find the coordinates of point P along the directed line segment AB from A(-2,-8) to B(3,-3) so that the ratio of AP to PB is 3 to 2

To find the coordinates of point \( P \) along the directed line segment \( AB \) from \( A(-2,-8) \) to \( B(3,-3) \) such that the ratio of \( AP \) to \( PB \) is \( 3:2 \), you can use the section formula. 

The section formula for a point \( P(x, y) \) dividing a line segment joining two points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m:n \) is given by:

\[
P(x, y) = \left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\]

Given:
- \( A(x_1, y_1) = (-2, -8) \)
- \( B(x_2, y_2) = (3, -3) \)
- Ratio \( m:n = 3:2 \)

Substituting the values into the section formula:

\[
P(x, y) = \left( \frac{3 \cdot 3 + 2 \cdot (-2)}{3+2}, \frac{3 \cdot (-3) + 2 \cdot (-8)}{3+2} \right)
\]

Calculating the coordinates of \( P \):

\[
x = \frac{9 - 4}{5} = \frac{5}{5} = 1
\]

\[
y = \frac{-9 - 16}{5} = \frac{-25}{5} = -5
\]

So, the coordinates of point \( P \) are \( \mathbf{(1, -5)} \).

Would you like further details on this solution or have any questions?

Here are some related questions:
1. How would the solution change if the ratio was \( 2:3 \) instead of \( 3:2 \)?
2. What are the coordinates of the midpoint of segment \( AB \)?
3. How do you find the distance between points \( A \) and \( B \)?
4. Can the section formula be used in three dimensions?
5. What is the geometric interpretation of the section formula?

**Tip:** The section formula is a powerful tool to find points that divide a segment in any given ratio, and it can be extended to three dimensions as well.

Learn how to find the coordinates of point P along the line segment AB from A(-2,-8) to B(3,-3) so that the ratio of AP to PB is 3:2 using the section formula. Detailed step-by-step solution provided.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation