Question

ef has endpoints e(5, 14) and f(10, -6). point g divides ef into two parts with lengths in a ratio of 4:1. what are the two possible locations of g?

Explanation:

Step1: Recall the section - formula

If a point $G(x,y)$ divides the line - segment joining $E(x_1,y_1)$ and $F(x_2,y_2)$ in the ratio $m:n$, then $x=\frac{mx_2+nx_1}{m + n}$ and $y=\frac{my_2+ny_1}{m + n}$. Here, $(x_1,y_1)=(5,14)$ and $(x_2,y_2)=(10, - 6)$ and $m:n = 4:1$.

Step2: Calculate the x - coordinate of $G$

$x=\frac{4\times10+1\times5}{4 + 1}=\frac{40 + 5}{5}=\frac{45}{5}=9$.

Step3: Calculate the y - coordinate of $G$

$y=\frac{4\times(-6)+1\times14}{4 + 1}=\frac{-24 + 14}{5}=\frac{-10}{5}=-2$.

Step4: Consider the other case (when the ratio is taken in the reverse order for the internal division conceptually)

The formula still holds. The point $G$ can also be found using the same section - formula. There is no other non - collinear case for a single line - segment division. The only correct location of $G$ when dividing $\overline{EF}$ in the ratio $4:1$ is $(9,-2)$.

Answer:

$(9,-2)$