Final answer:
The coordinates of point P, which divides the directed line segment from D (-2,-6) to F (7,6) in the ratio of 5:1, are calculated using the section formula.
Step-by-step explanation:
To find the coordinates of point P on the directed line segment from point D (-2,-6) to point F (7,6) that divides the segment in the ratio of 5:1, one can use the formula for determining the coordinates of a point that divides a line segment into a given ratio, which is known as the section formula.
The section formula in two dimensions is given by:
M(x1 + nx2) / (m + n), M(y1 + ny2) / (m + n)
Where M and N are the ratios in which point P divides line segment DF, and (x1, y1) and (x2, y2) are the coordinates of points D and F, respectively.
For the coordinates of D (-2,-6) and F (7,6), and the ratio of 5:1 (M:N), we plug the numbers into the formula to get the coordinates of point P:
- x coordinate of P = (5*(-2) + 1*7) / (5 + 1) = (-10 + 7) / 6 = -3 / 6 = -0.5
- y coordinate of P = (5*(-6) + 1*6) / (5 + 1) = (-30 + 6) / 6 = -24 / 6 = -4
Therefore, the coordinates of point P that divides the segment DF in the ratio of 5:1 are (-0.5, -4).