We are given that the points $L$, $M$, $N$, and $O$ all lie on the same line segment, with the ratio of the segments $LM : MN : NO = 1 : 6 : 1$, and that the total length of $LO = 16$.

To find the length of $NO$, we can use the following approach:

Step 1: Express the total length in terms of parts

The given ratio $LM : MN : NO = 1 : 6 : 1$ means that we can express each segment as a multiple of a common part, say $x$. So: $LM = 1x, \quad MN = 6x, \quad NO = 1x$ Thus, the total length of the segment $LO$ is: $LO = LM + MN + NO = 1x + 6x + 1x = 8x$

Step 2: Solve for $x$

We are told that $LO = 16$. Therefore: $8x = 16$ Solving for $x$, we get: $x = \frac{16}{8} = 2$

Step 3: Find $NO$

Since $NO = 1x$, we substitute $x = 2$ to get: $NO = 1 \times 2 = 2$

Thus, the length of $NO$ is $\boxed{2}$.

Would you like more details or further clarification on this?

Here are some related questions for further practice:

1. If $MN$ were given, how would you find the value of $LM$?
2. What if the total length $LO$ was different, say 24? How would that change the calculation of $NO$?
3. How can you generalize the approach to solve any ratio problem like this?
4. If the ratio had included another segment, say $NO : OP$, how would you solve for the length of $NO$?
5. How does the concept of ratios apply to real-world measurements?

Tip: Always check that the total length matches the sum of the segments when working with ratios to avoid calculation errors.