### Taxicab Geometry

1. Introduce taxicab geometry via the taxicab treasure hunt and moving in tivo. Highlight the possible number of intersections of taxicab circles for different examples.

2. US law is Euclidean.

3. SAS in taxicab geometry. Euclid's proof of SAS and what goes wrong in taxicab geometry.

We'll use the Taxicab distance file

4. Square Circles Under File release on New Sketch. Under Graph release on Show Grid. Using Graph, plot points, precisely plot the point (-6,4). Create a taxicab circle of radius 2 about the point (-6,4) by using the graph/plot point feature to create precise vertices of your taxicab circle. Use the point tool (not the graph/plot point feature) to create an approximate point on the boundary. Select the point (-6,4) and then the boundary point you created, and then click on All Steps from your script tool. Notice that the script will now run and it will calculate the taxicab distance for you. Move the point on the boundary to see that the taxicab distance to the center remains the same as the horizontal and vertical distances change.

5. Finding All the Points Equidistant from A=(0,0) and B=(3,3) in the Taxicab Metric Under File release on New Sketch and Under Graph release on Show Grid. Carefully create the point (3,3) using the Graph/plot points feature. Use the point tool to create a point that is NOT on the axes and is not directly above, directly below, or directly to the left/right of A or B. and use your Taxicab Script to measure the distances from this point to both A and B (click on 2 points and then All Steps in the script and then redo this). Move the point around and note the distance to A and B in order to find all of the points that are equidistant from A and B. Sketch a diagram of these equidistant points on a sheet of paper and show me when you think that you have found them all.

6. Precinct Problem In a town having perfect square blocks and equally spaced streets running north and south, east and west, two police stations are to be located at A=(0,0) and B=(3,3). The town officials want to divide the town into two precincts - Precinct 1 served by Station A and Precinct 2 served by Station B. What are the real-life issues that would go into deciding how the boundary should be drawn? Explain your ideas to me when you are ready.

7. 3 Noncollinear Points In Euclidean Geometry, 3 noncollinear points determine a unique circle. Is this true in taxicab geometry?

8. Equilateral triangles 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?

9. Pythagorean Theorem 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?

10. Equidistant Room 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 could R be located? (Give the coordinates).
Part B Is there more than one possible answer for Part A?
Part C: How many blocks does he have to walk from his room to each of the three points, assuming that he finds a shortest distance room that has the same walking distance to each of the points.
Part D: 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 E: 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?