The coordinates of the endpoints of EF are E(1,15) and F(15,8) Point G is on EF and divides it such that EG:FG is 5:2

Question

what are the coordinates of G
write your answer as integers or decimals

Answer 1

To find the coordinates of point G that divides the line segment EF in the ratio EG:FG = 5:2, we can use the section formula.

The section formula states that if a point G divides a line segment joining two points E(x1, y1) and F(x2, y2) in the ratio m:n, then the coordinates of point G (x, y) can be calculated as follows:

\[ x = \frac{mx_2 + nx_1}{m+n} \] \[ y = \frac{my_2 + ny_1}{m+n} \]

Given:

• E(1, 15) \( \Rightarrow (x_1, y_1) = (1, 15) \)
• F(15, 8) \( \Rightarrow (x_2, y_2) = (15, 8) \)
• Ratio \( EG:FG = 5:2 \) \( \Rightarrow m = 5 \) and \( n = 2 \)

Substituting the values into the formulas:

1. Calculate the x-coordinate of G: \[ x = \frac{5 \cdot 15 + 2 \cdot 1}{5 + 2} = \frac{75 + 2}{7} = \frac{77}{7} = 11 \]

2. Calculate the y-coordinate of G: \[ y = \frac{5 \cdot 8 + 2 \cdot 15}{5 + 2} = \frac{40 + 30}{7} = \frac{70}{7} = 10 \]

Thus, the coordinates of point G are:

\[ G(11, 10) \]

So, the final answer is: \[ \boxed{(11, 10)} \]