Question

The endpoints of segment RS are R(-4, 3) and S(8, -5). Complete each statement using a fraction.

a. (-1, 1) is the point ___ of the way from R to S.
b. (5, -3) is the point ___ of the way from R to S.

Answer 1

a. (-1 , 1) is the point at 1 : 3 of the way from R to S

b. (5 , -3) is the point at 3 : 1 of the way from R to S

Explanation:

* Lets explain how to solve the problem

- If point (x , y) divide 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, 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.

∵ RS is a line segment whose end points are R (-4 , 3) and S (8 , -5)

∵ Point (-1 , 1) divides it at ratio
`m_(1):m_(2)` from R

- By using the rule above

∴
`-1=(-4m_(2)+8m_(1))/(m_(1)+m_(2))`

- By using cross multiplication

∴
`(-1)m_(1)+(-1)m_(2)` =
`-4m_(2)+8m_(1)`

- By collecting
`m_(1)` in one side and
`m_(2)` in the

other side

∴
`3m_(2)=9m_(1)`

- Divide both sides by 3

∴
`m_(2)=3m_(1)`

∴
`m_(1)=(1)/(3)m_(2)`

∴ (-1 , 1) is the point at 1 : 3 of the way from R to S

b.

∵ RS is a line segment whose end points are R (-4 , 3) and S (8 , -5)

∵ Point (5 , -3) divides it at ratio
`m_(1):m_(2)` from R

- By using the rule above

∴
`5=(-4m_(2)+8m_(1))/(m_(1)+m_(2))`

- By using cross multiplication

∴
`5m_(1)+5m_(2)` =
`-4m_(2)+8m_(1)`

- By collecting
`m_(1)` in one side and
`m_(2)` in the

other side

∴
`9m_(2)=3m_(1)`

- Divide both sides by 3

∴
`3m_(2)=m_(1)`

∴
`(m_(1))/(m_(2))=3`

∴ (5 , -3) is the point at 3 : 1 of the way from R to S