To find the length of $AD$ in triangle $ABD$, where $AB$ is perpendicular to $BD$ and $AB = 4 \, \text{cm}$, $BC = 2 \, \text{cm}$, and $AC = BD$, we need to determine the length of $BD$ first.

Given that $AC = BD$, let's first calculate $AC$ using the Pythagorean theorem in triangle $ABC$ since $AB$ is perpendicular to $BC$.

In $\triangle ABC$: $AC^2 = AB^2 + BC^2$ $AC^2 = 4^2 + 2^2$ $AC^2 = 16 + 4 = 20$ $AC = \sqrt{20} = 2\sqrt{5} \, \text{cm}$

Since $AC = BD$, $BD = 2\sqrt{5} \, \text{cm}$.

Now, using the Pythagorean theorem in $\triangle ABD$: $AD^2 = AB^2 + BD^2$ $AD^2 = 4^2 + (2\sqrt{5})^2$ $AD^2 = 16 + 20 = 36$ $AD = \sqrt{36} = 6 \, \text{cm}$

Therefore, $AD = 6 \, \text{cm}$.

Would you like further details or have any questions about these calculations? Here are some additional questions to consider:

1. How does the Pythagorean theorem help in solving problems involving right triangles?
2. What are other examples where the Pythagorean theorem can be used outside of geometric problems?
3. How can you determine the height of a physical object using trigonometric ratios?
4. What are the properties of similar triangles and how can they be applied in real-world situations?
5. Can you explore the relationship between different types of triangles and the circle?

Tip: Always ensure you have identified the right type of triangle (whether it is a right triangle, isosceles, etc.) before applying geometric formulas and theorems.