Final answer:
The question asks for the coordinates of point C that would split the line segment AC into a 2:4 ratio with known points A and B. By applying the division formula in the given ratio, we can solve for C's coordinates.
Step-by-step explanation:
The question involves finding the coordinates of a point C that would split the segment AC into a 2:4 ratio, with A and B lying on a straight line and B being between A and C. We have A at (1, 5) and B at (5, 10).
To solve this, we can use the concept of internal division of a line segment in a given ratio. If B divides AC in the ratio m:n, then B's coordinates can be calculated with the formula ( (m*x2 + n*x1) / (m + n), (m*y2 + n*y1) / (m + n) ) where (x1, y1) are the coordinates of A and (x2, y2) are the coordinates of C.
Given the ratio of 2:4 (which simplifies to 1:2), and knowing the coordinates of A and B, we can set up a system of equations to solve for the coordinates of C:
For the x-coordinate: 5 = (1*x2 + 2*1) / (1 + 2)
For the y-coordinate: 10 = (1*y2 + 2*5) / (1 + 2)
By solving these equations, we find the coordinates of point C.