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To find the area of a quadrilateral with vertices \( A(13, 7) \), \( B(-4, 10) \), \( C(-3, -2) \), and \( D(8, -3) \), we can use the formula for the area of a polygon based on its vertices:

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

Using the vertices:
- \( A(13, 7) \): \( (x_1, y_1) = (13, 7) \)
- \( B(-4, 10) \): \( (x_2, y_2) = (-4, 10) \)
- \( C(-3, -2) \): \( (x_3, y_3) = (-3, -2) \)
- \( D(8, -3) \): \( (x_4, y_4) = (8, -3) \)

### Step-by-Step Calculation:

1. Compute \( x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 \):
   \[
   (13 \cdot 10) + (-4 \cdot -2) + (-3 \cdot -3) + (8 \cdot 7) = 130 + 8 + 9 + 56 = 203
   \]

2. Compute \( y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1 \):
   \[
   (7 \cdot -4) + (10 \cdot -3) + (-2 \cdot 8) + (-3 \cdot 13) = -28 - 30 - 16 - 39 = -113
   \]

3. Substitute these results into the formula:
   \[
   \text{Area} = \frac{1}{2} \left| 203 - (-113) \right| = \frac{1}{2} \times 316 = 158
   \]

### Answer
The area of the quadrilateral is \( \mathbf{158} \) square units.

Would you like further details on the process or have additional questions?

#### Related Questions:
1. How do you find the area of a polygon with more than four vertices?
2. What methods can be used to find the area if the quadrilateral is not on a coordinate plane?
3. How does the formula change for different shapes like triangles or pentagons?
4. Can this method be used in three-dimensional space, and how?
5. Why is the area formula structured with absolute values?

#### Tip:
When dealing with polygons in a coordinate plane, breaking them into triangles or using vertex-based formulas helps simplify area calculations.

Find the area of a quadrilateral with vertices A(13,7), B(-4,10), C(-3,-2), and D(8,-3) in a rectangular coordinate system.

Learn how to find the area of a quadrilateral using its vertices in coordinate geometry. This problem includes step-by-step calculations using the Shoelace theorem and is suitable for high school students in Grades 9-12.

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