Question

1.5 partitioning a line segment 1. given directed line segment \\(\overline{ab}\\), find the coordinates of \\(p\\) such that the ratio of \\(ap\\) to \\(pb\\) is \\(2:1\\). plot point \\(p\\).

Explanation:

To solve this, we first identify coordinates of \( A \) and \( B \). From the grid, let's assume \( A = (-8, -1) \) and \( B = (4, -1) \) (since they lie on the same horizontal line, \( y = -1 \)).

Step 1: Recall the section formula

For a point \( P(x, y) \) dividing segment \( AB \) in ratio \( m:n \) (here, \( m = 2 \), \( n = 1 \)), the coordinates are:
\[
x = \frac{m \cdot x_B + n \cdot x_A}{m + n}, \quad y = \frac{m \cdot y_B + n \cdot y_A}{m + n}
\]

Step 2: Substitute values

Given \( A(-8, -1) \), \( B(4, -1) \), \( m = 2 \), \( n = 1 \):

• For \( x \)-coordinate:

\[
x = \frac{2 \cdot 4 + 1 \cdot (-8)}{2 + 1} = \frac{8 - 8}{3} = 0
\]

• For \( y \)-coordinate:

\[
y = \frac{2 \cdot (-1) + 1 \cdot (-1)}{2 + 1} = \frac{-2 - 1}{3} = -1
\]

Thus, the coordinates of \( P \) are \( (0, -1) \).

Answer:

The coordinates of \( P \) are \(\boldsymbol{(0, -1)}\).