intitution versus ogic

question if the ring like mass m1 comes down a distance h what is the ratio of v1/v2
my method

i solved this problem mainly on intitution like i drew a erpendicular and the length comes out to be h cos@
now i want to know the reason behind this intitution like why the line meetng is peprendicular means what i hv done is found the pnt where the length wa constant and then found the extra distance moved so why is the line joining peprendicular

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