To find the area of the various fields (Field A, Field B, and Field C) on your worksheet, we will proceed with the following steps:
Step 1: Find the area of Field A
Vertices of Field A:
- (-4, 0)
- (0, 0)
- (7, 3)
To calculate the area of a polygon given its vertices, we can use the shoelace formula:
\[ \text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_1 - (y_1x_2 + y_2x_3 + y_3x_1) \right| \]
Plugging in the coordinates:
- \(x_1 = -4\), \(y_1 = 0\)
- \(x_2 = 0\), \(y_2 = 0\)
- \(x_3 = 7\), \(y_3 = 3\)
Calculating:
\[ \text{Area}_A = \frac{1}{2} \left| (-4)(0) + (0)(3) + (7)(0) - (0)(0) - (0)(7) - (3)(-4) \right| \]
\[ = \frac{1}{2} \left| 0 + 0 + 0 - (0 + 0 + (-12)) \right| \]
\[ = \frac{1}{2} \left| 12 \right| = 6 \text{ sq. units} \]
Step 2: Find the area of Field B
Vertices of Field B:
- (0, 0)
- (3, 0)
- (3, -3)
- (0, -3)
Using the shoelace formula again, we have:
\[ \text{Area}_B = \frac{1}{2} \left| 0\cdot0 + 3\cdot(-3) + 3\cdot(-3) + 0\cdot0 - (0\cdot3 + 0\cdot3 + (-3)\cdot0 + (-3)\cdot0) \right| \]
Calculating:
\[ = \frac{1}{2} \left| 0 - 9 - 9 + 0 \right| = \frac{1}{2} \left| -18 \right| = 9 \text{ sq. units} \]
Step 3: Find the area of Field C
Vertices of Field C:
- (4, 7)
- (-4, 4)
- (0, 4)
- (7, 3)
Using the shoelace formula:
\[ \text{Area}_C = \frac{1}{2} \left| 4\cdot4 + (-4)\cdot4 + 0\cdot3 + 7\cdot7 - (7\cdot(-4) + 4\cdot0 + 4\cdot7 + 3\cdot4) \right| \]
Calculating:
\[ = \frac{1}{2} \left| 16 - 16 + 0 + 49 - (-28 + 0 + 28 + 12) \right| \]
\[ = \frac{1}{2} \left| 49 - 12 \right| = \frac{1}{2} \times 37 = 18.5 \text{ sq. units} \]
Final areas:
- Area of Field A: 6 sq. units
- Area of Field B: 9 sq. units
- Area of Field C: 18.5 sq. units
If you need to convert the area from square units to acres or square yards, additional information about the original unit of measurement would be required.
For example, if these areas are given in square feet, you can convert to acres using:
- \(1 \text{ acre} = 43,560 \text{ sq. ft.}\)
For the square yards conversion, you can use the conversion factor where \(1 \text{ acre} = 4,840 \text{ sq. yards}\).
Let me know how you'd like to proceed!
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