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!