To find the point that divides the line segment from point A to point B in a 1:4 ratio, we will use the section formula. The coordinates of A and B are as follows:

• $A = (-4, -4)$
• $B = (6, 6)$

Section Formula:

If a point divides the line segment joining two points $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m:n$, the coordinates of the point are given by:

$P = \left( \frac{m x_2 + n x_1}{m+n}, \frac{m y_2 + n y_1}{m+n} \right)$

In this case:

• $m = 1$
• $n = 4$
• $x_1 = -4$, $y_1 = -4$ (coordinates of A)
• $x_2 = 6$, $y_2 = 6$ (coordinates of B)

Now, applying the formula:

$P_x = \frac{1 \cdot 6 + 4 \cdot (-4)}{1+4} = \frac{6 - 16}{5} = \frac{-10}{5} = -2$

$P_y = \frac{1 \cdot 6 + 4 \cdot (-4)}{1+4} = \frac{6 - 16}{5} = \frac{-10}{5} = -2$

Thus, the point that divides the line segment in the 1:4 ratio is $(-2, -2)$.

Would you like more details or have any questions?

Related Questions:

1. How would the coordinates change if the ratio was 2:3 instead of 1:4?
2. What is the distance between points A and B?
3. Can you find the midpoint of the line segment AB?
4. How would you generalize the section formula for a 3D coordinate system?
5. What would be the coordinates if the ratio was reversed (4:1)?

Tip:

Always ensure you assign the correct values for the ratio and the coordinates of the points when applying the section formula!