Question

A, B and C are collinear and B is between a and C. The ratio of AB to AC is 3:2. If A is at (4,8) and B is at (7,2) what are the coordinates of point C

Answer 1

The coordinates of point C are (9 , -2)

Explanation:

* Lets explain how to solve the problem

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

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

from the first point
`(x_(1),y_(1))`, 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 solve the problem

- A, B and C are collinear and B is between A and C

∴ Point A is
`(x_(1),y_(1))`

∴ Point C is
`(x_(2),y_(2))`

∴ Point B is (x , y)

- The ratio of AB to BC is 3 : 2

∴
`m_(1):m_(2)` = 3 : 2

* Lets use the rule above to find the coordinates of point C

∵ A = (4 , 8) and B = (7 , 2)

∵
`m_(1):m_(2)` = 3 : 2

∴
`7=((4)(2)+x_(2)(3))/(5)`

- Multiply each side by 5

∴
`35=8+3x_(2)`

- Subtract 8 from both sides

∴
`27=3x_(2)`

- Divide both sides by 3

∴
`9=x_(2)`

∴ The x-coordinate of point C is 9

∴
`2=((8)(2)+y_(2)(3))/(5)`

- Multiply each side by 5

∴
`10=16+3y_(2)`

- Subtract 16 from both sides

∴
`-6=3y_(2)`

- Divide both sides by 3

∴
`-2=y_(2)`

∴ The y-coordinate of point C is -2

* The coordinates of point C are (9 , -2)