Answer:
![\Delta d =\frac{y-x}{y}*d_{Hs}\](https://tex.z-dn.net/?f=%5CDelta%20d%20%3D%5Cfrac%7By-x%7D%7By%7D%2Ad_%7BHs%7D%5C)
Step-by-step explanation:
Fist the velocity of high-speed train will be given by:
![V_{Hs} =\frac{z}{x}](https://tex.z-dn.net/?f=V_%7BHs%7D%20%3D%5Cfrac%7Bz%7D%7Bx%7D)
And the velocity of the regular will be given by:
![V_{R} =\frac{z}{y}](https://tex.z-dn.net/?f=V_%7BR%7D%20%3D%5Cfrac%7Bz%7D%7By%7D)
The position equations of each movement will be given by:
![d_{Hs} =\frac{z}{x} t](https://tex.z-dn.net/?f=d_%7BHs%7D%20%3D%5Cfrac%7Bz%7D%7Bx%7D%20t)
![d_{R} =z-\frac{z}{y} t](https://tex.z-dn.net/?f=d_%7BR%7D%20%3Dz-%5Cfrac%7Bz%7D%7By%7D%20t)
We can get the encounter point isolating t in the first equation and reepalcing it in the second one:
First isolating:
![d_{Hs}\frac{x}{z} = t](https://tex.z-dn.net/?f=d_%7BHs%7D%5Cfrac%7Bx%7D%7Bz%7D%20%3D%20t)
Second reeplacing t:
![d_{R} =z-\frac{z}{y}*d_{Hs}\frac{x}{z}](https://tex.z-dn.net/?f=d_%7BR%7D%20%3Dz-%5Cfrac%7Bz%7D%7By%7D%2Ad_%7BHs%7D%5Cfrac%7Bx%7D%7Bz%7D)
![d_{R}=z-\frac{x}{y}*d_{Hs}\](https://tex.z-dn.net/?f=d_%7BR%7D%3Dz-%5Cfrac%7Bx%7D%7By%7D%2Ad_%7BHs%7D%5C)
In the moment of the encounter the high-speed train have gone round
and the regular one ![\frac{x}{y}*d_{Hs}\](https://tex.z-dn.net/?f=%5Cfrac%7Bx%7D%7By%7D%2Ad_%7BHs%7D%5C)
Then the difference between the distance traveled the high-speed train and the regular one is:
![\Delta d =d_{Hs}-\frac{x}{y}*d_{Hs}\](https://tex.z-dn.net/?f=%5CDelta%20d%20%3Dd_%7BHs%7D-%5Cfrac%7Bx%7D%7By%7D%2Ad_%7BHs%7D%5C)
![\Delta d =\frac{y-x}{y}*d_{Hs}\](https://tex.z-dn.net/?f=%5CDelta%20d%20%3D%5Cfrac%7By-x%7D%7By%7D%2Ad_%7BHs%7D%5C)
Because depending on the decimal it can be converted into a fraction. for example,
0.5=1/2
The scale factor is 5
In the first image, you can see that the pre image square it 1 unit long and the image is 5. How do you get 1 to 5, you multiply by 5. So the scale factor is 5.