The endpoints of the directed line segment GR are G(-5,10) and R(15,-20) what are the coordinates of point x such that it partitions segment gr into a ratio of 3:2

To find the coordinates of the point \( X(x_1, y_1) \) that partitions the line segment \( GR \) into a ratio of 3:2, we can use the section formula for a directed line segment.

Given:
- \( G(x_1, y_1) = (-5, 10) \)
- \( R(x_2, y_2) = (15, -20) \)
- Ratio \( m:n = 3:2 \)

The section formula states:
\[
X\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\]
Plugging in the values:
\[
X\left( \frac{3(15) + 2(-5)}{3+2}, \frac{3(-20) + 2(10)}{3+2} \right)
\]

Calculating the coordinates:

- \( x_1 = \frac{3(15) + 2(-5)}{5} = \frac{45 - 10}{5} = \frac{35}{5} = 7 \)
- \( y_1 = \frac{3(-20) + 2(10)}{5} = \frac{-60 + 20}{5} = \frac{-40}{5} = -8 \)

Therefore, the coordinates of point \( X \) are \( (7, -8) \).

Would you like more details or have any questions?

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Here are 5 related questions you might find interesting:
1. How do you apply the section formula to a line segment in 3D?
2. What is the midpoint formula, and how is it derived?
3. How can you find the ratio if the coordinates of the points on the segment are known?
4. How does the section formula change if the point divides the segment externally?
5. What is the significance of directed line segments in vector geometry?

**Tip:** When working with ratios, remember that the section formula can be used for both internal and external division of line segments.

Discover how to find the coordinates of point X that divides the line segment GR, with endpoints G(-5, 10) and R(15, -20), into a ratio of 3:2. This mathematical problem involves using the section formula for internal division of line segments.

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