Final answer:
To find the area of the rhombus, we use the Pythagorean theorem to find the length of the other diagonal. Once we have the lengths of both diagonals, we can use the formula for the area to calculate it. The area of the rhombus is 24 square feet.
Step-by-step explanation:
To find the area of a rhombus, we need to know the lengths of its diagonals. In this case, we know the perimeter is 20 feet, so each side of the rhombus is 5 feet (since 20/4 = 5). We also know that one diagonal measures 6 feet. Draw a rhombus with side lengths of 5 feet and a diagonal of 6 feet. Now, we can use the Pythagorean theorem to find the length of the other diagonal.
Let's label the length of the other diagonal as 'd'. We have a right triangle formed by one side, half of the first diagonal (which is 3 feet), and half of the other diagonal (which is d/2 feet). Using the Pythagorean theorem, we can write the equation:
(d/2)^2 + 3^2 = 5^2
Simplifying the equation:
(d^2/4) + 9 = 25
Multiplying both sides by 4 to eliminate the fraction:
d^2 + 36 = 100
Subtracting 36 from both sides:
d^2 = 64
Taking the square root of both sides:
d = 8
So, the other diagonal measures 8 feet. Now that we know the lengths of both diagonals, we can use the formula for the area of a rhombus: Area = (diagonal1 * diagonal2) / 2.
Substituting the values:
Area = (6 * 8) / 2
Area = 48 / 2
Area = 24
Therefore, the area of the rhombus is 24 square feet.