To find the coordinates of a point that is a certain fraction of the way from point $A(x_1, y_1)$ to point $B(x_2, y_2)$, we can use the section formula.

Given:

• Point $A(-4, -6)$
• Point $B(11, 4)$
• The point is three-tenths of the way from $A$ to $B$

We use the formula for the point dividing the line segment in the ratio $m:n$:

$P\left( \frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n} \right)$

Here, $m = 3$ (the portion of the way), and $n = 7$ (the remaining part to make the full length).

Now, substituting the values:

$P_x = \frac{3 \cdot 11 + 7 \cdot (-4)}{3 + 7} = \frac{33 - 28}{10} = \frac{5}{10} = 0.5$ $P_y = \frac{3 \cdot 4 + 7 \cdot (-6)}{3 + 7} = \frac{12 - 42}{10} = \frac{-30}{10} = -3$

Thus, the coordinates of the point three-tenths of the way from $A$ to $B$ are:

$P(0.5, -3)$

Would you like more details or have any questions?

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Here are 5 related questions for deeper understanding:

1. How do you calculate the midpoint of a line segment between two points?
2. Can you find the coordinates of the point two-thirds of the way from $A$ to $B$?
3. How would the coordinates change if the point were seven-tenths of the way from $A$ to $B$?
4. How does the section formula apply to 3D coordinates?
5. What is the geometric meaning of dividing a line segment in a certain ratio?

Tip: The section formula can also be used to divide a segment externally, not just internally, depending on the ratio!