Question

Given the points C(-1,-5) and D(-7,11) find the coordinates of point E on CD such that the ratio of CE to ED is 5:3.

Answer 1

Answer:

The coordinates of point E on CD are $\left(-\frac{19}{4},5 \right)$.

Step-by-step explanation:

Let be $C = (-1,-5)$, $D = (-7,11)$ and $\frac{CE}{ED} = \frac{5}{3}$. The given ratio can be translated vectorially into this:

$\overrightarrow {CE} = \frac{5}{3} \cdot \overrightarrow{ED}$

And let consider that each point is a vector with respect to origin:

$\vec C = (-1, -5)$ and $\vec {D} = (-7,11)$

Then,

$\vec E -\vec C = \frac{5}{3}\cdot (\vec D -\vec E)$

$\vec E +\frac{5}{3}\cdot \vec E = \frac{5}{3}\cdot \vec D + \vec C$

$\frac{8}{3}\cdot \vec E = \frac{5}{3}\cdot \vec D + \vec C$

$\vec E = \frac{5}{8}\cdot \vec D + \frac{3}{8}\cdot \vec C$

$\vec E = \frac{5}{8}\cdot (-7,11)+\frac{3}{8}\cdot (-1,-5)$

$\vec E = \left(-\frac{35}{8},\frac{55}{8} \right)+\left(-\frac{3}{8},-\frac{15}{8} \right)$

$\vec E = \left(-\frac{35}{8}-\frac{3}{8},\frac{55}{8}-\frac{15}{8} \right)$

$\vec{E} = \left(-\frac{19}{4}, 5\right)$

The coordinates of point E on CD are $\left(-\frac{19}{4},5 \right)$.