# Loss due to impedance matching in a filter

I have a quite basic question, after starting to re-read some RF theory. If a voltage source has a resistance of 50 ohms, and the equivalent impedance of a connected bandpass filter and its load are 50 ohms for a perfect match, won't there be significant loss - so much so that the passband will disappear entirely? If the bandwidth of a filter is defined as the portion where the gain is above 1/sqrt2 ~ 0.707 V, and the highest the output can ever get is 0.5 V due to the voltage divider rule, is it impossible to have an impedance matched band pass filter? Why would someone want to have a matched preselection filter between an antenna and an LNA if matching losses might cause the weak signal to be out of amplification range?

The $\frac{1}{\sqrt{2}}$ point is defined relative to the magnitude, so if your magnitude in the passband is 0.5, then the -3dB point is at $\frac{0.5}{\sqrt{2}}\approx0.354$.