3. SHOW YOUR WORK: Use the Pythagorean Theorem to find the area of Field C in acres: (2 points) (4,7) (7,3)
To find the area of Field C, we need to find the length of the base and the height of the triangle formed by the given coordinates.
Using the distance formula, we can find the length of the base:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Distance = √((7 - 4)^2 + (3 - 7)^2)
Distance = √(3^2 + (-4)^2)
Distance = √(9 + 16)
Distance = √25
Distance = 5
So, the length of the base is 5 units.
Next, we can find the height of the triangle. Since the triangle is a right triangle, we can use the Pythagorean Theorem.
C^2 = A^2 + B^2
C^2 = 5^2 + 4^2
C^2 = 25 + 16
C^2 = 41
Taking the square root of both sides, we get:
C = √41
Now, we need to convert the length and height from units to acres. Let's assume that 1 acre is equivalent to 10,000 square feet.
To convert the length, 5 units, to acres, we can use the conversion factor:
1 acre = 43,560 square feet
Length in acres = 5 units * (43,560 square feet / 10,000 square feet)
Length in acres = 21,780 / 10,000 acres
Length in acres = 2.178 acres
To convert the height, √41 units, to acres, we can use the conversion factor:
Height in acres = √41 * (43,560 square feet / 10,000 square feet)
Height in acres = √(41 * 4.356)
Height in acres ≈ √178.596
Height in acres ≈ 13.375 acres
Finally, we can calculate the area of Field C by multiplying the length and height:
Area = Length * Height
Area = 2.178 acres * 13.375 acres
Area ≈ 29.145 acres
Therefore, the area of Field C is approximately 29.145 acres.
To find the area of Field C in acres using the Pythagorean Theorem, we first need to calculate the length of the sides of the triangle formed by the given points (4,7) and (7,3).
Step 1: Find the length of the base of the triangle:
The base of the triangle is the horizontal distance between the two points (4,7) and (7,3). By subtracting the x-coordinates, we get:
Base = 7 - 4 = 3 units
Step 2: Find the length of the height of the triangle:
The height of the triangle is the vertical distance between the two points (4,7) and (7,3). By subtracting the y-coordinates, we get:
Height = 7 - 3 = 4 units
Step 3: Calculate the hypotenuse of the triangle using the Pythagorean Theorem:
The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. In our case, the base and height are the other two sides. We can calculate the hypotenuse as follows:
Hypotenuse^2 = Base^2 + Height^2
Hypotenuse^2 = 3^2 + 4^2
Hypotenuse^2 = 9 + 16
Hypotenuse^2 = 25
Hypotenuse = √25
Hypotenuse = 5 units
Step 4: Calculate the area of the triangle:
The area of a triangle can be calculated using the formula A = 0.5 * base * height. Using the values we found:
Area = 0.5 * 3 * 4
Area = 6 square units
To convert the area from square units to acres, we need to consider the conversion factor. 1 acre is equal to 43,560 square feet. Therefore, to convert the area from square units to acres, we divide the area by 43,560:
Area in acres = 6 / 43,560
Area in acres ≈ 0.000138 acres
So, the area of Field C is approximately 0.000138 acres.