To find the areas of the fields based on the provided coordinates, let's calculate each one step by step.
1. Area of Field A
Vertices of Field A: (-4, 4), (0, 4), (0, 0), (-4, 0)
Field A forms a square. The length of one side can be determined by the distance between two vertices.
- The distance from (-4, 4) to (0, 4) is \(0 - (-4) = 4\).
Thus, the area \(A\) of Field A is: \[ A = \text{side}^2 = 4^2 = 16 \text{ square units (or acres)} \]
Answer:
Area = 16 acres
2. Area of Field B
Vertices of Field B: (0, 0), (3, 0), (3, -3), (0, -3)
Field B also forms a square. The length of one side can be determined in a similar manner.
- The distance from (0, 0) to (3, 0) is \(3 - 0 = 3\).
Thus, the area \(B\) of Field B is: \[ B = \text{side}^2 = 3^2 = 9 \text{ square units (or acres)} \]
Answer:
Area = 9 acres
3. Area of Field C
Vertices of Field C: (0, 0), (4, 7), (7, 3), (3, 0)
Field C does not form a square but a quadrilateral. To find the area, we can divide it into two triangles or apply the shoelace formula.
Let's use the shoelace formula: \[ \text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right| \]
Substituting the coordinates into the formula where:
- \((x_1, y_1) = (0, 0)\)
- \((x_2, y_2) = (4, 7)\)
- \((x_3, y_3) = (7, 3)\)
- \((x_4, y_4) = (3, 0)\)
Calculating: \[ \text{Area} = \frac{1}{2} \left| 0 \cdot 7 + 4 \cdot 3 + 7 \cdot 0 + 3 \cdot 0 - (0 \cdot 4 + 7 \cdot 7 + 3 \cdot 3 + 0 \cdot 0) \right| \] \[ = \frac{1}{2} \left| 0 + 12 + 0 + 0 - (0 + 49 + 9 + 0) \right| \] \[ = \frac{1}{2} \left| 12 - 58 \right| \] \[ = \frac{1}{2} \left| -46 \right| = \frac{46}{2} = 23 \text{ square units (or acres)} \]
Answer:
Area = 23 acres
Summary of Areas:
- Area of Field A = 16 acres
- Area of Field B = 9 acres
- Area of Field C = 23 acres