As alun said antenna theory isn't that simple and can't be explaned in a few rows of text.
Dipoles are always 1/2 wave, increased frequency is equal to shorter antenna, if you cut the dipole to 1/4 wave length you no longer have the 73 ohms (theoretically) besides the impedance of a dipole changes with height and how it is mounted against ground. The thicker the wire/radiator is the shorter the physical length is for a given frequency and impedance will change as well, thicker wire/radiator= lower impedance and lower Q= increased bandwidth.
For the 50 ohms there is no need to build a yagi, just form the 1/2 dipole as a inverted vee (up side down V) and the impedance will decrease to aprox. 50-60 ohms but bandwith will suffer and the resonant frequency will be lowered compared to a horizontal mounted dipole.
A 1/4 wave vertical with 45 deg groundplanes will give about 50 ohms also.
Almost any antenna can be matched to 50 ohms by using different matching techniques.
If you calculate a 1/2 wave dipole for say 100mhz the formula is 150/100Mhz which will give you 1.5 meters totally, I can guarantee that the antenna is to long meaning that the resonant frequency is lower then 100Mhz, that's because of a factor lambda/d (lambda=wave length d=diameter of the wire/radiator).
Wow it's a lot to say about this