To find the area of the polygon [tex]\(WXYZ\)[/tex] with vertices at [tex]\(W(-3,-2)\)[/tex], [tex]\(X(-3,5)\)[/tex], [tex]\(Y(2,5)\)[/tex], and [tex]\(Z(2,-2)\)[/tex], you can use the Shoelace Theorem, also known as Gauss's area formula for polygons. This method involves the coordinates of the vertices.
The Shoelace formula states that for a polygon with vertices [tex]\((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\)[/tex], the area [tex]\(A\)[/tex] is given by:
[tex]\[
A = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + \cdots + x_{n-1}y_n + x_ny_1 - (y_1x_2 + y_2x_3 + y_3x_4 + \cdots + y_{n-1}x_n + y_nx_1) \right|
\][/tex]
Let's apply this to the coordinates:
1. List the coordinates in order and repeat the first vertex at the end to close the polygon:
[tex]\[
W(-3,-2), X(-3,5), Y(2,5), Z(2,-2), W(-3,-2)
\][/tex]
2. Apply the Shoelace formula:
[tex]\[
A = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right|
\][/tex]
Plug in the coordinates:
[tex]\[
A = \frac{1}{2} \left| (-3 \cdot 5) + (-3 \cdot 5) + (2 \cdot -2) + (2 \cdot -2) - ((-2 \cdot -3) + (5 \cdot 2) + (5 \cdot 2) + (-2 \cdot -3)) \right|
\][/tex]
3. Calculate the products for each term:
[tex]\[
A = \frac{1}{2} \left| (-15) + (-15) + (-4) + (-4) - (6 + 10 + 10 + 6) \right|
\][/tex]
4. Add and subtract the products:
[tex]\[
A = \frac{1}{2} \left| -38 - 32 \right|
\][/tex]
5. Simplify:
[tex]\[
A = \frac{1}{2} \cdot \left| -70 \right|
\][/tex]
6. This simplifies to:
[tex]\[
A = \frac{1}{2} \cdot 70 = 35 \text{ square units}
\][/tex]
So the area of the polygon [tex]\(WXYZ\)[/tex] is [tex]\(35\)[/tex] square units. The correct answer is:
A) 35 square units
