To calculate the area of polygon ABCD given its vertices A(-4, 2), B(3, 2), C(3, -2), and D(-4, -2), we can use the Shoelace theorem. The Shoelace theorem is a mathematical algorithm to determine the area of a simple polygon when the coordinates of its vertices are known.
Here's the step-by-step process:
1. List the vertices in order: Ensure the vertices of the polygon are listed in a specific order, either clockwise or counterclockwise. Given vertices A, B, C, D, we will proceed in the order: A(-4, 2), B(3, 2), C(3, -2), and D(-4, -2), closing the polygon by returning to vertex A.
2. Apply the Shoelace theorem formula:
[tex]\[
\text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right|
\][/tex]
Using the vertices:
- [tex]\( (x_1, y_1) = (-4, 2) \)[/tex]
- [tex]\( (x_2, y_2) = (3, 2) \)[/tex]
- [tex]\( (x_3, y_3) = (3, -2) \)[/tex]
- [tex]\( (x_4, y_4) = (-4, -2) \)[/tex]
3. Plug in the coordinates:
[tex]\[
\begin{aligned}
\text{Area} &= \frac{1}{2} \left| (-4 \cdot 2) + (3 \cdot -2) + (3 \cdot -2) + (-4 \cdot 2) \right. \\
&\quad - \left. \left( (2 \cdot 3) + (2 \cdot 3) + (-2 \cdot -4) + (-2 \cdot -4) \right) \right|
\end{aligned}
\][/tex]
4. Calculate the individual terms:
[tex]\[
(-4 \cdot 2) = -8
\][/tex]
[tex]\[
(3 \cdot -2) = -6
\][/tex]
[tex]\[
(3 \cdot -2) = -6
\][/tex]
[tex]\[
(-4 \cdot -2) = 8
\][/tex]
Summing these up:
[tex]\[
-8 -6 -6 + 8 = -12
\][/tex]
And for the other terms:
[tex]\[
(2 \cdot 3) = 6
\][/tex]
[tex]\[
(2 \cdot 3) = 6
\][/tex]
[tex]\[
(-2 \cdot -4) = 8
\][/tex]
[tex]\[
(-2 \cdot -4) = 8
\][/tex]
Summing these up:
[tex]\[
6 + 6 + 8 + 8 = 28
\][/tex]
5. Calculate the area:
[tex]\[
\begin{aligned}
\text{Area} &= \frac{1}{2} \left| -12 - 28 \right| \\
&= \frac{1}{2} \left| -40 \right| \\
&= \frac{1}{2} \times 40 \\
&= 20
\end{aligned}
\][/tex]
Thus, the area of polygon ABCD is 28.0 square units.
