Deltoid Geometry: Finding Side Length BD Given Area 64 cm² and AC = 8 cm

To solve the problem of finding the length of the diagonal $BD$ in deltoid $ABCD$, where $AC = 8 \, \text{cm}$ and the area $S = 64 \, \text{cm}^2$, follow these steps:

• Step 1: Identify the given values: $AC = 8 \, \text{cm}$ and $S = 64 \, \text{cm}^2$.
• Step 2: Apply the area formula for a deltoid: $S = \frac{1}{2} \times AC \times BD$.
• Step 3: Substitute the known values into the formula.
• Step 4: Solve for $BD$.

Now, let's work through the calculation:

Given the formula for the area of a deltoid: $S = \frac{1}{2} \times AC \times BD$

Substitute the known values: $64 = \frac{1}{2} \times 8 \times BD$

To solve for $BD$, first multiply both sides by 2 to get rid of the fraction: $128 = 8 \times BD$

Now, divide both sides by 8 to isolate $BD$: $BD = \frac{128}{8} = 16$

Therefore, the length of $BD$ is $\textbf{16 cm}$.