Deltoid ABCD: Finding Side Length BD Given Area 27.5 cm² and AC = 5.5 cm

To solve this problem, we'll proceed as follows:

• Step 1: Identify given information and formula to use: The area of a deltoid is given as $\text{Area} = \frac{1}{2} \times d_1 \times d_2$.
• Step 2: Plug in the known values: $27.5 = \frac{1}{2} \times 5.5 \times BD$.
• Step 3: Calculate the unknown length $BD$.

Now, let's work through each step:

Step 1: Given $\text{Area} = 27.5$ cm$^2$, $AC = 5.5$ cm, and the formula for the area of a deltoid: $\text{Area} = \frac{1}{2} \times d_1 \times d_2$ where $d_1 = AC$ and $d_2 = BD$.

Step 2: Use the formula with the given values:
$27.5 = \frac{1}{2} \times 5.5 \times BD$.

Step 3: Solve for $BD$:
Multiply both sides by 2 to eliminate the fraction:
$55 = 5.5 \times BD$.
Now, divide both sides by $5.5$:
$BD = \frac{55}{5.5}$.

Simplify $\frac{55}{5.5}$:
$BD = 10$ cm.

Therefore, the length of side $BD$ is $10$ cm.