When learning a new language, I'm always looking for similarities between word roots - this helps me in remembering them. When learning Russian, I was surprised by the similarity of the words дорогой (expensive) and дорога (road). Do they have a common origin, or is this just a coincidence?
Are "dear" and "deer", or "drag" and "drug" related in English? No they are not. It's just happens sometime that some words sounds similar.
"Дорога" is related to "дёрн" (turf), "драть" (tear), "дёргать" and "дерево" (tree) and initially meant something like a path in a forest. Surprisingly, "дорога" is related to English verb drag.
Here you can find out more:
Происходит от индоевроп. *dorgh- (связано с дёргать и означает «продранное в лесу пространство»);
As of "дорогой", it's related to English word "dear", here's a quote:
From Middle English dere, from Old English dēore, from Proto-Germanic *diurijaz. Cognate with Dutch duur (“costly, precious”), German teuer (“costly, precious”), Icelandic dýr (“expensive”), Norwegian dyr, Swedish dyr (“expensive”).