posted by Paula .
why does a cool, distant, supergiant star such as Betelgeuse have such a high apparent magnitude?
Because it is relatively close to us and large in size. It is actually large and close enough for its disc to be resolved by modern telescopes, using interferometry.
The apparent brightness of a star depends upon its size (diameter) , surface temperature and distance. Betelgeuse, although relatively cool, scores on two out of three.
Betelguese actually qualifies as a supergiant. If you Google the words "Betelgeuse image" you will see what it looks like, based upon recent state-of-the-art imaging techniques.
You're welcome! I love astronomy questions.