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The problem involves finding the coordinates of a point \( K \) that lies on the segment \( XY \) such that \( K \) divides the segment \( XY \) in the ratio \( 1:3 \). The coordinates of \( X \) and \( Y \) are given as \( X(-1, -4) \) and \( Y(5, 2) \), respectively.

To solve this, we can use the section formula, which states that if a point \( K \) divides the line segment joining two points \( X(x_1, y_1) \) and \( Y(x_2, y_2) \) in the ratio \( m:n \), then the coordinates of \( K(x, y) \) are given by:

\[
x = \frac{m \cdot x_2 + n \cdot x_1}{m+n}
\]

\[
y = \frac{m \cdot y_2 + n \cdot y_1}{m+n}
\]

For this problem:
- \( x_1 = -1 \)
- \( y_1 = -4 \)
- \( x_2 = 5 \)
- \( y_2 = 2 \)
- \( m = 1 \)
- \( n = 3 \)

Now, substituting the values into the formulas:

\[
x = \frac{1 \cdot 5 + 3 \cdot (-1)}{1+3} = \frac{5 - 3}{4} = \frac{2}{4} = 0.5
\]

\[
y = \frac{1 \cdot 2 + 3 \cdot (-4)}{1+3} = \frac{2 - 12}{4} = \frac{-10}{4} = -2.5
\]

Thus, the coordinates of point \( K \) are \( \left(0.5, -2.5\right) \).

Would you like any further details or have any questions?

Here are 5 related questions that expand upon this concept:
1. How would the coordinates of point \( K \) change if the ratio were \( 2:3 \) instead of \( 1:3 \)?
2. Can you apply the section formula if the ratio is negative, and what would it represent?
3. How can you determine the length of the segment \( XY \) using the distance formula?
4. What is the midpoint of segment \( XY \) using the midpoint formula?
5. How would the problem change if the coordinates of \( X \) and \( Y \) were swapped?

**Tip:** Always verify if the section formula can be applied directly or if an alternative approach (like using the midpoint formula) would be more suitable for certain ratios.

Learn how to find the coordinates of point K that divides segment XY in a 1:3 ratio using the section formula. This problem involves coordinate geometry with given points X(-1, -4) and Y(5, 2). Suitable for high school students.

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