Area of a rhombus is 72 ft. and the product of the diagonals is 144 ft. What is the length of each diagonal?
I do not see that this can be solved with a unique solution.
For instance, 12 x 12 (square) works; 36 x 4 also works with the same area.
To find the length of each diagonal of a rhombus, we can use the formula:
Area = (1/2) * d1 * d2
where d1 and d2 are the lengths of the diagonals.
Given that the area is 72 ft and the product of the diagonals is 144 ft, we can substitute these values into the formula:
72 = (1/2) * d1 * d2
144 = d1 * d2
From the second equation, we can solve for d1 or d2 by dividing both sides of the equation by d2:
144/d2 = d1
Now, we can substitute this expression for d2 into the first equation:
72 = (1/2) * (144/d2) * d2
To simplify, we can cancel out d2 on the right side of the equation:
72 = (1/2) * 144
Now, we can solve for d1 by multiplying both sides of the equation by 2:
2 * 72 = 144
144 = 144
This means that d1 = d2 = 144.
Therefore, the length of each diagonal is 144 ft.
To find the length of each diagonal of a rhombus, we can start by using the formula for the area of a rhombus:
Area = (d1 * d2) / 2
Given that the area is 72 ft^2, we can substitute this value into the formula:
72 = (d1 * d2) / 2
Next, we are given that the product of the diagonals is 144 ft. We can substitute this value as well:
144 = d1 * d2
Now we have a system of two equations:
72 = (d1 * d2) / 2
144 = d1 * d2
We can solve this system of equations to find the values of d1 and d2, which represent the lengths of the diagonals.
To solve this system of equations, let's rearrange the first equation:
d1 * d2 = 72 * 2
d1 * d2 = 144
Now divide both sides of the second equation by d2:
d1 = 144 / d2
Substitute this value of d1 into the first equation:
d1 * d2 = 144
(144 / d2) * d2 = 144
Cancel out the d2 terms:
144 = 144
This equation is always true, which means any value of d2 will satisfy it. Therefore, the length of each diagonal can be any value that satisfies the equation d1 * d2 = 144 ft.