Area of a Deltoid: Calculation using percentages

To solve the problem, we need to find the lengths of the diagonals and then calculate the area of the deltoid using these lengths.

• First, find the length of diagonal $AC$.

Given that $AC$ is 40% longer than $DB$, which is 10 cm:

$AC = 10 + 0.4 \times 10 = 10 + 4 = 14 \, \text{cm}$

• Now, apply the formula for the area of the deltoid:

The area $A$ of a deltoid can be calculated using the formula:

$A = \frac{1}{2} \times d_1 \times d_2$

Substituting the given values ($d_1 = DB = 10 \, \text{cm}$ and $d_2 = AC = 14 \, \text{cm}$):

$A = \frac{1}{2} \times 10 \times 14 = \frac{1}{2} \times 140 = 70 \, \text{cm}^2$

Therefore, the area of the deltoid ABCD is 70 cm², which matches the given correct answer.

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.