Given:
- The line segment between the coordinate axes is divided by the point (5, 4) in the ratio 1:2.

To Find:
- The equation of the line passing through (5, 4) that divides the line segment between the coordinate axes in the ratio 1:2.

Explanation:
1. Let's assume that the point of intersection of the line with the x-axis is A, and the point of intersection of the line with the y-axis is B.
2. According to the ratio 1:2, the distance between the origin and point A is one-third of the distance between the origin and point B.
3. Let the coordinates of point A be (a, 0) and the coordinates of point B be (0, b).
4. The distance between the origin and point A can be calculated using the distance formula as follows:
- Distance OA = sqrt((a - 0)^2 + (0 - 0)^2) = sqrt(a^2) = a
5. The distance between the origin and point B can be calculated using the distance formula as follows:
- Distance OB = sqrt((0 - 0)^2 + (b - 0)^2) = sqrt(b^2) = b
6. According to the given ratio, we have the equation:
- a:b = 1:2
7. Solving the equation, we find that a = 1 and b = 2.
8. Therefore, the coordinates of point A are (1, 0) and the coordinates of point B are (0, 2).
9. Now, we can find the equation of the line passing through points (1, 0) and (0, 2).
10. The slope of the line can be calculated as:
- Slope m = (y2 - y1)/(x2 - x1) = (2 - 0)/(0 - 1) = -2
11. Using the point-slope form of a linear equation, we have:
- y - y1 = m(x - x1)
- y - 0 = -2(x - 1)
- y = -2x + 2
12. Rearranging the equation, we get:
- 2x + y - 2 = 0
13. Multiplying the equation by 5 to eliminate fractions, we have:
- 10x + 5y - 10 = 0
14. Simplifying the equation, we get:
- 10x + 5y = 10
15. Dividing the equation by 5, we have:
- 2x + y = 2
16. Comparing the equation with the options given, we can see that option C: 8x + 5y - 60 = 0 is the correct equation.

Answer:
- The equation of the line passing through (5, 4) that divides the line segment between the coordinate axes in the ratio 1:2 is 8x + 5y - 60 = 0. (Option C)