To calculate the area of quadrilateral $ABCD$, we will apply the shoelace theorem (also known as Gauss's area formula for polygons). This method allows us to determine the area of any polygon given its vertices on the coordinate plane.

Steps:

1. Identify the coordinates of the vertices of quadrilateral $ABCD$:
   • $A(x_1, y_1)$
   • $B(x_2, y_2)$
   • $C(x_3, y_3)$
   • $D(x_4, y_4)$

From the image:

• $A(-5, 4)$
• $B(3, 8)$
• $C(5, 1)$
• $D(-3, -3)$

2. Apply the shoelace formula:

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

Substitute the coordinates of $A, B, C, D$:

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

3. Simplify the expression: $\text{Area} = \frac{1}{2} \left| -40 + 3 - 15 - 12 - (12 + 40 - 3 + 15) \right|$ $\text{Area} = \frac{1}{2} \left| -64 - 64 \right|$ $\text{Area} = \frac{1}{2} \times 128 = 64 \text{ square units}$

Thus, the area of quadrilateral $ABCD$ is 64 square units.

Would you like more details or have any questions about the method?

Related questions:

1. How does the shoelace theorem work for polygons with more than four sides?
2. What is the importance of the absolute value in the shoelace formula?
3. How would the calculation change if one of the vertices was at the origin?
4. Can you use this method for irregular shapes like triangles?
5. What are the limitations of the shoelace method?

Tip:

When applying the shoelace formula, carefully organize the coordinates to avoid mix-ups, and double-check your multiplication steps!