Given A(−1,2) and B(3,6) , find the coordinates of point X on AB¯¯¯¯¯¯¯¯ such that the ratio of AX to XB is 3:5.

To find the coordinates of point \( X \) on line segment \( \overline{AB} \) such that the ratio of \( AX \) to \( XB \) is 3:5, we can use the section formula.

The section formula states that if a point divides a line segment in a given ratio, the coordinates of that point can be found using:

\[
X = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right)
\]

Where:
- \( (x_1, y_1) \) and \( (x_2, y_2) \) are the coordinates of points \( A \) and \( B \),
- \( m:n \) is the given ratio of \( AX \) to \( XB \).

In this case:
- \( A(-1, 2) \), \( B(3, 6) \),
- \( m:n = 3:5 \).

Now, applying the section formula:

\[
X = \left( \frac{5(3) + 3(-1)}{3+5}, \frac{5(6) + 3(2)}{3+5} \right)
\]

First, calculate the \( x \)-coordinate:

\[
x = \frac{5(3) + 3(-1)}{8} = \frac{15 - 3}{8} = \frac{12}{8} = 1.5
\]

Next, calculate the \( y \)-coordinate:

\[
y = \frac{5(6) + 3(2)}{8} = \frac{30 + 6}{8} = \frac{36}{8} = 4.5
\]

Thus, the coordinates of point \( X \) are \( (1.5, 4.5) \).

### Summary:
The coordinates of point \( X \) are \( \boxed{(1.5, 4.5)} \).

Would you like any further details or explanations?

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Here are some related questions for practice:
1. Find the coordinates of point \( X \) dividing the line segment in the ratio 2:3.
2. How would the section formula change if \( AX \) to \( XB \) was 5:3?
3. What are the coordinates of the midpoint of \( \overline{AB} \)?
4. How do you apply the section formula when the ratio involves negative numbers?
5. Can you derive the section formula from basic geometry?

**Tip:** Always label your points clearly in geometry problems, as it helps to avoid confusion when applying formulas.

Given A(−1,2) and B(3,6), find the coordinates of point X on AB¯¯¯¯¯¯¯¯ such that the ratio of AX to XB is 3:5.

Learn how to find the coordinates of a point on a line segment using the section formula with a ratio of 3:5. This problem involves key concepts in coordinate geometry, including the section formula and ratio proportion, suitable for students in Grades 9-10. Given points A(-1, 2) and B(3, 6), find point X such that AX:XB = 3:5.

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