Question

Determine the coordinates of point  G , that divides the line  AB  into two segments with lengths in a ratio of 3:5.(1 point)G is located at (,).
point (0,0) (0,8)

Answer 1

To find the coordinates of the point \( G \) that divides the line segment \( AB \) into two segments in the ratio \( 3:5 \), we can use the section formula.

Given points:
- \( A(0, 0) \)
- \( B(0, 8) \)

Let \( G(x, y) \) divide the line segment \( AB \) in the ratio \( m:n = 3:5 \).

The coordinates of point \( G \) can be calculated using the section formula:

\[
G\left( \frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n} \right)
\]

Substituting the values:
- \( m = 3 \)
- \( n = 5 \)
- \( x_1 = 0 \) (from point A)
- \( y_1 = 0 \) (from point A)
- \( x_2 = 0 \) (from point B)
- \( y_2 = 8 \) (from point B)

Now substitute these into the formula:

1. For \( x \)-coordinate:
\[
x_G = \frac{3 \cdot 0 + 5 \cdot 0}{3 + 5} = \frac{0 + 0}{8} = 0
\]

2. For \( y \)-coordinate:
\[
y_G = \frac{3 \cdot 8 + 5 \cdot 0}{3 + 5} = \frac{24 + 0}{8} = \frac{24}{8} = 3
\]

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

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