Question

Point b on a segment with endpoints a(2, -2) and c(4, 1) partitions the segment in a 1:3 ratio. Find the coordinates of point b.

1) (2.5, -1.25)
2) (0.5, 0.75)
3) (-1.25, 2.5)
4) (0.75, 0.5)

Answer 1

The coordinates of point b that partitions the segment AC in a 1:3 ratio are found using the section formula and are (2.5, -1.25), which corresponds to option (1).

Step-by-step explanation:

To find the coordinates of point b on a segment with endpoints A(2, -2) and C(4, 1) that partitions the segment in a 1:3 ratio, we use the section formula. The coordinates of point B can be calculated as follows:

1. Let x1 = 2, y1 = -2 (coordinates of point A), x2 = 4, y2 = 1 (coordinates of point C), and let the ratio be m:n = 1:3.
2. Plug these values into the section formula to find the coordinates of point B:
3. x = (mx2 + nx1) / (m + n)
4. y = (my2 + ny1) / (m + n)
5. Thus, Bx = (1*4 + 3*2) / (1 + 3) = (4 + 6) / 4 = 10 / 4 = 2.5
6. By = (1*1 + 3*(-2)) / (1 + 3) = (1 - 6) / 4 = -5 / 4 = -1.25
7. Therefore, the coordinates of point B are (2.5, -1.25).

Hence, option 1) (2.5, -1.25) is the correct answer.