Question

In 3 to 5 steps explain how to find the coordinate of point P.

What are the coordinates of the point P on the directed line segment from S(-2, 4) to T(7, -2) that partitions the segment into a ratio of 1:2?

Answer 1

To find the coordinates of point P that partitions the directed line segment from S(-2, 4) to T(7, -2) in the ratio 1:2, follow these steps:

1. **Determine the ratio**: The ratio of 1:2 means that point P divides the segment into 1 part from S to P and 2 parts from P to T.

2. **Find the coordinates using the section formula**: The section formula for finding a point P that divides a line segment joining two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the ratio \( m:n \) is given by:
\[
P\left( \frac{m x_2 + n x_1}{m + n}, \frac{m y_2 + n y_1}{m + n} \right)
\]
Here, \( S(-2, 4) = (x_1, y_1) \) and \( T(7, -2) = (x_2, y_2) \), with \( m = 1 \) and \( n = 2 \).

3. **Plug in the values**:
\[
P_x = \frac{1 \cdot 7 + 2 \cdot (-2)}{1 + 2} = \frac{7 - 4}{3} = \frac{3}{3} = 1
\]
\[
P_y = \frac{1 \cdot (-2) + 2 \cdot 4}{1 + 2} = \frac{-2 + 8}{3} = \frac{6}{3} = 2
\]

4. **Write the coordinates of point P**: The coordinates of point P are \( (1, 2) \).

Thus, point P that partitions the segment from S to T in the ratio 1:2 is at coordinates \( (1, 2) \).