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:
\(\[
\text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right|
\]\)
Using the vertices:
- \(\( (x_1, y_1) = (-4, 2) \)\)
- \(\( (x_2, y_2) = (3, 2) \)\)
- \(\( (x_3, y_3) = (3, -2) \)\)
- \(\( (x_4, y_4) = (-4, -2) \)\)
3. Plug in the coordinates:
\(\[
\begin{aligned}
\text{Area} &= \frac{1}{2} \left| (-4 \cdot 2) + (3 \cdot -2) + (3 \cdot -2) + (-4 \cdot 2) \right. \\
&\quad - \left. \left( (2 \cdot 3) + (2 \cdot 3) + (-2 \cdot -4) + (-2 \cdot -4) \right) \right|
\end{aligned}
\]\)
4. Calculate the individual terms:
\(\[
(-4 \cdot 2) = -8
\]\)
\(\[
(3 \cdot -2) = -6
\]\)
\(\[
(3 \cdot -2) = -6
\]\)
\(\[
(-4 \cdot -2) = 8
\]\)
Summing these up:
\(\[
-8 -6 -6 + 8 = -12
\]\)
And for the other terms:
\(\[
(2 \cdot 3) = 6
\]\)
\(\[
(2 \cdot 3) = 6
\]\)
\(\[
(-2 \cdot -4) = 8
\]\)
\(\[
(-2 \cdot -4) = 8
\]\)
Summing these up:
\(\[
6 + 6 + 8 + 8 = 28
\]\)
5. Calculate the area:
\(\[
\begin{aligned}
\text{Area} &= \frac{1}{2} \left| -12 - 28 \right| \\
&= \frac{1}{2} \left| -40 \right| \\
&= \frac{1}{2} \times 40 \\
&= 20
\end{aligned}
\]\)
Thus, the area of polygon ABCD is 28.0 square units.
