Area of a Deltoid: Using Pythagoras' theorem

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

• Step 1: Calculate the side length of the square using the given area.
• Step 2: Determine the length of $BD$ using the side length.
• Step 3: Use the kite area formula with diagonals $BD$ and $AC=2x$.

Now, let's work through each step:

Step 1: The area of the square is 36 cm². The side length of the square, denoted as $s$, can be calculated by taking the square root of the area:

$s = \sqrt{36} = 6 \text{ cm}$

Step 2: To find diagonal $BD$, we use the relationship for the diagonal of a square in terms of its side:

$BD = s \sqrt{2}$. Given $s = 6$, we compute:

$BD = 6\sqrt{2} \text{ cm}$

Step 3: Now, we apply the formula for the area of a kite, which is $\frac{1}{2} \times d1 \times d2$, where $d1 = BD = 6\sqrt{2}$ and $d2 = AC = 2x$:

The area $A$ of the kite is:

$A = \frac{1}{2} \times 6\sqrt{2} \times 2x = 6\sqrt{2}x \text{ cm}^2$

Therefore, the area of the kite in terms of $x$ is $6\sqrt{2}x$ cm².