We can make a proportion relating the lengths of a line segment given in the attachment. When the ratio of the entire segment to the longer portion, a/b, is equal to the ratio of the longer portion to the smaller portion, b/c, we would have:
a/b = b/c
Cross multiplying and solving for b, b would be considered the golden ratio and also considered the geometric mean of a and c.
It is called "golden" simply because the ancient people, Greeks, Romans, one of them back there, considered it a "beautiful" ratio. And, they considered gold beautiful, also. So, when something came up that was beautiful, just like we do today, we would do something like call it "the golden ratio".
Good luck, schooledmom. I hope this assist.