The problem asks to find the length of $AB$ in triangle $\Delta ABC$, where $BD$ is the altitude on the hypotenuse.

Given:

• $AD = 3\sqrt{3} \, \text{cm}$
• $DC = \sqrt{75} \, \text{cm}$

We can solve this using the geometric mean theorem for right triangles. This theorem tells us that:

$AB^2 = AD \cdot AC$

To use this, we first need to find $AC$, the entire hypotenuse, using:

$AC = AD + DC$

Let’s proceed with the calculations.

Step 1: Calculate $AC$

$AC = 3\sqrt{3} + \sqrt{75}$

Simplify $\sqrt{75}$: $\sqrt{75} = \sqrt{25 \cdot 3} = 5\sqrt{3}$

Thus, $AC = 3\sqrt{3} + 5\sqrt{3} = 8\sqrt{3}$

Step 2: Apply the geometric mean theorem

$AB^2 = AD \cdot AC = (3\sqrt{3}) \cdot (8\sqrt{3}) = 3 \cdot 8 \cdot 3 = 72$

Step 3: Solve for $AB$

$AB = \sqrt{72} = \sqrt{36 \cdot 2} = 6\sqrt{2} \, \text{cm}$

Thus, the length of $AB$ is $6\sqrt{2} \, \text{cm}$.

Let me know if you'd like further clarifications or details!

Here are some related questions:

1. How do you apply the geometric mean theorem in different triangle configurations?
2. What are the properties of right triangles regarding their altitudes?
3. How can the Pythagorean theorem assist in solving triangle-related problems?
4. What are the steps to simplify square roots, like $\sqrt{75}$?
5. How would you approach this problem if only one leg of the triangle was known?

Tip: Always look for special relationships between triangle sides, such as geometric means, especially when working with right triangles!