Calculate Deltoid Perimeter: Given Area 16cm² with Height 3 and 5 Units

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

• Step 1: Calculate the full length of diagonal $BD$ using the area formula.

• Step 2: Use the Pythagorean theorem to find the side lengths of the deltoid.

• Step 3: Calculate the perimeter by summing all the side lengths.

Let's work through the solution:

Step 1: The area formula for a deltoid/kite with diagonals $d_1$ and $d_2$ is: $Area = \frac{1}{2} \times d_1 \times d_2$ Given that $d_1 = AE + EC = 3 + 5 = 8$ cm, we can solve for $d_2 = BD$.

Since the area is 16 cm$^2$, substitute the values: $16 = \frac{1}{2} \times 8 \times BD$$BD = 4 \, \text{cm}$

Step 2: The frame of the deltoid breaks into two congruent right triangles due to symmetry.
Let's consider triangle $ABE$: - $AE = 3$ cm and $BE = BD/2 = 2$ cm (as $BD$ bisects $AC$).
- Using the Pythagorean theorem: $AB^2 = AE^2 + BE^2 = 3^2 + 2^2 = 9 + 4 = 13$$AB = \sqrt{13} \text{ cm}$

For triangle $CED$:
- $CE = 5$ cm and again $DE = 2$ cm.
- Therefore, for side $CD$: $CD^2 = CE^2 + DE^2 = 5^2 + 2^2 = 25 + 4 = 29$$CD = \sqrt{29} \text{ cm}$

Step 3: The deltoid ABCD is symmetric, thus $AB = CD$ and $BC = AD$, so: $\text{Perimeter} = 2AB + 2CD = 2\sqrt{13} + 2\sqrt{29} \, \text{cm}$

Therefore, the perimeter of the deltoid ABCD is $2\sqrt{13}+2\sqrt{29}$ cm.