To find the total area of the garden plot, we can treat the coordinates of the poles as vertices of a quadrilateral. We will calculate the area using the Shoelace Formula, which is suitable for polygons defined by their vertices.

The Shoelace Formula

For a quadrilateral with vertices $(x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4)$, the area is given by:

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

Step 1: Write the Coordinates in Order

The vertices are:

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

We list them in the correct order (clockwise or counterclockwise). The vertices in clockwise order are: $(0, 0) \to (6, 0) \to (4, 5) \to (2, 4)$

Step 2: Apply the Shoelace Formula

Substitute the values into the formula:

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

Simplify step by step:

1. Calculate the first term: $0 \cdot 0 + 6 \cdot 5 + 4 \cdot 4 + 2 \cdot 0 = 0 + 30 + 16 + 0 = 46$

2. Calculate the second term: $0 \cdot 6 + 0 \cdot 4 + 5 \cdot 2 + 4 \cdot 0 = 0 + 0 + 10 + 0 = 10$

3. Subtract the second term from the first: $46 - 10 = 36$

4. Take the absolute value and divide by 2: $\text{Area} = \frac{1}{2} \cdot 36 = 18$

Final Answer:

The total area of the garden plot is 18 square feet.

Would you like a breakdown of any specific steps, or have additional questions?

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1. What is the Shoelace Formula, and how does it apply to polygons?
2. How can you determine the correct order of vertices for a quadrilateral?
3. What happens if you list the vertices in the wrong order?
4. How would the calculation differ for a non-convex quadrilateral?
5. Can the Shoelace Formula be used for polygons with more than 4 sides?

Tip: Always ensure vertices are listed in the correct sequential order (clockwise or counterclockwise) for accurate area calculations.