Need Girl Help. Really Shy?

I'm in high school and never have had a girlfriend. I have an eye on a girl, but I'm too shy to talk to her. I would say I'm pretty good looking, as it's a family trait, but around girls, especially really pretty ones, I find myself unable to speak. I'm a very passive, easy-going, and not aggressive person. The upside to my personality is I'm a kind, caring person, and my mom says any girl who got to know me would adore me. The downside is, I'm a little introverted and shy. All my friends say to go up and talk, but to tell the truth, I feel as if I'd rather do just about anything else. I got her to dance with me at a high school dance this year, but she seemed taken aback and distant. We never said a word, which was really awkward, but at least I tried for once. The hard part is, we have no classes together, and even when she's at lunch, she's surrounded by her friends and being shy, I'd be terrified to try to talk to her with all them around. Another problem is, I am known for my shy/quietness and any girl would be surprised I would strike up a conversation outside of free time discussion between everyone in class. I play basketball and maybe football, but that really won't help since she doesn't play sports. I really don't know what to do. She is also shy which makes it harder. Apart from being in the same school, I don't even know if we have anything in common. Any help would be appreciated. Thanks and God Bless.