1. **Stating the problem:** We have four points on a horizontal line: $A(1,2)$, $M(a,b)$, $N(c,d)$, and $B(10,17)$ arranged from left to right. We want to understand the relationship between these points, especially the coordinates of $M$ and $N$ on the line segment $AB$. 2. **Key concept:** Since $A$, $M$, $N$, and $B$ lie on a horizontal line from left to right, their $y$-coordinates must be the same for all points because a horizontal line has constant $y$. 3. **Check $y$-coordinates:** Given $A(1,2)$ and $B(10,17)$, their $y$-coordinates differ ($2$ and $17$), so the line segment $AB$ is not horizontal. This contradicts the problem statement that the points lie on a horizontal line. 4. **Conclusion:** Since $A$ and $B$ do not share the same $y$-coordinate, the line $AB$ is not horizontal. Therefore, points $M$ and $N$ cannot lie on a horizontal line with $A$ and $B$ unless their $y$-coordinates are adjusted. 5. **If the line is not horizontal:** The points $M$ and $N$ lie somewhere on the line segment $AB$. The coordinates of $M$ and $N$ can be expressed using the section formula or parametric form: $$M = (a,b) = (1 + t_1(10 - 1), 2 + t_1(17 - 2)) = (1 + 9t_1, 2 + 15t_1)$$ $$N = (c,d) = (1 + t_2(10 - 1), 2 + t_2(17 - 2)) = (1 + 9t_2, 2 + 15t_2)$$ where $0 < t_1 < t_2 < 1$ because $M$ and $N$ lie between $A$ and $B$. 6. **Summary:** The points $M$ and $N$ lie on the line segment $AB$ and their coordinates depend on parameters $t_1$ and $t_2$ as above. **Final answer:** $$a = 1 + 9t_1, \quad b = 2 + 15t_1$$ $$c = 1 + 9t_2, \quad d = 2 + 15t_2$$ with $0 < t_1 < t_2 < 1$. This expresses $M$ and $N$ as points dividing the segment $AB$ internally.