https://file.aiquickdraw.com/mathbot/file/1725852135547h6gx3gjg.jpg

The problem asks to find the coordinates of point \( S \) such that the ratio of \( AS \) to \( ST \) is 3:1, where \( A \) and \( T \) are endpoints of a line segment.

Let's break it down step by step:

1. **Identify the coordinates of \( A \) and \( T \)**:
   - From the image, \( A \) appears to have coordinates \( (2, 3) \).
   - \( T \) appears to have coordinates \( (10, 7) \).

2. **Use the section formula**:
   The section formula in geometry helps to find the coordinates of a point dividing a line segment in a given ratio. For two points \( A(x_1, y_1) \) and \( T(x_2, y_2) \), and a point \( S \) dividing the line segment in the ratio \( m:n \), the coordinates of \( S \) are given by:

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

   Here, \( m:n = 3:1 \), so \( m = 3 \) and \( n = 1 \).

3. **Apply the section formula**:
   \[
   x_S = \frac{3 \cdot 10 + 1 \cdot 2}{3+1} = \frac{30 + 2}{4} = \frac{32}{4} = 8
   \]
   \[
   y_S = \frac{3 \cdot 7 + 1 \cdot 3}{3+1} = \frac{21 + 3}{4} = \frac{24}{4} = 6
   \]

4. **Conclusion**:
   The coordinates of point \( S \) are \( (8, 6) \).

Would you like further details or have any questions about this solution?

### Related questions:
1. How does the section formula work in a 3D coordinate system?
2. How can you use the section formula when the ratio involves fractions?
3. What are alternative methods to find the coordinates of a dividing point without using the section formula?
4. How would the solution change if the ratio was reversed to 1:3?
5. Can this method be extended to finding points in a curve or non-linear segment?

### Tip:
Always double-check the coordinates from the graph, as reading errors can lead to incorrect results in geometric problems.

Given a line segment from A to T, find the coordinates of point S such that the ratio of AS to ST is 3:1, using the section formula.

Discover how to determine the coordinates of point S that divides the line segment from A to T in the ratio 3:1 using the section formula. This problem is ideal for students in grades 8 to 10, exploring concepts in geometry and coordinate geometry.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation