Question

How are the distance formula, the Pythagorean Theorem, and equations of circles all related? How does one help you understand the others?

Answer 1

If we place the triangle in the coordinate plane, having A and B coordinates of (x1,y1) and (x2,y2) respectively, it is clear that the length of AC is |x2 - x1| and the length of BC is |x2 - x1|.& We are finding the length, which means that we want a positive value; the absolute value signs guarantee that the result of the operation is always positive. But in the final equation,c^2 = |x2 - x1|² + |y2-y1|², the absolute value sign is not needed since we squared all the terms, and squared numbers are always positive. Getting the square root of both sides we have,

c = √(|x2 - x1|² + |y2-y1|²).

We say that c is the distance between A and B, and we call the formula above, the distance formula.

If we want coordinates of B(x,y) where x and y are variables and the distance of B from A constant, say r,& then moving point B about point D maintaining the distance r forms a circle. If A has coordinates (h,k), then
& r² = (x-h)² + (y-k)²

which means that

r = √( (x-h)² + (y-k)² ).

Observe that the two& equations above are all of the same form, they are all consequences of the& Pythagorean Theorem.