Final answer:
The length of the longer diagonal of the kite is found to be 10 cm by using the ratio of the diagonals and the area attribute related to the longer diagonal. The formula for the area of a kite was instrumental in solving for the length of the longer diagonal. Therefore, the length of the longer diagonal is 10 cm.
Step-by-step explanation:
The question involves finding the length of the longer diagonal of a kite given that the ratio of the diagonals is 2:5 and the area formed by the longer diagonal is 45 cm².
To solve this, we understand that the area of a kite can be found using the formula:
Area = (Diagonal1 × Diagonal2) / 2
Let's denote the shorter diagonal as 2x and the longer diagonal as 5x, based on the given ratio. We are given that the area associated with the longer diagonal is 45 cm², which means (5x × shorter diagonal) / 2 = 45 cm². However, since we are dealing only with the longer diagonal and its relation to the area, we have enough information to calculate its length.
Since the area is given by the product of half the diagonals, we have:
(5x × shorter diagonal) = 45 cm² × 2
We know we are only focusing on the longer diagonal (5x), which means the shorter diagonal portion of the formula will be some constant value k. Hence, 5x = 90/k. We don't need to find the exact value of k, as we are not asked for the shorter diagonal's length. We can determine the length of the longer diagonal which is sufficient to solve the problem.
We can now use the 45 cm² area to solve for x, which is part of the length of the longer diagonal. Since we know that the area that corresponds to the longer diagonal is 45 cm²:
45 cm² = (5x/2) × k
Multiplying both sides by 2 to eliminate the division by 2 and divide both sides by 45 to solve for x, we get:
x = 90/45
x = 2 cm
Now, since the longer diagonal is 5x, we multiply x by 5:
Longer diagonal = 5 × 2 cm
Longer diagonal = 10 cm, which corresponds to option D.
Therefore, the length of the longer diagonal is 10 cm.