# Greatest and smallest possible distance between Y and Z

The distance from point X to point Y is 20 miles, and the distance from point X to point Z is 12 miles. If d is the distance, in miles, between points Y and Z, then what is the possible range of possible values ​​for d?

I find, as straight distance is the shortest distance, $20-12= 8$ is the smallest possible distance of Y and Z. But, I don't understand if it is necessarily true that straight distance of Z from x, in opposite direction of Y $(20+12=32)$, is the greatest possible distance or not. Can anyone explain it?

To find the maximum distance between $Y$ and $Z$, assume they are in opposite directions from $X$. Then a straight line from $Y$ to $Z$ passes first from $Y$ to $X$ (20 miles) and then from $X$ to $Z$ (12 miles), and has a total distance of 32 miles. You have the right idea for minimizing the distance between the two points. This occurs when $Z$ is "on the way" to $Y$ from $X$. In this case the distance between $Y$ and $Z$ is the difference of their distances from $X$, as you wrote.