Point R that lies along the directed segment from J (10, -5) to K (-2, -3) and partitions the segment in a 2 to 7 ratio.

We can solve this problem using the section formula, which helps us find a point that divides a line segment into a given ratio. The formula for a point \( R(x, y) \) dividing a line segment \( JK \) in the ratio \( m:n \) is:

\[
R(x, y) = \left( \frac{m \cdot x_2 + n \cdot x_1}{m + n}, \frac{m \cdot y_2 + n \cdot y_1}{m + n} \right)
\]

Here, the coordinates of point \( J \) are \( (x_1, y_1) = (10, -5) \), and the coordinates of point \( K \) are \( (x_2, y_2) = (-2, -3) \). The ratio \( m:n \) is \( 2:7 \).

Now, applying the section formula:

\[
R(x) = \frac{2 \cdot (-2) + 7 \cdot (10)}{2 + 7} = \frac{-4 + 70}{9} = \frac{66}{9} = 7.33
\]

\[
R(y) = \frac{2 \cdot (-3) + 7 \cdot (-5)}{2 + 7} = \frac{-6 - 35}{9} = \frac{-41}{9} \approx -4.56
\]

Thus, the coordinates of point \( R \) are approximately \( R(7.33, -4.56) \).

Would you like further details on this, or do you have any questions?

Here are five related questions:
1. How does the section formula work for internal division?
2. How would the coordinates change if the ratio was \( 3:5 \)?
3. What if the point \( R \) divided the line segment externally?
4. Can the section formula be applied to 3D coordinates?
5. How can this method be used in vector representation?

**Tip:** The section formula is versatile and can be used to find both internal and external divisions of a line segment.

Learn how to find the point R that divides a line segment between points J (10, -5) and K (-2, -3) in a 2:7 ratio using the section formula. This detailed step-by-step guide covers key concepts in coordinate geometry and ratio division, suitable for students in Grades 9-12.

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