Question

the planners at the local department at transportation office want to build a highway with two rest stops between the towns. the rest stops will divide the highway into a three equal parts. what are the coordinates of the points at which the rest stop should be build?

Answer 1

The coordinates of the rest stops are (-1 , -1/3) and (1 , 4/3)

Explanation:

* Lets explain how to solve the problem

- The rest spots between the two towns will divide the highway

into three equal parts

- Each part of the highway will be 1/3 the distance between the two

towns, so the first rest stop will divide the highway at ratio 1 : 2 and

the second rest stop will be the midway between the first rest stop

and the second town

- The location of the first town is (-3 , -2) and the location of the

second town is (3 , 3)

- Assume that the location of the first rest stop is (x , y)

- If point (x , y) divide a 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))`

- Let (-3 , -2) is
`(x_(1),y_(1))` and (3 , 3) is
`(x_(2),y_(2))`

and
`(m_(1):m_(2))` = 1 : 2

∴
`x=((-3)(2)+(3)(1))/(1+2)=(-6+3)/(3)=(-3)/(3)=-1`

∴
`x=((-2)(2)+(3)(1))/(1+2)=(-4+3)/(3)=(-1)/(3)`

∴ The location of the first rest stop is (-1 , -1/3)

∵ The second rest stop is the midway between the first rest stop

and the second town

∴ The second stop location M is the mid-point between (x , y) and

`(x_(2),y_(2))`

- The mid-point M is
`M=((x+x_(2))/(2),(y+y_(2))/(2))`

∴
`M=((-1+3)/(2),((-1)/(3)+3)/(2))=((2)/(2),(8)/(6))=(1,(4)/(3))`

∴ The location of the second rest stop is (1 , 4/3)

* The coordinates of the rest stops are (-1 , -1/3) and (1 , 4/3)