### Project 6

 Problem 1 Part A: Use this figure (a square pyramid inside a cube) and Geoblocks (you can sign a kit out from the math/science education center on the second floor) or models that you build to give a plausible explanation of why the volume of a square pyramid is one-third the height times the base. Part B: Can you find a similar decomposition relating a rectangular box and a rectangular pyramid? Explain.

Part C: Can you find a similar decomposition relating a rectangular box and a triangular prism? Explain.

For the following problems, you may do by-hand work and/or use Sketchpad. Be sure to show work and explain (if you use Sketchpad then print out your Sketchpad work and explain it).

Problem 2 Look at the triangle whose vertices are A=(2,2), B=(-1,1), and C=(1,-1).

Part A: Show that this triangle is equilateral under the taxicab metric.

Part B: Is the triangle equilateral or isosceles under the Euclidean metric?

Problem 3 In order to show that the Pythagorean Theorem sometimes but not always holds in taxicab geometry...

Part A: Look at the triangle whose vertices are A=(-3,9), B=(12,4) and C=(0,0). Notice that the slope of AC = -3 while the slope of BC = 1/3 and so AC is perpendicular to BC. Hence this is a right triangle. Let c be the hypotenuse (opposite vertex C) and let a be the side opposite vertex A and b be the side opposite vertex B. Compute a, b and c under the taxicab metric. Compare a2 + b2 with c2. Does the Pythagorean theorem hold for this right triangle in taxicab geometry?

Part B: Look at the triangle whose vertices are A=(0,0), B=(4,3) and C=(4,0). Notice that this is a right triangle. Compare a2 + b2 with c2. Does the Pythagorean theorem hold for this right triangle in taxicab geometry?

Problem 4 A man has a newspaper stand at W=(1,0), eats regularly at a cafeteria located at E=(8,3), and does his laundry at a laundromat at L=(7,2)

Part A: If he wants to find a room R using the taxicab metric so as to be at the same walking distance from each of these points, where should R be located? (Give the coordinates).

Part B: How many blocks does he have to walk from his room to each of the three points, assuming that he finds such a room?

Part C: If all conceivable shortcuts are possible (ie, using the Euclidean metric), where should the man's room be so that it is equidistant? (Hint: use the perpendicular bisectors).

Part D: What is the Euclidean measurement of the distance that he has to walk from his room in Part C to each of the three points?