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.

((Yes im this stupid, please help-))

Answer 1

The coordinates of point E on CD are
`\left(-(19)/(4),5 \right)`.

Explanation:

Let be
`C = (-1,-5)`,
`D = (-7,11)` and
`(CE)/(ED) = (5)/(3)`. The given ratio can be translated vectorially into this:

`\overrightarrow {CE} = (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 = (5)/(3)\cdot (\vec D -\vec E)`

`\vec E +(5)/(3)\cdot \vec E = (5)/(3)\cdot \vec D + \vec C`

`(8)/(3)\cdot \vec E = (5)/(3)\cdot \vec D + \vec C`

`\vec E = (5)/(8)\cdot \vec D + (3)/(8)\cdot \vec C`

`\vec E = (5)/(8)\cdot (-7,11)+(3)/(8)\cdot (-1,-5)`

`\vec E = \left(-(35)/(8),(55)/(8) \right)+\left(-(3)/(8),-(15)/(8) \right)`

`\vec E = \left(-(35)/(8)-(3)/(8),(55)/(8)-(15)/(8) \right)`

`\vec{E} = \left(-(19)/(4), 5\right)`

The coordinates of point E on CD are
`\left(-(19)/(4),5 \right)`.