This is a right-angled triangle with the following known elements:

• One side (adjacent) = $16$ units (base $ON$).
• Hypotenuse = $25$ units ($OP$).
• A right angle at vertex $N$.
• Unknown side ($x$) is the vertical side ($NP$).

We will use the Pythagorean theorem to solve for $x$:

$OP^2 = ON^2 + NP^2$

Step 1: Substitute known values:

$25^2 = 16^2 + x^2$

$625 = 256 + x^2$

Step 2: Solve for $x^2$:

$x^2 = 625 - 256 = 369$

Step 3: Find $x$:

$x = \sqrt{369} \approx 19.21 \, \text{units.}$

Final Answer:

The length of $NP$ is approximately $19.21$ units.

Would you like me to verify the result further, provide more explanation, or solve related problems?

Here are 5 related questions to explore:

1. What are the trigonometric ratios ($\sin$, $\cos$, $\tan$) for this triangle?
2. How would the solution change if the hypotenuse were longer or shorter?
3. Can we calculate the angles other than the right angle in the triangle?
4. How is this problem connected to real-world applications of the Pythagorean theorem?
5. What is the perimeter of the triangle?

Tip: Always remember that the hypotenuse is the longest side in any right-angled triangle.