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

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 $(x_{1},y_{1})$

 and $(x_{2},y_{2})$ at ratio $(m_{1}:m_{2})$ , then

 $x=\frac{x_{1}m_{2}+x_{2}m_{1}}{m_{1}+m_{2}}$ , and

 $y=\frac{y_{1}m_{2}+y_{2}m_{1}}{m_{1}+m_{2}}$

- Let (-3 , -2) is $(x_{1},y_{1})$ and (3 , 3) is $(x_{2},y_{2})$

 and $(m_{1}:m_{2})$ = 1 : 2

∴ $x=\frac{(-3)(2)+(3)(1)}{1+2}=\frac{-6+3}{3}=\frac{-3}{3}=-1$

∴ $x=\frac{(-2)(2)+(3)(1)}{1+2}=\frac{-4+3}{3}=\frac{-1}{3}$

∴ 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 $M=(\frac{x+x_{2}}{2},\frac{y+y_{2}}{2})$

∴ $M=(\frac{-1+3}{2},\frac{\frac{-1}{3}+3}{2})=(\frac{2}{2},\frac{8}{6})=(1,\frac{4}{3})$

∴ 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)