Area of a rhombus is 72 ft. and the product of the diagonals is 144 ft. What is the length of each diagonal?
the two pieces of data are redundant, since the area is always 1/2 the product of the diagonals.
There is not enough info here to pin down the lengths. All we have is pq=72.
Also, area and the product are ft^2, not ft.
To find the length of each diagonal, we can use the formula for the area of a rhombus:
Area = (diagonal1 * diagonal2) / 2
Given that the area is 72 ft, we can set up the equation:
72 = (diagonal1 * diagonal2) / 2
Multiplying both sides of the equation by 2:
144 = diagonal1 * diagonal2
Since we also know that the product of the diagonals is 144 ft, we can conclude that diagonal1 * diagonal2 = 144.
To find the length of each diagonal, we need to find the factors of 144. The possible pairs of factors are:
(1, 144), (2, 72), (3, 48), (4, 36), (6, 24), (8, 18), (9, 16), (12, 12)
Since a rhombus has congruent diagonals, we can eliminate pairs where the diagonals are different lengths. Therefore, we are left with (12, 12).
Thus, the length of each diagonal is 12 feet.
To find the length of each diagonal of a rhombus, we can use the given information about the area and product of the diagonals.
1. Let's start by using the formula for the area of a rhombus: Area = (diagonal1 * diagonal2) / 2.
Since the area is given as 72 ft, we have: 72 = (diagonal1 * diagonal2) / 2.
2. We are also given that the product of the diagonals is 144 ft. This means that diagonal1 * diagonal2 = 144.
3. Now that we have two equations, we can solve them simultaneously to find the lengths of the diagonals.
Let's simplify the equation for the area to isolate one of the diagonals:
72 * 2 = diagonal1 * diagonal2.
144 = diagonal1 * diagonal2.
4. Since the product of the diagonals is equal to 144, we can substitute this value into the area equation:
144 = diagonal1 * diagonal2.
5. Now we have a system of equations:
diagonal1 * diagonal2 = 144 ----------- Equation 1
diagonal1 * diagonal2 = 144 ----------- Equation 2
6. Since both equations are the same, it implies that diagonal1 = diagonal2. So, we can replace diagonal2 with diagonal1 in Equation 1:
diagonal1 * diagonal1 = 144.
7. Taking the square root of both sides, we get:
diagonal1 = √144 = 12 ft.
8. Therefore, the length of each diagonal of the rhombus is 12 ft.