To find the area of a quadrilateral given its vertices, we can use the shoelace theorem (or Gauss's area formula), which is applied to coordinates $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, and $(x_4, y_4)$.

The formula for the area of a quadrilateral with vertices $A(x_1, y_1), B(x_2, y_2), C(x_3, y_3), D(x_4, y_4)$ is:

$\text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right|$

Given the vertices from the image:

• $A(-4, 8)$
• $B(-3, -4)$
• $C(0, -5)$
• $D(5, 6)$

Let's plug these into the formula:

$\text{Area} = \frac{1}{2} \left| (-4)(-4) + (-3)(-5) + (0)(6) + (5)(8) - (8)(-3) - (-4)(0) - (-5)(5) - (6)(-4) \right|$

Simplifying step by step:

$\text{Area} = \frac{1}{2} \left| 16 + 15 + 0 + 40 - (-24 - 0 + 25 - (-24)) \right|$ $\text{Area} = \frac{1}{2} \left| 71 - (-25) \right|$ $\text{Area} = \frac{1}{2} \left| 71 + 25 \right|$ $\text{Area} = \frac{1}{2} \times 96 = 48 \, \text{cm}^2$

So, the area of quadrilateral ABCD is 48 cm².

Would you like further details or clarifications on any step?

Related questions:

1. What is the shoelace theorem and why does it work for polygon areas?
2. Can this method be applied to find the area of polygons with more than four sides?
3. What happens if the vertices are listed in a different order?
4. How can the formula be adapted for concave quadrilaterals?
5. How can you calculate the area if the vertices are given in 3D space?

Tip: Ensure to arrange the vertices in a sequential order (clockwise or counterclockwise) for correct application of the shoelace theorem.