Plies on a line segment AB and AP = 3 7 AB.

A is at (-2,-2) and B is at (2, -4).

Find the coordinates of P.

P =

Given:
- Coordinates of \( A \): \( (-2, -2) \)
- Coordinates of \( B \): \( (2, -4) \)
- Point \( P \) divides \( AB \) such that \( AP = \frac{3}{7} AB \)

This means the ratio in which \( P \) divides \( AB \) is \( 3 : 4 \) (since \( AP = \frac{3}{7} AB \), \( PB = \frac{4}{7} AB \)).

Using the section formula, if a point \( P(x, y) \) divides a line segment joining points \( A(x_1, y_1) \) and \( B(x_2, y_2) \) in the ratio \( m : n \), the coordinates of \( P \) are given by:
\[ x = \frac{mx_2 + nx_1}{m+n} \]
\[ y = \frac{my_2 + ny_1}{m+n} \]

Here:
- \( x_1 = -2 \), \( y_1 = -2 \)
- \( x_2 = 2 \), \( y_2 = -4 \)
- \( m = 3 \), \( n = 4 \)

Calculating the coordinates of \( P \):
\[ x = \frac{3 \cdot 2 + 4 \cdot (-2)}{3 + 4} = \frac{6 - 8}{7} = \frac{-2}{7} \]
\[ y = \frac{3 \cdot (-4) + 4 \cdot (-2)}{3 + 4} = \frac{-12 - 8}{7} = \frac{-20}{7} \]

So, the coordinates of \( P \) are:
\[ P \left( \frac{-2}{7}, \frac{-20}{7} \right) \]

Would you like more details or have any questions?

Here are 5 questions for you to ask:
1. Can you explain the section formula in more detail?
2. How do you convert a ratio like \( AP : PB \) to \( m : n \)?
3. Can you show another example of a point dividing a line segment in a given ratio?
4. What happens if the point \( P \) divides \( AB \) externally?
5. Can you explain the midpoint formula as well?

**Tip:** When working with ratios, ensure you clearly distinguish between the different parts of the segment being compared to avoid mistakes.

Learn how to find the coordinates of point P on line segment AB when given that AP divides AB in the ratio 3:4. Detailed solution using the section formula in coordinate geometry. Suitable for high school students studying coordinate geometry.

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