The four vertices of the rectangle are given as [tex]\(1+4i\)[/tex], [tex]\(-2+4i\)[/tex], [tex]\(-2-3i\)[/tex], and [tex]\(1-3i\)[/tex] in the complex plane. To find the area of the rectangle, we need to calculate the lengths of its sides.
1. Determine the height of the rectangle:
- The imaginary parts of [tex]\(1+4i\)[/tex] and [tex]\(1-3i\)[/tex] (vertices with the same real part) are 4 and -3, respectively.
- The height is the distance between 4 and -3. This is calculated as:
[tex]\[
\text{Height} = |4 - (-3)| = |4 + 3| = 7 \text{ units}
\][/tex]
2. Determine the width of the rectangle:
- The real parts of [tex]\(1+4i\)[/tex] and [tex]\(-2+4i\)[/tex] (vertices with the same imaginary part) are 1 and -2, respectively.
- The width is the distance between 1 and -2. This is calculated as:
[tex]\[
\text{Width} = |1 - (-2)| = |1 + 2| = 3 \text{ units}
\][/tex]
3. Calculate the area of the rectangle:
- The area [tex]\(A\)[/tex] of a rectangle is given by multiplying the height by the width.
[tex]\[
A = \text{Height} \times \text{Width} = 7 \times 3 = 21 \text{ square units}
\][/tex]
Therefore, the area of the rectangle is [tex]\(\boxed{21}\)[/tex] square units.
