Answer:
The coordinates of the rest stops are (-1 , -1/3) and (1 , 4/3)
Step-by-step 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})](https://tex.z-dn.net/?f=%28x_%7B1%7D%2Cy_%7B1%7D%29)
and
at ratio
, then
, and
![y=\frac{y_{1}m_{2}+y_{2}m_{1}}{m_{1}+m_{2}}](https://tex.z-dn.net/?f=y%3D%5Cfrac%7By_%7B1%7Dm_%7B2%7D%2By_%7B2%7Dm_%7B1%7D%7D%7Bm_%7B1%7D%2Bm_%7B2%7D%7D)
- Let (-3 , -2) is
and (3 , 3) is ![(x_{2},y_{2})](https://tex.z-dn.net/?f=%28x_%7B2%7D%2Cy_%7B2%7D%29)
and
= 1 : 2
∴ ![x=\frac{(-3)(2)+(3)(1)}{1+2}=\frac{-6+3}{3}=\frac{-3}{3}=-1](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B%28-3%29%282%29%2B%283%29%281%29%7D%7B1%2B2%7D%3D%5Cfrac%7B-6%2B3%7D%7B3%7D%3D%5Cfrac%7B-3%7D%7B3%7D%3D-1)
∴ ![x=\frac{(-2)(2)+(3)(1)}{1+2}=\frac{-4+3}{3}=\frac{-1}{3}](https://tex.z-dn.net/?f=x%3D%5Cfrac%7B%28-2%29%282%29%2B%283%29%281%29%7D%7B1%2B2%7D%3D%5Cfrac%7B-4%2B3%7D%7B3%7D%3D%5Cfrac%7B-1%7D%7B3%7D)
∴ 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})](https://tex.z-dn.net/?f=%28x_%7B2%7D%2Cy_%7B2%7D%29)
- The mid-point M is
∴ ![M=(\frac{-1+3}{2},\frac{\frac{-1}{3}+3}{2})=(\frac{2}{2},\frac{8}{6})=(1,\frac{4}{3})](https://tex.z-dn.net/?f=M%3D%28%5Cfrac%7B-1%2B3%7D%7B2%7D%2C%5Cfrac%7B%5Cfrac%7B-1%7D%7B3%7D%2B3%7D%7B2%7D%29%3D%28%5Cfrac%7B2%7D%7B2%7D%2C%5Cfrac%7B8%7D%7B6%7D%29%3D%281%2C%5Cfrac%7B4%7D%7B3%7D%29)
∴ 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)