Use the Pythagorean theorem to find the area of field C (4,7) (0,4) (3,0) (7,3)
To find the area of field C using the Pythagorean theorem, we need to find the length of its sides first.
Using the distance formula, the lengths of the sides can be determined as follows:
Side AB = sqrt((7-4)^2 + (0-3)^2) = sqrt(3^2 + (-3)^2) = sqrt(18) = 3√2
Side BC = sqrt((3-0)^2 + (7-0)^2) = sqrt(3^2 + 7^2) = sqrt(58)
Side CD = sqrt((7-4)^2 + (3-0)^2) = sqrt(3^2 + 3^2) = sqrt(18) = 3√2
Side DA = sqrt((0-3)^2 + (4-0)^2) = sqrt((-3)^2 + 4^2) = sqrt(25) = 5
Now that we have the lengths of all four sides, we can use the formula for the area of a quadrilateral:
Area of field C = (1/4) * sqrt((ab^2 + ac^2 - bc^2 + bd^2) * (-(ab^2) + ac^2 + bc^2 + bd^2))
Area of field C = (1/4) * sqrt((3√2)^2 + (5)^2 - (sqrt(58))^2 + (3√2)^2) * (-(3√2)^2 + (5)^2 + (sqrt(58))^2 + (3√2)^2))
Area of field C = (1/4) * sqrt(18 + 25 - 58 + 18) * (-(18) + 25 + 58 + 18))
Area of field C = (1/4) * sqrt(3) * 79
Area of field C ≈ 4.414
To find the area of field C using the Pythagorean theorem, we need to calculate the lengths of its sides.
Step 1: Find the length of the sides using the distance formula:
- Side AB: The distance between points A(4,7) and B(0,4). Using the distance formula √[(x2 - x1)^2 + (y2 - y1)^2], we have:
AB = √[(0 - 4)^2 + (4 - 7)^2]
= √[(-4)^2 + (-3)^2]
= √(16 + 9)
= √25
= 5
- Side BC: The distance between points B(0,4) and C(3,0). Using the distance formula, we have:
BC = √[(3 - 0)^2 + (0 - 4)^2]
= √[3^2 + (-4)^2]
= √(9 + 16)
= √25
= 5
- Side CD: The distance between points C(3,0) and D(7,3). Using the distance formula, we have:
CD = √[(7 - 3)^2 + (3 - 0)^2]
= √[(4)^2 + (3)^2]
= √(16 + 9)
= √25
= 5
- Side DA: The distance between points D(7,3) and A(4,7). Using the distance formula, we have:
DA = √[(4 - 7)^2 + (7 - 3)^2]
= √[(-3)^2 + (4)^2]
= √(9 + 16)
= √25
= 5
Step 2: Determine if the field is a rectangle or another shape based on the lengths of its sides.
Since all sides of the field have the same length (5 units), and the opposite sides are parallel, field C is a rectangle.
Step 3: Calculate the area of the rectangle.
Now that we know field C is a rectangle, we can use the formula to find its area:
Area = length * width
= AB * BC
= 5 * 5
= 25
So, the area of field C is 25 square units.
To find the area of field C using the Pythagorean theorem, we first need to determine the length of all the sides of the field.
Let's label the four points on the field as follows:
A(4,7) is the top right corner.
B(0,4) is the top left corner.
C(3,0) is the bottom left corner.
D(7,3) is the bottom right corner.
To find the length of each side, we need to calculate the distance between each pair of adjacent points. The distance between two points (x1, y1) and (x2, y2) can be calculated using the distance formula:
Distance = √((x2 - x1)^2 + (y2 - y1)^2)
Now, let's calculate the distances:
AB = √((0 - 4)^2 + (4 - 7)^2) = √((-4)^2 + (-3)^2) = √(16 + 9) = √25 = 5
BC = √((3 - 0)^2 + (0 - 4)^2) = √(3^2 + (-4)^2) = √(9 + 16) = √25 = 5
CD = √((7 - 3)^2 + (3 - 0)^2) = √(4^2 + 3^2) = √(16 + 9) = √25 = 5
DA = √((7 - 4)^2 + (3 - 7)^2) = √(3^2 + (-4)^2) = √(9 + 16) = √25 = 5
Now that we have the lengths of all the sides (AB, BC, CD, and DA), we can determine the type of shape formed by the four points: a rectangle.
Since the opposite sides of a rectangle are equal, we can use any two adjacent sides to calculate the area of the rectangle.
Let's use AB and BC.
The area of the rectangle is calculated by multiplying the length and width. In this case, the length is AB and the width is BC.
Area of the rectangle = AB * BC = 5 * 5 = 25 square units.
Therefore, the area of field C is 25 square units.