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.