We are given two points, $A(-4, 7)$ and $B(5, -2)$, and the segment joining these points is divided in a ratio where one part is five times as long as the other. This means the ratio of division is $5:1$.

To find the point of division, we can use the section formula, which provides the coordinates of a point dividing a line segment in the ratio $m:n$.

Section Formula:

If a point $P(x_1, y_1)$ divides the line joining $A(x_1, y_1)$ and $B(x_2, y_2)$ in the ratio $m:n$, then the coordinates of the point of division $P(x, y)$ are given by: $x = \frac{mx_2 + nx_1}{m + n}, \quad y = \frac{my_2 + ny_1}{m + n}$

Given:

• $A(-4, 7)$ and $B(5, -2)$
• Ratio of division = $5:1$ (i.e., $m = 5$ and $n = 1$)

Step-by-step Calculation:

1. For the x-coordinate: $x = \frac{5(5) + 1(-4)}{5 + 1} = \frac{25 - 4}{6} = \frac{21}{6} = 3.5$

2. For the y-coordinate: $y = \frac{5(-2) + 1(7)}{5 + 1} = \frac{-10 + 7}{6} = \frac{-3}{6} = -0.5$

Thus, the point dividing the line in the ratio $5:1$ is $\left( 3.5, -0.5 \right)$.

Final Answer:

The point dividing the segment joining $(-4, 7)$ and $(5, -2)$ in the ratio 5:1 is $\left( 3.5, -0.5 \right)$.

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

Here are 5 related questions to expand on this:

1. How does the section formula apply when the ratio is reversed (e.g., $1:5$)?
2. What happens when a segment is divided externally instead of internally?
3. Can you use the section formula for three-dimensional points?
4. What is the geometric significance of the ratio in dividing a line segment?
5. How would the midpoint of the segment be calculated?

Tip: Always double-check your ratio and point placements when using the section formula for accuracy!