Answer:
8 ft and 14 ft
Step-by-step explanation:
Let one diagonal be x then the other diagonal is 2x - 2
The area (A) of the rhombus is calculated using
A = [tex]\frac{1}{2}[/tex] product of the diagonals, that is
A = [tex]\frac{1}{2}[/tex] x(2x - 2) = 56
Multiply both sides by 2
x(2x - 2) = 112 ← distribute left side
2x² - 2x = 112 ( subtract 112 from both sides )
2x² - 2x - 112 = 0 ← in standard form ( divide through by 2 )
x² - x - 56 = 0
To factor the quadratic
Consider the factors of the constant term (- 56) which sum to give the coefficient of the x- term (- 1)
The factors are - 8 and + 7, since
- 8 × 7 = - 56 and - 8 + 7 = - 1, thus
(x - 8)(x + 7) = 0
Equate each factor to zero and solve for x
x - 8 = 0 ⇒ x = 8
x + 7 = 0 ⇒ x = - 7
However, x > 0 ⇒ x = 8
One diagonal = 8 ft and the other = 2x - 2 = (2 × 8) - 2 = 16 - 2 = 14 ft
