To calculate the total area of the garden plot formed by the four poles, we will treat the coordinates as vertices of a quadrilateral. The vertices are:

• $(0, 0)$
• $(6, 0)$
• $(4, 5)$
• $(2, 4)$

The formula for the area of a polygon given its vertices $(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)$ is the Shoelace Theorem (or Gauss's area formula), which is:

$\text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + \dots + x_ny_1 - (y_1x_2 + y_2x_3 + \dots + y_nx_1) \right|$

Step 1: Arrange the coordinates in order

We arrange the coordinates in a clockwise or counter-clockwise order. Here’s one possible ordering:

$(0, 0), (6, 0), (4, 5), (2, 4)$

Now, apply the Shoelace Theorem.

Step 2: Set up the calculation

Using the vertices $(x_1, y_1) = (0, 0)$, $(x_2, y_2) = (6, 0)$, $(x_3, y_3) = (4, 5)$, and $(x_4, y_4) = (2, 4)$, the formula becomes:

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

Step 3: Calculate the sums

First, calculate the sum of the products of the coordinates going clockwise:

$0 \times 0 + 6 \times 5 + 4 \times 4 + 2 \times 0 = 0 + 30 + 16 + 0 = 46$

Next, calculate the sum of the products of the coordinates going counter-clockwise:

$0 \times 6 + 0 \times 4 + 5 \times 2 + 4 \times 0 = 0 + 0 + 10 + 0 = 10$

Step 4: Calculate the area

Now, substitute the sums into the formula:

$\text{Area} = \frac{1}{2} \left| 46 - 10 \right| = \frac{1}{2} \times 36 = 18 \, \text{square feet}$

Thus, the total area of the garden plot is 18 square feet.

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Would you like more details or have any questions?

Here are some related questions:

1. How do you calculate the area of a polygon with irregular sides?
2. What is the significance of the Shoelace Theorem in geometry?
3. How does the orientation of the points affect the area calculation?
4. What are other methods to find the area of a quadrilateral?
5. How can this formula be applied to polygons with more than four vertices?

Tip: Always ensure that the points are listed in a consistent clockwise or counter-clockwise order when applying the Shoelace Theorem.