To find the area of a polygon given its vertices, we can use the Shoelace formula (also known as Gauss's area formula). This formula is useful for polygons defined by their vertices in a coordinate plane.
The Shoelace formula states that for a polygon with vertices \((x_1, y_1)\), \((x_2, y_2)\), \(\ldots\), \((x_n, y_n)\) listed in a consistent order (either clockwise or counterclockwise), the area \(A\) is:
[tex]\[
A = \frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i y_{i+1} - y_i x_{i+1}) + (x_n y_1 - y_n x_1) \right|
\][/tex]
Given the vertices of the polygon \(K(-3,3)\), \(L(3,3)\), \(M(3,-1)\), and \(N(-3,-1)\), we can apply the Shoelace formula step-by-step:
1. List the coordinates in a consistent order (let's consider counterclockwise order):
[tex]\[
(x_1, y_1) = (-3, 3), \quad (x_2, y_2) = (3, 3), \quad (x_3, y_3) = (3, -1), \quad (x_4, y_4) = (-3, -1)
\][/tex]
2. Identify the vertices coordinates:
[tex]\[
x_1 = -3, \quad y_1 = 3
\][/tex]
[tex]\[
x_2 = 3, \quad y_2 = 3
\][/tex]
[tex]\[
x_3 = 3, \quad y_3 = -1
\][/tex]
[tex]\[
x_4 = -3, \quad y_4 = -1
\][/tex]
3. Substitute these into the formula:
[tex]\[
\text{Area} = \frac{1}{2} \left| (-3 \cdot 3 + 3 \cdot -1 + 3 \cdot -1 + -3 \cdot 3) - (3 \cdot 3 + 3 \cdot 3 + (-1) \cdot -3 + (-1) \cdot -3) \right|
\][/tex]
Simplify step-by-step:
[tex]\[
= \frac{1}{2} \left| (-9 - 3 - 3 - 9) - (9 + 9 + 3 + 3) \right|
\][/tex]
[tex]\[
= \frac{1}{2} \left| -24 - 24 \right|
\][/tex]
[tex]\[
= \frac{1}{2} \left| -48 \right|
\][/tex]
[tex]\[
= \frac{1}{2} \cdot 48
\][/tex]
[tex]\[
= 24
\][/tex]
Therefore, the area of the polygon with vertices [tex]\(K(-3,3)\)[/tex], [tex]\(L(3,3)\)[/tex], [tex]\(M(3,-1)\)[/tex], and [tex]\(N(-3,-1)\)[/tex] is [tex]\(24\)[/tex] square units.
