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To solve this problem, we are tasked with finding the coordinates of point \( B \) that divides the line segment \( \overline{AC} \) in the ratio \( 2:1 \).

### Given:
- Point \( A(2, -8) \)
- Point \( C(-4, 7) \)
- The ratio \( AB : BC = 2:1 \)

To find the coordinates of point \( B \), we can use the section formula, which determines the coordinates of a point that divides a line segment in a given ratio.

### Section Formula:
If a point \( B(x, y) \) divides a line joining two points \( A(x_1, y_1) \) and \( C(x_2, y_2) \) in the ratio \( m:n \), the coordinates of \( B \) are given by:

\[
B\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\]

### Applying the formula:
Here, the ratio \( m:n = 2:1 \), so \( m = 2 \) and \( n = 1 \), and the points \( A(2, -8) \) and \( C(-4, 7) \) have the coordinates:

- \( A(2, -8) \) → \( x_1 = 2 \), \( y_1 = -8 \)
- \( C(-4, 7) \) → \( x_2 = -4 \), \( y_2 = 7 \)

Now we apply the section formula:

1. \( x_B = \frac{(2)(-4) + (1)(2)}{2+1} = \frac{-8 + 2}{3} = \frac{-6}{3} = -2 \)
2. \( y_B = \frac{(2)(7) + (1)(-8)}{2+1} = \frac{14 - 8}{3} = \frac{6}{3} = 2 \)

Thus, the coordinates of point \( B \) are \( B(-2, 2) \).

### Final Answer:
\[
B = (-2, 2)
\]

Would you like more details on this solution or have any questions?

Here are some related questions:
1. How do we generalize the section formula for different ratios?
2. What would happen if the ratio were reversed (i.e., \( AB:BC = 1:2 \))?
3. How can we verify that point \( B(-2, 2) \) divides \( AC \) correctly in the ratio \( 2:1 \)?
4. What is the equation of the line passing through points \( A \) and \( C \)?
5. Can we use the midpoint formula to check if point \( B \) is the midpoint for a different ratio?

**Tip:** The section formula works for both internal and external division, depending on how the ratio is applied.

El punto A está en (2, -8) y el punto C está en (-4, 7). Determina las coordenadas del punto B en AC, tales que la razón de AB a BC sea 2:1.

Learn how to determine the coordinates of point B on line AC, where A is at (2, -8) and C is at (-4, 7), given the ratio of AB to BC as 2:1. Using the section formula, this problem involves basic geometry and ratio concepts.

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