Final answer:
To determine the sum of the diagonals of a kite with an area of 30 cm² with diagonals measuring x and x-4, we set up and solve the quadratic equation x(x-4) = 60. The sum of the diagonals is found to be 16 cm.
Step-by-step explanation:
The area of a kite can be calculated using the formula ½(d1 × d2), where d1 and d2 are the lengths of the diagonals. Given the area 30 cm² and that one diagonal is x and the other is x-4, we have ½(x × (x-4)) = 30. Simplifying the equation, we get x(x-4) = 60. To find the value of x, we solve this quadratic equation.
Expanding the equation gives us x² - 4x - 60 = 0. Factoring, we find that (x - 10)(x + 6) = 0, so either x = 10 or x = -6. Since a diagonal cannot have a negative length, we conclude that x = 10 cm and therefore, the other diagonal is 10 - 4 = 6 cm. The sum of the diagonals will be 10 + 6 = 16 cm.