Deltoid ABCD: Calculate DB When AC is 75% of DB and Area is 108X cm²

To solve this problem, let's proceed through the steps:

Step 1: Formula for the Area of a Deltoid

The area of a deltoid can be calculated through its diagonals using the formula:

$\text{Area} = \frac{1}{2} \times \text{Diagonal 1} \times \text{Diagonal 2}$

In this problem, the diagonals are $AC$ and $DB$.

Step 2: Express $AC$ in terms of $DB$

We are told that $AC$ is $75\%$ of $DB$. Therefore:

$AC = 0.75 \times DB$

Step 3: Substitute into the Area Formula

Substitute $AC = 0.75 \times DB$ into the area formula:

$\frac{1}{2} \times (0.75 \times DB) \times DB = 108X$

Simplifying gives:

$0.375 \times DB^2 = 108X$

Step 4: Solve for $DB$

Substituting $DB = X$ and rearranging the equation:

$0.375 \times X^2 = 108X$

Divide both sides by $X$ (assuming $X \neq 0$):

$0.375 X = 108$

Solving for $X$:

$X = \frac{108}{0.375} = 288$

Thus, the length of diagonal $DB$ is $\boxed{288}$ cm.