Question

The four vertices of a rectangle drawn on a complex plane are defined by 1 + 4i, -2 + 4i, -2 – 3i, and 1 – 3i. The area of the rectangle is square units.

Answer 1

The rectangle is 1-(-2) = 3 units in the real direction and 4i-(-3i) = 7 units in the imaginary direction. Its area is 3×7 = 21 square units.

Answer 2

Answer:

The area of rectangle =21 sq units .

Step-by-step explanation:

Given the four vertices of rectangle are 1+4i,-2+4i, -2-3i and 1-3i.

Consider ABCD is  a rectangle and its vertices are 1+4i,-2+4i,-2-3i and 1-3i.

First we find sides of rectangle

AB=vertices of B- vertices of A

AB= -2+4i-(1+4i)=-3

If complex number=a+bi

Then modulus=$\sqrt{a^2+b^2}$

Length of AB= $\sqrt{(-3)^2}$=3 ( Because magnitude = length always positive )

BC= Vertices of C - vertices of B

BC=-2-3i-(-2+4i)=-7i

Length of BC=$\sqrt{(-7)^2}$=7 ( Magnitude always positive)

CD= vertices of D- vertices of C

CD= 1-3i-(-2-3i)=3

Length of CD= $\sqrt{3^2}$=3 ( Magnitude  always positive)

DA= vertices of A - vertices of D

DA= 1+4i-(1-3i) =7i

Length of DA= $\sqrt{7^2}$=7 ( Magnitude always positive)

AB=CD and DA= BC

Length BC=7 units

Breadth AB=3 units

Area of the rectangle = $length\times breadth$

Area of rectangle =$AB\times BC$

Area of rectangle= $3\times 7$=21 sq units .