Question

find the coordinates of point e on (overline{df}) such that the ratio of de to df is 1:4. d(4, - 7) f(2,10) show your work here

Explanation:

Step1: Recall the section - formula

If a point $E(x,y)$ divides the line - segment joining $D(x_1,y_1)$ and $F(x_2,y_2)$ in the ratio $m:n$, then the coordinates of $E$ are given by $x=\frac{mx_2+nx_1}{m + n}$ and $y=\frac{my_2+ny_1}{m + n}$. Here, $D(4,-7)$, $F(2,10)$, and $m = 1$, $n = 3$ (since the ratio of $DE$ to $DF$ is $1:4$, so the ratio of $DE$ to $EF$ is $1:3$).

Step2: Calculate the $x$ - coordinate of $E$

Substitute $x_1 = 4$, $x_2 = 2$, $m = 1$, and $n = 3$ into the formula for $x$.
$x=\frac{1\times2+3\times4}{1 + 3}=\frac{2 + 12}{4}=\frac{14}{4}=\frac{7}{2}=3.5$.

Step3: Calculate the $y$ - coordinate of $E$

Substitute $y_1=-7$, $y_2 = 10$, $m = 1$, and $n = 3$ into the formula for $y$.
$y=\frac{1\times10+3\times(-7)}{1 + 3}=\frac{10-21}{4}=\frac{-11}{4}=-2.75$.

Answer:

The coordinates of point $E$ are $(3.5,-2.75)$