Final answer:
To find the point that divides line segment AB into a 3:4 ratio, we need the coordinates of A and B to apply the section formula. As the coordinates for A and B have not been given, we cannot accurately solve for the dividing point with the provided information.
Step-by-step explanation:
The student has asked which point divides the directed line segment AB into a 3:4 ratio. To solve this problem, we use the section formula, which is used to find a point P that divides the line segment joining A(x1, y1) and B(x2, y2) in the ratio m:n. The formula for the coordinates of point P is given by:
P(x, y) = ((mx2 + nx1) / (m + n), (my2 + ny1) / (m + n))
Without the coordinates for points A and B, we can't calculate the exact coordinates for the point that divides the line into a 3:4 ratio. However, the student has provided multiple-choice options: (4, 3), (7, 3), (9, 3), (12, 3). To choose from these, we would need to see which option, when placed into the section formula with assumed coordinates for A and B that match the y-values of the options, results in a 3:4 ratios on both the x and y axes. Since we do not have sufficient information to solve the problem completely, it is best to consider this question as incomplete or to ask for additional information regarding points A and B to find the correct answer.