Final answer:
The coordinates of point B on the line segment AC, where AB is 6 times as long as BC and points A and C are (1,5) and (-5,-7) respectively, are found to be (-29/7, -37/7).
Step-by-step explanation:
To find the coordinates of point B on line segment AC, where AB is 6 times as long as BC, we can use the section formula. Given that point A is at (1,5) and point C is at (-5,-7), let the coordinates of B be (x, y), with AB:BC being 6:1.
The section formula for internal division states that:
- x = (m * x2 + n * x1) / (m + n)
- y = (m * y2 + n * y1) / (m + n)
Where m:n is the ratio, in this case 6:1, x1 and y1 are the coordinates of A, and x2 and y2 are the coordinates of C.
Plugging in the values, we get:
- x = (6 * (-5) + 1 * 1) / (6 + 1) = (-30 + 1) / 7 = -29 / 7
- y = (6 * (-7) + 1 * 5) / (6 + 1) = (-42 + 5) / 7 = -37 / 7
So the coordinates of point B are (-29/7, -37/7).