The segment joining (-4, 7), (5,-2) is divided into two, one of which is five times as long as the other. Find the points of division

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!

The segment joining (-4, 7), (5, -2) is divided into two, one of which is five times as long as the other. Find the points of division.

Learn how to find the point dividing the line segment joining (-4, 7) and (5, -2) in a 5:1 ratio. This solution uses the section formula to calculate the coordinates of the division point, making it perfect for high school students in Grades 9-12 studying coordinate geometry.

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