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Determine the coordinates of point G , that divides the line AB into two segments with lengths in a ratio of 3:5.

To determine the coordinates of point \( G \), which divides the line segment \( AB \) into two segments with a ratio of 3:5, we can apply the **section formula**. Since the points lie on the \( x \)-axis, only the \( x \)-coordinates will change, while the \( y \)-coordinate remains 0.

### Given:
- Coordinates of \( A = (0, 0) \)
- Coordinates of \( B = (8, 0) \)
- Ratio \( AG : GB = 3:5 \)

The section formula for a point dividing a line segment in the ratio \( m:n \) is given by:

\[
x_G = \frac{m \cdot x_B + n \cdot x_A}{m + n}
\]
\[
y_G = \frac{m \cdot y_B + n \cdot y_A}{m + n}
\]

### Applying the values:

- \( m = 3 \), \( n = 5 \)
- \( x_A = 0 \), \( x_B = 8 \)
- \( y_A = 0 \), \( y_B = 0 \)

Substituting these into the formulas:

\[
x_G = \frac{3 \cdot 8 + 5 \cdot 0}{3 + 5} = \frac{24 + 0}{8} = \frac{24}{8} = 3
\]
\[
y_G = \frac{3 \cdot 0 + 5 \cdot 0}{3 + 5} = 0
\]

Thus, the coordinates of point \( G \) are \( (3, 0) \).

Would you like further explanation on the section formula or anything else?

### Related Questions:
1. What would the coordinates of \( G \) be if the ratio were 2:3?
2. How do you apply the section formula to three-dimensional coordinates?
3. What if the points \( A \) and \( B \) weren't on the \( x \)-axis—how would the formula change?
4. How would you find the midpoint of \( AB \) without using ratios?
5. Can the section formula be applied to vectors? If so, how?

### Tip:
When using the section formula, always double-check that you assign the correct values for the ratio \( m:n \), as reversing them will give the wrong division point!

Determine the coordinates of point G, that divides the line AB into two segments with lengths in a ratio of 3:5.

Learn how to find the coordinates of point G, which divides the line segment AB into two parts in the ratio 3:5 using the section formula. This problem involves applying basic coordinate geometry and is suitable for Grades 8-10.

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