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How could scientists know how far a star or galaxy is from us?

How do astronomers measure how far a star (or galaxy) is away from the earth? How would they know that it has taken 13 million years for light to travel in space before it reaches us?
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In relativity, what is the difference between a rod that is perpendicular to direction of motion and a rod parallel to the direction of motion?

In Feynman's Lectures on Physics , chapter 15, page 6 he writes about 2 identical, synchronized light signal clocks. These are clocks that consist of a rod (meter stick) with a mirror at each end, and ...
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How do you measure distance to stars within the galaxy?

I know that for close by stars (<50 LY) we can use the parallax effect. And for distant galaxies we use red-shift (& hubble's constant). So how do we measure how far is a star lets say 50,000 ...
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Most accurate ways to find the average distance between stars in Milky way galaxy

I've already posted here on quora. But, I'm not totally sure if it's the most reasonable method. Would anyone care to elaborate on how to find the average distance between stars in a given galaxy ...