Use the Pythagorean theorem to find the area of field C in acres (4,7) (0,4) (3,0) (7,3)
To find the area of field C, we can divide it into two triangles.
First, we need to find the length of one side of each triangle.
For the first triangle, we can use the points (4, 7), (0, 4), and (3, 0) to form a right triangle. The length of the base is 4 - 0 = 4 units, and the length of the height is 7 - 0 = 7 units.
Using the Pythagorean theorem, we have:
hypotenuse^2 = base^2 + height^2.
hypotenuse^2 = 4^2 + 7^2.
hypotenuse^2 = 16 + 49.
hypotenuse^2 = 65.
To simplify the calculation of area, let's take the square root of both sides:
hypotenuse = √65.
The area of the triangle is given by:
area1 = 1/2 * base * height.
area1 = 1/2 * 4 * 7.
area1 = 14 square units.
For the second triangle, we can use the points (3, 0), (7, 3), and (4, 7) to form another right triangle. The length of the base is 7 - 3 = 4 units, and the length of the height is 7 - 0 = 7 units.
Again, using the Pythagorean theorem, we have:
hypotenuse^2 = base^2 + height^2.
hypotenuse^2 = 4^2 + 7^2.
hypotenuse^2 = 16 + 49.
hypotenuse^2 = 65.
Taking the square root of both sides:
hypotenuse = √65.
The area of the second triangle is:
area2 = 1/2 * base * height.
area2 = 1/2 * 4 * 7.
area2 = 14 square units.
Finally, to find the total area of field C, we sum the areas of both triangles:
total area = area1 + area2.
total area = 14 + 14.
total area = 28 square units.
Now, to convert the area from square units to acres, we need to divide by the conversion factor. Since 1 acre is equal to 43,560 square feet, we have:
total area in acres = 28 / 43,560.
total area in acres ≈ 0.0006432714740207 acres.
Therefore, the area of field C is approximately 0.0006 acres.
To find the area of a quadrilateral using the Pythagorean theorem, you would first need to calculate the lengths of the four sides of the quadrilateral. Then you can use the formula for the area of a quadrilateral:
Area = (1/2) * d1 * d2
where d1 and d2 are the diagonals of the quadrilateral. In this case, we have a quadrilateral with vertices (4,7), (0,4), (3,0), and (7,3).
Let's calculate the lengths of the four sides:
Side 1: Distance between (4,7) and (0,4)
Using the distance formula: d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
d1 = sqrt((0 - 4)^2 + (4 - 7)^2) = sqrt(16 + 9) = sqrt(25) = 5
Side 2: Distance between (0,4) and (3,0)
d2 = sqrt((3 - 0)^2 + (0 - 4)^2) = sqrt(9 + 16) = sqrt(25) = 5
Side 3: Distance between (3,0) and (7,3)
d3 = sqrt((7 - 3)^2 + (3 - 0)^2) = sqrt(16 + 9) = sqrt(25) = 5
Side 4: Distance between (7,3) and (4,7)
d4 = sqrt((4 - 7)^2 + (7 - 3)^2) = sqrt(9 + 16) = sqrt(25) = 5
Now that we have the lengths of all four sides, we can calculate the diagonals:
Diagonal 1: Distance between (4,7) and (3,0)
d5 = sqrt((3 - 4)^2 + (0 - 7)^2) = sqrt(1 + 49) = sqrt(50)
Diagonal 2: Distance between (0,4) and (7,3)
d6 = sqrt((7 - 0)^2 + (3 - 4)^2) = sqrt(49 + 1) = sqrt(50)
Now, let's calculate the area using the formula:
Area = (1/2) * d5 * d6
Area = (1/2) * sqrt(50) * sqrt(50)
Area = (1/2) * 50
Area = 25 acres
Therefore, the area of field C is 25 acres.
To find the area of field C using the Pythagorean theorem, we need to determine the lengths of the sides of the quadrilateral.
Step 1: Find the distance between points (4,7) and (0,4).
- Using the distance formula: distance = √((x2 - x1)^2 + (y2 - y1)^2)
- Plugging in the values: distance = √((0 - 4)^2 + (4 - 7)^2)
- Simplifying: distance = √((-4)^2 + (-3)^2)
- Evaluating: distance = √(16 + 9)
- Calculating: distance = √25
- Final result: distance = 5
Step 2: Find the distance between points (0,4) and (3,0).
- Using the distance formula: distance = √((x2 - x1)^2 + (y2 - y1)^2)
- Plugging in the values: distance = √((3 - 0)^2 + (0 - 4)^2)
- Simplifying: distance = √((3)^2 + (-4)^2)
- Evaluating: distance = √(9 + 16)
- Calculating: distance = √25
- Final result: distance = 5
Step 3: Find the distance between points (3,0) and (7,3).
- Using the distance formula: distance = √((x2 - x1)^2 + (y2 - y1)^2)
- Plugging in the values: distance = √((7 - 3)^2 + (3 - 0)^2)
- Simplifying: distance = √((4)^2 + (3)^2)
- Evaluating: distance = √(16 + 9)
- Calculating: distance = √25
- Final result: distance = 5
Step 4: Find the distance between points (7,3) and (4,7).
- Using the distance formula: distance = √((x2 - x1)^2 + (y2 - y1)^2)
- Plugging in the values: distance = √((4 - 7)^2 + (7 - 3)^2)
- Simplifying: distance = √((-3)^2 + (4)^2)
- Evaluating: distance = √(9 + 16)
- Calculating: distance = √25
- Final result: distance = 5
Step 5: Divide the quadrilateral into two triangles.
- Connect the points (4,7), (0,4), and (3,0) to form one triangle.
- Connect the points (4,7), (3,0), and (7,3) to form the other triangle.
Step 6: Calculate the area of each triangle.
- Using the formula: Area = 0.5 * base * height
- For the triangle formed by points (4,7), (0,4), and (3,0):
- Base = distance between (4,7) and (0,4) = 5
- Height = distance between (0,4) and (3,0) = 5
- Area of the first triangle = 0.5 * 5 * 5 = 12.5 acres
- For the triangle formed by points (4,7), (3,0), and (7,3):
- Base = distance between (7,3) and (4,7) = 5
- Height = distance between (3,0) and (4,7) = 5
- Area of the second triangle = 0.5 * 5 * 5 = 12.5 acres
Step 7: Calculate the total area of field C.
- Total area = area of the first triangle + area of the second triangle
- Total area = 12.5 acres + 12.5 acres
- Total area = 25 acres
Therefore, the area of field C is 25 acres.