A surveyor needs to determine the area of a land show in Fig below. The coordinates of the four vertices of the land are as follows: A (8,13), B (3,10), C (4,4), D (16,5) 

To determine the area of the land with vertices \(A(8,13)\), \(B(3,10)\), \(C(4,4)\), and \(D(16,5)\), we can use the Shoelace Theorem (also known as Gauss's area formula for polygons).

The Shoelace Theorem formula for the area \(A\) of a polygon with vertices \((x_1, y_1)\), \((x_2, y_2)\), ..., \((x_n, y_n)\) is given by:
\[
A = \frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i y_{i+1} - y_i x_{i+1}) + (x_n y_1 - y_n x_1) \right|
\]

Applying this to our vertices:

1. List the coordinates in order, repeating the first vertex at the end:
   \[
   (8,13), (3,10), (4,4), (16,5), (8,13)
   \]

2. Compute the sum of the products of the coordinates as specified:
   \[
   \sum_{i=1}^{n-1} (x_i y_{i+1}) = 8 \cdot 10 + 3 \cdot 4 + 4 \cdot 5 + 16 \cdot 13 = 80 + 12 + 20 + 208 = 320
   \]
   \[
   \sum_{i=1}^{n-1} (y_i x_{i+1}) = 13 \cdot 3 + 10 \cdot 4 + 4 \cdot 16 + 5 \cdot 8 = 39 + 40 + 64 + 40 = 183
   \]

3. Calculate the area:
   \[
   A = \frac{1}{2} \left| 320 - 183 \right| = \frac{1}{2} \left| 137 \right| = \frac{137}{2} = 68.5
   \]

Thus, the area of the land is \(68.5\) square units.

Do you want details or have any questions? Here are five questions you might ask next:
1. How do you use the Shoelace Theorem for polygons with more vertices?
2. Can you explain how to find the centroid of a polygon?
3. What are some other methods for finding the area of irregular polygons?
4. How do you apply the distance formula in coordinate geometry?
5. Can you show how to use determinants to find the area of a triangle?

**Tip:** Always plot the points to visualize the polygon and ensure the vertices are listed in the correct order (clockwise or counterclockwise) to apply the Shoelace Theorem correctly.

Learn how to calculate the area of a polygon with vertices (8,13), (3,10), (4,4), (16,5) using the Shoelace Theorem. This problem covers coordinate geometry concepts and provides a step-by-step solution using the formula for polygonal area calculation. Suitable for students in Grades 9-12.

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