To find the area of the field, we first need to clarify the dimensions and shape of the field. The description implies that the field is a square, based on the mention of corners and points.
Let's identify the coordinates provided:
- Point A: (3, 0) - one corner of the field with a value of 3.
- Point B: (0, 4) - the other corner of that side, with a value of 4.
- Point C: (4, 7) - one of the other corners.
- Point D: (7, 3) - the final corner.
Step 1: Determine the Length of a Side
To find the length of the square's side, we can calculate the distance between two known points. Let’s find the distance between points A and B (the coordinates (3, 0) and (0, 4)) using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of point A \((3, 0)\) and point B \((0, 4)\): \[ d = \sqrt{(0 - 3)^2 + (4 - 0)^2} \] \[ d = \sqrt{(-3)^2 + (4)^2} \] \[ d = \sqrt{9 + 16} \] \[ d = \sqrt{25} \] \[ d = 5 \]
Step 2: Calculate the Area of the Field
Since we have confirmed the side length of the square is 5, we can calculate the area \(A\) of the square using the formula: \[ A = \text{side length}^2 \] \[ A = 5^2 \] \[ A = 25 \]
Conclusion
The area of the field is \( \boxed{25} \).
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