Calculate the Perimeter of a Kite: Given Area 36cm² and Height 3 Units

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

• Step 1: Identify the given information about the diagonals and area.

• Step 2: Use the area formula to find the other diagonal.

• Step 3: Use the Pythagorean theorem to find the side lengths of the kite.

• Step 4: Add the side lengths to find the perimeter.

Now, let's work through each step:

Step 1: We are given that the kite has an area of 36 cm². The diagonal AE is 3 cm, and BD is given to be 8 cm.
The formula for the area of a kite is $\frac{1}{2} \times d_1 \times d_2 = 36$. Here, $d_1$ and $d_2$ are the diagonals AE and DT respectively.

Step 2: Substitute the known values into the formula:
$\frac{1}{2} \times 3 \times x = 36$
$\Rightarrow \frac{3x}{2} = 36$
$\Rightarrow 3x = 72$
$\Rightarrow x = 24$.
Thus, the length of diagonal DT is 24 cm.

Step 3: The kite can be divided into two pairs of right triangles; each formed with half the diagonals. The first triangle has sides AE = 3 cm and $\frac{BD}{2} = 4$ cm (since BD = 8 cm and each triangle shares half). We calculate the hypotenuse AD using the Pythagorean theorem:
$AD = \sqrt{\left(\frac{BD}{2}\right)^2 + AE^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$ cm.

In similar manner, calculate another pair of triangles constituted by CD and another half diagonal arrangement, with DE = 3 cm and CE = 12 cm each.
$CD = \sqrt{\left(\frac{BD}{2}\right)^2 + DE^2} = \sqrt{12^2 + 3^2} = \sqrt{144 + 9} = \sqrt{153} = \sqrt{73}$ cm.

Step 4: Calculate the perimeter by adding all four side lengths:
$P = 2 \times AD + 2 \times CD = 2 \times 5 + 2 \times \sqrt{73} = 10 + 2\sqrt{73}$ cm.

Therefore, the solution to the problem is $10 + 2\sqrt{73}$ cm.