Question

A,B,and C are collinear, and B is between A and C. The ratio of AB to AC is 2:7.

If A is at (0,-8) and B is at (2,-4), what are the coordinates of point C?

C=( , )

Answer 1

The coordinates of point C are (7 , 6)

Explanation:

* Lets explain how to solve the problem

- If point (x , y) divides a line segment whose endpoints are

`(x_(1),y_(1))` and
`(x_(2),y_(2))` at ratio
`m_(1):m_(2)`

from the first point, then
`x=(x_(1)m_(2)+x_(2)m_(1))/(m_(1)+m_(2))`

and
`y=(y_(1)m_(2)+y_(2)m_(1))/(m_(1)+m_(2))`

* Lets use this rule to solve the problem

- A, B, and C are col-linear, and B is between A and C

∴ A is
`(x_(1),y_(1))`

∴ C is
`(x_(2),y_(2))`

∴ B is (x , y)

- The ratio of AB to AC is 2 : 7

∵ AB : AC = 2 : 7

∴ AB is 2 parts of AC and BC is (7 - 2) = 5 parts of AC

∴ AB : BC = 2 : 5

∴
`m_(1):m_(2)=2:5`

∵ A = (0 , -8)

∴
`(x_(1),y_(1))` = (0 , -8)

∵ B = (2 , -4)

∴ (x , y) = (2 , -4)

∵
`2=((0)(5)+(2)x_(2))/(2+5)`

∴
`2=(0+(2)x_(2))/(7)`

∴
`2=((2)x_(2))/(7)`

- Multiply both sides by 7

∴
`14=(2)x_(2)`

- Divide both sides by 2

∴
`x_(2)=7`

* The x-coordinate of point C is 7

∵
`-4=((-8)(5)+(2)y_(2))/(2+5)`

∴
`-4=(-40+(2)y_(2))/(7)`

- Multiply both sides by 7

∴
`-28=-40+(2)y_(2)`

- Add 40 to both sides

∴
`12=(2)y_(2)`

- Divide both sides by 2

∴
`y_(2)=6`

* The y-coordinate of point C is 6

∴ The coordinates of point C are (7 , 6)