Calculate the Diagonal Length: Deltoid with Area 39 cm² and Side 13 cm

To find the length of diagonal $BD$, we will apply the formula for the area of a deltoid:

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

In this problem, Diagonal 1 is $AC = 13$ cm, and Diagonal 2 is $BD$, which we are trying to find. The area is given as $39$ cm². Substituting these values into the formula, we get:

$39 = \frac{1}{2} \times 13 \times BD$

To solve for $BD$, multiply both sides by 2 to eliminate the fraction:

$78 = 13 \times BD$

Now, solve for $BD$ by dividing both sides by 13:

$BD = \frac{78}{13}$

Simplify to find:

$BD = 6$

Therefore, the length of diagonal $BD$ is $6$ cm.