {9 + 16} \\
   = \sqrt {25} \\
   = 5 \\
 \]
Length of side \[BC = \left| {(7 + 7i) - (3 + 4i)} \right|\]
\[
  BC = \sqrt {{{\left( {7 - 3} \right)}^2} + {{\left( {7 - 4} \right)}^2}} \\
   = \sqrt {{{(4)}^2} + {{(3)}^2}} \\
   = \sqrt {16 + 9} \\
   = \sqrt {25} \\
   = 5 \\
 \]
Length of side \[CD = \left| {(4 + 3i) - (7 + 7i)} \right|\]
\[
  CD = \sqrt {{{\left( {4 - 7} \right)}^2} + {{\left( {3 - 7} \right)}^2}} \\
   = \sqrt {{{( - 3)}^2} + {{( - 4)}^2}} \\
   = \sqrt {9 + 16} \\
   = \sqrt {25} \\
   = 5 \\
 \]
Length of side \[DA = \left| {(0) - (4 + 3i)} \right|\]
\[
  DA = \sqrt {{{\left( {0 - 4} \right)}^2} + {{\left( {0 - 3} \right)}^2}} \\
   = \sqrt {{{( - 4)}^2} + {{( - 3)}^2}} \\
   = \sqrt {16 + 9} \\
   = \sqrt {25} \\
   = 5 \\
 \]
Length of diagonal \[AC = \left| {(7 + 7i) - (0)} \right|\]
\[
  AC = \sqrt {{{\left( {7 - 0} \right)}^2} + {{\left( {7 - 0} \right)}^2}} \\
   = \sqrt {{{(7)}^2} + {{(7)}^2}} \\
   = \sqrt {49 + 49} \\
   = \sqrt {2(49)} \\
   = 7\sqrt 2 \\
 \]
Length of diagonal \[BD = \left| {(4 + 3i) - (3 + 4i)} \right|\]
\[
  BD = \sqrt {{{\left( {4 - 3} \right)}^2} + {{\left( {3 - 4} \right)}^2}} \\
   = \sqrt {{{(1)}^2} + {{(1)}^2}} \\
   = \sqrt {1 + 1} \\
   = \sqrt 2 \\
 \]
It can be observed that the sides of the given quadrilateral are equal in length
i.e. \[AB = BC = CD = DA = 5\]
But the diagonals are not equal in length
i.e. \[AC = 7\sqrt 2 \] and \[BD = \sqrt 2 \]
Therefore, the given quadrilateral is a rhombus (since all four sides are equal in length but diagonals are not).

Option D is the correct answer.

Note: In questions where the nature of the quadrilateral is to be determined, the length of the side is the key. For finding the distance of the sides, the formula should be used carefully as it involves finding the difference between the coordinates and wrong calculation results in the wrong distance. Also, many a times a rhombus is confused with a square, but keep in mind every square is a rhombus(with all interior angles as right angles) but every rhombus is not a square because a square has all four interior angles as right angles but rhombus has no compulsion about angles.