Question

If the vertices of a quadrilateral be \[A = 1 + 2i,B = - 3 + i,C = - 2 - 3i\;and\;D = 2 - 2i,\] then the quadrilateral is
E. Parallelogram
F. Rectangle
G. Square
H. Rhombus

Answer 1

Hint: in this question we have to find which type of quadrilateral is formed by given vertices of quadrilateral. Find the length of each sides and diagonal of quadrilateral then use the property of each types of quadrilateral to get the answer.

Formula Used: Modulus of complex number \[z = x + iy\] is given by
\[\left| z \right| = \sqrt {{x^2} + {y^2}} \]
Where
z is a complex number
x represent real part of complex number
iy is a imaginary part of complex number
i is iota

Complete step by step solution: Given: vertices of quadrilateral
Length of AB is given as
\[AB = \sqrt {16 + 1} = \sqrt {17} \]
Similarly length of other sides of quadrilateral is
\[BC = \sqrt {16 + 1} = \sqrt {17} \]
\[CD = \sqrt {16 + 1} = \sqrt {17} \]
\[DA = \sqrt {16 + 1} = \sqrt {17} \]
Length of Diagonals of quadrilateral
\[AC = \sqrt {9 + 25} = \sqrt {34} \]
\[AC = \sqrt {9 + 25} = \sqrt {34} \]
We get all sides are equal to each other and diagonal are also equal
This is a property of square

Option ‘C’ is correct

Note: We must remember the property of square. Square is a quadrilateral having all sides equal to each other and also diagonals are bisect at right angle and equal to each other.
Sometimes student get confused between square and rhombus due to similarity in property. The only difference between square and rhombus is that in square length diagonals are equal to each other but in rhombus length of diagonals are different.
Whereas in rectangle and parallelogram only opposite sides are equal to each other. In rectangle adjacent side is perpendicular to each other.