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 a land plot with vertices given by their coordinates, we can use the formula for the area of a polygon when the coordinates of its vertices are known. This formula is known as the Shoelace formula or Gauss's area formula. For a polygon with vertices \((x_1, y_1), (x_2, y_2), \ldots, (x_n, y_n)\), the area \(A\) is given by:

\[ A = \frac{1}{2} \left| \sum_{i=1}^{n-1} (x_i y_{i+1} - y_i x_{i+1}) + (x_n y_1 - y_n x_1) \right| \]

For the given vertices \(A(8, 13)\), \(B(3, 10)\), \(C(4, 4)\), and \(D(16, 5)\), we will calculate the area as follows:

1. List the coordinates cyclically:

   \[
   \begin{array}{cccccc}
   (8, 13) & (3, 10) & (4, 4) & (16, 5) & (8, 13) \\
   \end{array}
   \]

2. Apply the Shoelace formula:

   \[
   A = \frac{1}{2} \left| 8 \cdot 10 + 3 \cdot 4 + 4 \cdot 5 + 16 \cdot 13 - (13 \cdot 3 + 10 \cdot 4 + 4 \cdot 16 + 5 \cdot 8) \right|
   \]

   \[
   = \frac{1}{2} \left| 80 + 12 + 20 + 208 - (39 + 40 + 64 + 40) \right|
   \]

   \[
   = \frac{1}{2} \left| 320 - 183 \right|
   \]

   \[
   = \frac{1}{2} \left| 137 \right|
   \]

   \[
   = \frac{137}{2} = 68.5
   \]

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

Would you like further details or have any questions?

Here are five related questions you might find interesting:

1. How would the area calculation change if the coordinates were given in a different order?
2. What is the significance of the absolute value in the Shoelace formula?
3. How can the Shoelace formula be derived?
4. Can the Shoelace formula be used for polygons with more than four sides?
5. What are some real-life applications of the Shoelace formula?

**Tip:** Always double-check your vertices order when applying the Shoelace formula to ensure accurate area calculation.

Learn how to calculate the area of a land plot with given vertices using the Shoelace formula (Gauss's area formula). This problem demonstrates the application of coordinate geometry and includes a step-by-step solution for determining the area based on the coordinates of the vertices. Suitable for high school students.

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