Question

The points $E(6,4)$ and $F(14,12)$ lie in the standard $(x,y)$ coordinate plane shown below. Point $D$ lies on $\overline{EF}$ between $E$ and $F$ such that the length of $\overline{EF}$ is $4$ times the length of $\overline{DE}$. What are the coordinates of $D$?

• A. $(7,5)$
• B. $(8,6)$
• C. $(8,8)$
• D. $(10,8)$
• E. $(12,10)$

Answer 1

The correct option is B $(8,6)$

Using the section formula, if a point $(x,y)$ divides the line joining the points $(x_1,y_1)$ and $(x_2,y_2)$ in the ratio $m:n$, then

$(x,y)=(\frac{m{x}_2+n{x}_1}{m+n},\frac{m{y}_2+n{y}_1}{m+n})$

From the given condition we find that $DE:EF=1:3$. Let the coordinate of D be $(x,y)$. Then
$D(x,y)=(\frac{1\left(14\right)+3\left(6\right)}{4},\frac{1\left(12\right)+3\left(4\right)}{4})=(\frac{32}{4},\frac{24}{4})=(8,6)$