Area of a rhombus is 72 ft. and the product of the diagonals is 144 ft. What is the length of each diagonal?
To find the length of each diagonal of a rhombus, we can use the formula:
Area of a rhombus = (diagonal1 * diagonal2) / 2
Given:
Area = 72 ft²
Product of the diagonals = 144 ft
Let's solve for the length of each diagonal step-by-step:
Step 1: Multiply the area by 2.
2 * Area = 2 * 72 ft² = 144 ft²
Step 2: Substitute the values into the formula.
144 ft² = diagonal1 * diagonal2
Step 3: Use the fact that the product of the diagonals is 144 ft to find the values of diagonal1 and diagonal2.
By finding two numbers whose product is 144, we can solve this equation.
Let's list the factors of 144:
1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144
Since we're looking for the length of each diagonal, diagonal1 and diagonal2 must be positive integers. We'll check the factors to find the pair that satisfies this condition.
After checking the factors, we can see that the pair that satisfies the condition is:
diagonal1 = 12 ft
diagonal2 = 12 ft
Therefore, the length of each diagonal is 12 ft.
To find the length of each diagonal of a rhombus, we can use the formula:
Diagonal 1 = 2 * (Area of the rhombus / Length of the other diagonal)
Given:
Area of the rhombus = 72 ft²
Product of the diagonals = 144 ft
Let's find the length of the first diagonal:
Diagonal 1 = 2 * (72 / Diagonal 2)
Diagonal 1 = 144 / Diagonal 2
Now, we have a equation:
Diagonal 1 * Diagonal 2 = 144 ft
To find the value of each diagonal, we need to find the pair of numbers whose product is 144 and one of the numbers is twice the other number.
The possible pairs of numbers whose product is 144 are:
1 * 144 = 144
2 * 72 = 144
3 * 48 = 144
4 * 36 = 144
6 * 24 = 144
8 * 18 = 144
9 * 16 = 144
12 * 12 = 144
16 * 9 = 144
18 * 8 = 144
24 * 6 = 144
36 * 4 = 144
48 * 3 = 144
72 * 2 = 144
144 * 1 = 144
Out of these pairs, the pair that satisfies the condition of one number being twice the other number is (8, 18).
Therefore, the length of each diagonal is:
Diagonal 1 = 8 ft
Diagonal 2 = 18 ft
let the diagonals be a and b respectively.
given : ab = 144
but the area of a rhomus = (1/2) product of diagonals
= 1/2 (144) = 72
and that was also given.
So we have redundant information, but not enough to find the values of a and b
a and b could be any values, so that ab = 144
e.g.
a = 4 , b = 36
a = 12, b = 12 , making it a square
a = 9, b = 16
etc