Question

de has endpoints d(4, 11) and e(8, 9). point f divides de into two parts with lengths in a ratio of 3:1. what are the two possible locations of f? (7.5, 8) (3, 5) (7, 9.5) (5.5, 8) (5, 10.5) (6, 10)

Explanation:

Step1: Recall section - formula

If a point $F(x,y)$ divides the line - segment joining $D(x_1,y_1)$ and $E(x_2,y_2)$ in the ratio $m:n$, then the coordinates of $F$ are given by $x=\frac{mx_2+nx_1}{m + n}$ and $y=\frac{my_2+ny_1}{m + n}$. Here, $D(4,11)$, $E(8,9)$ and the ratio $m:n = 3:1$.

Step2: Calculate the x - coordinate of $F$

$x=\frac{3\times8 + 1\times4}{3 + 1}=\frac{24 + 4}{4}=\frac{28}{4}=7$.

Step3: Calculate the y - coordinate of $F$

$y=\frac{3\times9+1\times11}{3 + 1}=\frac{27 + 11}{4}=\frac{38}{4}=9.5$.

Step4: Consider the other case

If the ratio is considered as $m:n=1:3$, then $x=\frac{1\times8+3\times4}{1 + 3}=\frac{8 + 12}{4}=\frac{20}{4}=5$ and $y=\frac{1\times9+3\times11}{1 + 3}=\frac{9+33}{4}=\frac{42}{4}=10.5$.

Answer:

$(7,9.5),(5,10.5)$