Explorations in Geometry

with the

Geometer’s Sketchpad

Exploration 6

- Construct a triangle, ABC, and select a point inside it and call the point H. Measure the distance from point H to the three vertices, A, B, and C. Move H around and find the point where the sum of the distances is a minimum. What do you notice about the figure at this point.
- Repeat the experiment with a quadrilateral, DEFG and an interior point I. Where can you put I to make the sum of the distances to the vertices a minimum?
- Assume that Points D,E, F, and G are cities, and you need to connect build a roadway so that people can travel from any of the cities to any of the other cities. You have a limited amount of money, so you have to make the roadway as short as possible. Can you build a road with even less distance than the answer in part 2? How ?
- Can you see a way that the answers to parts 1) and 3) are related? What other patterns and relations can you find among these ideas?
- Write a brief paragraph explaining what you have found in this exploration. Remember to answer all the specific questions above.