# Midpoints

1. Find the distance between points A and B: 2. Find 4 other points with an equivalent distance from point A in problem 1.
3. How many total points are an equivalent distance from point A in problem 2. Is there a quick way to find all equally distant points?
4. Find the points halfway between A and B, B and C, A and C, and B and D. Make sure you can prove that these are, in fact, the midpoints.
5. Have you invented a strategy to quickly identify the midpoint between any two points? Explain
6. Mathematicians may think of the midpoint problem as finding the point that splits segment AD into two congruent segments, AE and ED (see below). Where would the points be that divide AD into four congruent segments? 7. Once again, were you able to develop a technique to find a solution to problem 6? Explain.
8. Mathematicians would call point E from problem 6 equidistant from points A and D. What does this mean? Find three other points that are equidistant from point A and D.
9. Looking at problem 8, how many total points are equidistant from points A and D? Is there a quick way to find all possibilities?
10. Can you find a point that is equidistant to points B, C, and D? What about A, B, C, and D? 11. What is the midpoint between point N (2, 5, 10) and M(-3, -4, -2)?
12. Thinking back to problem 10 and including the third dimension, is it now possible to find a point equidistant to A, B, C, and D? Explain.