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.