To find the coordinates of the point $P(x, y)$ that divides the line segment joining points $A(2, -2)$ and $B(4, 5)$ in the ratio $1:3$, we can use the section formula. For a point dividing a line in the ratio $m:n$, the coordinates are given by:

$P(x, y) = \left( \frac{m \cdot x_2 + n \cdot x_1}{m+n}, \frac{m \cdot y_2 + n \cdot y_1}{m+n} \right)$

Here, $A(x_1, y_1) = (2, -2)$, $B(x_2, y_2) = (4, 5)$, and the ratio is $m:n = 1:3$.

Substitute the values:

$P(x, y) = \left( \frac{1 \cdot 4 + 3 \cdot 2}{1+3}, \frac{1 \cdot 5 + 3 \cdot (-2)}{1+3} \right)$

Let's compute this step by step:

$P(x) = \frac{4 + 6}{4} = \frac{10}{4} = 2.5$ $P(y) = \frac{5 - 6}{4} = \frac{-1}{4} = -0.25$

Thus, the coordinates of the point are $P(2.5, -0.25)$.

Would you like more details or have any questions?

Here are some related problems for you to consider:

1. Find the point dividing the line segment joining $(1, 2)$ and $(5, 6)$ in the ratio 2:3.
2. Calculate the midpoint of the segment joining $(3, -2)$ and $(7, 4)$.
3. Determine the distance between points $(0, 0)$ and $(5, 12)$.
4. What is the ratio in which the point $(2, 3)$ divides the segment joining $(1, 2)$ and $(5, 6)$?
5. Find the area of the triangle with vertices at $(0, 0)$, $(4, 0)$, and $(4, 3)$.

Tip: The section formula can be used for both internal and external division. Make sure to adjust the signs accordingly for external division!