To find the length of $DE$ in the cuboid using Pythagoras' theorem, we approach this in two steps.

The cuboid has three distinct dimensions:

• Height = 5 cm (BC)
• Length = 19 cm (AD)
• Width = 21 cm (EH)

1. First Step: Consider the right-angled triangle $ADE$, where $AD = 19 \, \text{cm}$ and $AE = 21 \, \text{cm}$. We will use the Pythagorean theorem in this plane to find the diagonal $DE$.

The Pythagorean theorem states: $DE^2 = AD^2 + AE^2$ Substitute the known values: $DE^2 = 19^2 + 21^2 = 361 + 441 = 802$ $DE = \sqrt{802} \approx 28.3 \, \text{cm}$

So, the length of $DE$ is approximately 28.3 cm to 1 decimal place.

Would you like further details on the steps or have any questions? Here are some related questions to explore further:

1. What are the properties of a cuboid and how do they relate to calculating distances inside it?
2. How can you extend Pythagoras' theorem to find distances in 3D shapes?
3. What are other ways to calculate the diagonal in different shapes?
4. What are the different types of right-angled triangles, and how do they relate to Pythagoras’ theorem?
5. How does the Pythagorean theorem apply to real-world situations, such as construction or design?

Tip: Make sure to square both dimensions when applying the Pythagorean theorem to avoid any mistakes in the calculation.