Why is partitioning a directed line segment with a ratio of 1: 2 not the same as finding half the length of the directed line segment

The ratio would need to be 1:1 in order to split the line in half. Think of it like having two cookies. If you split those cookies between you and your friend, then you get 1 and s/he gets the other. So the ratio of your cookie count to your friend's count is 1:1

With the ratio 1:2, you would have 1 cookie while your friend has 2 cookies. This is no longer a fair split. Things are not 50/50 anymore. You have 1/3 of the cookies while your friend has 2/3 of the cookies. Your friend would have twice as much.

So that applies to the segment lengths as well. The line segment would be split up into two parts A and B. Part A is the smaller segment that is half as long as part B. Or put another way, part B is twice as long as part A