1. What are the four ways astronomers can detect extrasolar planets?
The four ways to find a planet are radial velocity, astrometry, transit method, and optical detection.

2. Choose one of the four ways and explain it in detail.
Transit method is when a planet passes in front of its star. this causes some of the light form the star which reduces its apparent magnitude. This can tell you how long the planet takes to move in front of the star.

3. In your answer to #2 you chose one way that astronomers detect extrasolar planets. Search the Internet to find how many extrasolar planets have been detected using this method.
45 extrasolar planets have been detected using the transit method.