Deltoid Geometry: Finding Side Length BD Given Area 40 cm² and AC = 10 cm

To solve this problem, we'll follow these steps:

• Step 1: Utilize the formula for the area of a kite or deltoid, $S = \frac{1}{2} \cdot d_1 \cdot d_2$.
• Step 2: Substituting the known values into this formula.
• Step 3: Solve for the unknown diagonal $BD$.

Now, let's work through each step:
Step 1: The given side $AC$ acts as the first diagonal $d_1 = 10$ cm. The area $S = 40$ cm².
Step 2: Plug these values into the formula $S = \frac{1}{2} \cdot d_1 \cdot d_2$ which becomes $40 = \frac{1}{2} \cdot 10 \cdot BD$.
Step 3: Solving for $BD$ involves rearranging the equation: $40 = \frac{1}{2} \cdot 10 \cdot BD \implies 40 = 5 \cdot BD \implies BD = \frac{40}{5} = 8 \text{ cm}$

Therefore, the length of the side $BD$ is $8 \text{ cm}$.