Final answer:
The partition point with a ratio 3 to 1 of the segment from (-6, -1) to (6, -9) is (3, -7), which can be found using the section formula. The calculated point is not among the options provided, suggesting an error in the question or options.
Step-by-step explanation:
The coordinates of the point that partitions the line segment from (-6, -1) to (6, -9) in a 3 to 1 ratio can be found using the section formula, which is an application of the weighted average.
For a line segment with endpoints (x1, y1) and (x2, y2) and a ratio of m:n, the coordinates (x, y) of the partitioning point are calculated using the formula:
x = (mx2 + nx1) / (m + n)
y = (my2 + ny1) / (m + n)
Plugging the given values and ratio 3:1 into the formula:
x = (3*6 + 1*(-6)) / (3 + 1)
y = (3*(-9) + 1*(-1)) / (3 + 1)
After solving, we get:
x = (18 - 6) / 4 = 12 / 4 = 3
y = (-27 -1) / 4 = -28 / 4 = -7
Therefore, the coordinates are (3, -7). However, this point is not among the options provided, which indicates there might be a mistake in the question or the choices given.