Question

2. Given a directed line segment with endpoints A(3, 2) and B(6, 11), what is the point that

divides AB two-thirds from A to B?

Answer 1

The point of division is (5 , 8)

Explanation:

* Lets explain how to solve the problem

- If point (x , y) divides the line whose endpoints are
`(x_(1),y_(1))`

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

- The directed line segment with endpoints A (3 , 2) and B (6 , 11)

- There is a point divides AB two-thirds from A to B

∵ The coordinates of the endpoints of the directed line segments

are A = (3 , 2) and B = (6 , 11)

∴
`(x_(1),y_(1))` is (3 , 2)

∴
`(x_(2),y_(2))` is (6 , 11)

∵ Point (x , y) divides AB two-thirds from A to B

- That means the distance from A to the point (x , y) is 2/3 from

the distance of the line AB, and the distance from the point (x , y)

to point B is 1/3 from the distance of the line AB

∴
`m_(1):m_(2)` = 2 : 1

∵
`x=((3)(1)+(6)(2))/(2+1)=(3+12)/(3)=(15)/(3)=5`

∴ The x-coordinate of the point of division is 5

∵
`y=((2)(1)+(11)(2))/(2+1)=(2+22)/(3)=(24)/(3)=8`

∴ The y-coordinate of the point of division is 8

∴ The point of division is (5 , 8)