To find the coordinates of point G, which divides the line segment EF in the ratio \( EG:FG = 3:1 \), we can use the section formula.
The section formula states that if a point \( G \) divides the line segment joining points \( E(x_1, y_1) \) and \( F(x_2, y_2) \) in the ratio \( m:n \), then the coordinates of point \( G(x, y) \) can be found using the formulas:
\[ x = \frac{mx_2 + nx_1}{m+n} \] \[ y = \frac{my_2 + ny_1}{m+n} \]
In this case, we have:
- \( E(-16, -12) \) → \( x_1 = -16, y_1 = -12 \)
- \( F(-4, 0) \) → \( x_2 = -4, y_2 = 0 \)
- Ratio \( EG:FG = 3:1 \) → \( m = 3, n = 1 \)
Now we can plug the values into the formulas.
Calculate the x-coordinate:
\[ x = \frac{3(-4) + 1(-16)}{3+1} = \frac{-12 - 16}{4} = \frac{-28}{4} = -7 \]
Calculate the y-coordinate:
\[ y = \frac{3(0) + 1(-12)}{3+1} = \frac{0 - 12}{4} = \frac{-12}{4} = -3 \]
Thus, the coordinates of point G are: \[ G(-7, -3) \]
So, the coordinates of G are \((-7, -3)\).
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