Use the Pythagorean theorem to find the area of field C (4,7) (0,4) (3,0) (7,3)

Answer 1

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