To solve for $CN$ in the given cuboid:

Given Information:

• $AB = 50 \, \text{cm}$, $BC = 25 \, \text{cm}$, and $CG = 16 \, \text{cm}$.
• The diagonal $CN$ is the line connecting $C$ and $N$, where $N$ is the midpoint of $AB$.

Steps to Find $CN$:

1. Find the coordinates of $N$: Since $N$ is the midpoint of $AB$, its coordinates are: $N = \left( \frac{50}{2}, 0, 0 \right) = (25, 0, 0)$

2. Determine the coordinates of $C$: Based on the cuboid dimensions, $C = (50, 25, 0)$.

3. Calculate the vector $\overrightarrow{CN}$: $\overrightarrow{CN} = \text{Coordinates of } N - \text{Coordinates of } C$ $\overrightarrow{CN} = (25, 0, 0) - (50, 25, 0) = (-25, -25, 0)$

4. Calculate the magnitude of $\overrightarrow{CN}$: The length $CN$ is the magnitude of $\overrightarrow{CN}$, given by: $CN = \sqrt{(-25)^2 + (-25)^2 + 0^2}$ $CN = \sqrt{625 + 625} = \sqrt{1250} = 35.36 \, \text{cm} \, (\text{rounded to 2 decimal places})$

Thus, the correct length of $CN$ is approximately 35.4 cm.

Would you like a step-by-step derivation or more details on the concepts?

Questions to Consider:

1. How can the midpoint formula be generalized for different dimensions?
2. What is the significance of diagonals in a cuboid for calculating distances?
3. How do coordinate geometry principles simplify such 3D problems?
4. Can the problem be approached differently (e.g., vector algebra)?
5. How can such geometric problems be applied in real-world contexts?

Tip:

Always double-check for consistent unit usage and ensure