Find the coordinate of the point that divides A(2,-2) and B(4,5) into 1:3.

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!

Discover how to find the coordinate of the point that divides the line segment joining A(2,-2) and B(4,5) in a 1:3 ratio. This problem uses the section formula from coordinate geometry, providing a detailed solution for students in Grades 9-11.

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