Final answer:
The coordinates of point P on line segment ST with given ratio AP:PB of 4:3 are found using the internal division formula, resulting in (approximately) (-2.14, -2).
Step-by-step explanation:
To find the coordinates of point P that lies on line segment ST with points S(-1, 2) and T(-3, -5) and given that the ratio of AP to PB is 4:3, we can use the section formula which is derived from the concept of similar triangles.
Since the ratio is 4:3, we can say that AP is 4x and PB is 3x for some value x. Therefore, the entire segment ST is 7x. This information can be used to apply the internal division formula:
- M_x = \(\frac{x_1 \cdot n + x_2 \cdot m}{m + n}\)
- M_y = \(\frac{y_1 \cdot n + y_2 \cdot m}{m + n}\)
Where ('M_x', 'M_y') are the coordinates of P, (x_1, y_1) are the coordinates of S, (x_2, y_2) are those of T, and m:n is the ratio of AP:PB respectively.
By plugging in the known values:
- M_x = \(\frac{-1 \cdot 3 + (-3) \cdot 4}{4 + 3}\) = \(\frac{-3 - 12}{7}\) = \(\frac{-15}{7}\)
- M_y = \(\frac{2 \cdot 3 + (-5) \cdot 4}{4 + 3}\) = \(\frac{6 - 20}{7}\) = \(\frac{-14}{7}\)
After simplification, we find that the coordinates of point P are (\(\frac{-15}{7}\), \(\frac{-14}{7}\)) or approximately (-2.14, -2).