Here's an activity that might help students appreciate the usefulness of the distance formula. I think this would be considered an application, but tell me if it's not:)
Students can work in pairs or small groups. Each group is given a coordinate plan with a line drawn on it and is asked to find the length of the line. They should be able to determine the coordinates of the endpoints and the line should not be vertical or horizontal (because that'd be too easy!). As the students are working, the teacher can observe the various methods that students use. If no one has thought of it after awhile, the teacher could suggest that the students create a right triangle with the original line as the hypotenuse. Ask them, "how might this drawing allow them to find the length of the line?" Hopefully some students will use the Pythagorean theorem, but if not, the teacher can introduce this idea.
Then the teacher can lead the class in deriving the distance formula by generalizing the process that they just used with the Pythagorean theorem.
Students can go back to the original problem and apply the distance formula directly to see that it works.
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That is a neat activity! I like math and even started a blog about it. It can be found at http://mathvariety.blogspot.com so check it out. I have explanations, practice problems, and solutions pertaining to a wide range of math topics.
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