Taxicab Geometry. 1. Taxicab Routes. Suppose you live in Taxicab City where all the streets run straight north and south or straight east and west.

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1 1 Taxicab Geometry 1. Taxicab Routes Suppose you live in Taxicab City where all the streets run straight north and south or straight east and west. One night the 911 dispatcher for Taxicab City receives a report of an accident at location X = 1,4. There are two police cars in the area, car C at 2,1 and car D at 2,0. Which car should be sent to the scene of the accident to arrive most quickly?

2 2 2. Euclidean vs. Taxicab Distances Consider the points in the following graph: (a) Calculate the following distances in both Euclidean and Taxicab geometries. Give decimal approximations to 2 decimal places. Euclidean Distance Taxicab Distance from A to B from A to C from A to D from A to E (b) Is the Euclidean distance between two points always less than or equal to the Taxicab distance? Is it possible for the Euclidean distance to be greater than the Taxicab distance? (c) Given two points A = a, b and B = c, d write a formula for the Euclidean distance from A to B given by d A, B. Next, write a formula for the Taxicab distance from A to B given by d!"#$ A, B. d!"#$ A, B = d A, B =

3 3 3. Taxicab Distances Using Taxicab geometry, consider the points A = 3,2 and B = 3,0. (a) Is the point 2, 3 closer to A or to B? (b) Is the point 1, 2 closer to A or to B? (c) Find one point that is exactly the same distance from A as it is from B. Mark it on the graph. (d) Find another such point. Mark it on the graph. (e) Mark all points on the graph that are equally distant from A to B. [Remember, this includes all points with non- integer coordinates.]

4 4 4. Taxicab Circles (a) In Taxicab City, George works at City Hall located at M = 1,2. He goes out to eat for lunch once a week, but he is usually short on time, so George likes to walk exactly 3 blocks from City Hall to make it back to work on time. Where in the city are restaurants at which George can eat? Draw their locations on the graph.

5 5 4. Taxicab Circles (continued) (b) On the graph below, draw Taxicab circles around the point M = 1,2 of radii 1, 2, 3, and 4. (c) Describe a quick technique for drawing a Taxicab circle of radius r around a point P. (d) Use your Taxicab circles in part (a) to complete the following table: radius Taxicab Circumference Taxicab Area Formula: (e) What is the value of π in Taxicab geometry? (f) Find an equation of a Taxicab circle with center a, b and radius r.

6 6 5. The set of points equidistant from two given points A and B In Euclidean Geometry, the set of all points equidistant from two given points A and B is the perpendicular bisector of AB. Here we investigate this concept in Taxicab geometry? Question: Alice and Bob live in Taxicab City and are looking for an apartment. The following points are the coordinates of their workplaces. Where should they live if they would like to live the same distance from their jobs? a. Alice (- 2, 1) Bob (4, 1) b. Alice (- 1, 3) Bob (- 1, - 1)

7 7 c. Alice (- 2, 1) Bob (4, 1) d. Alice (- 1, 3) Bob (- 1, - 1)

8 8 6. Ellipse: In Euclidean Geometry, an ellipse is the set of points that the sum of the distances to the two fixed points (the fuci) is constant. Here we investigate ellipse in Taxicab geometry. (a) Anna and Brian want to meet, but they only want to walk a combined distance of 9 blocks. The following points are the coordinates of their homes. Where are all locations where they might meet? Anna (- 3, 1) Brian (2, 1)

9 9 (b) Draw the taxicab ellipse with foci (2,- 3) and (2,2), so that the total distance from each point to the foci is 7. (c) Draw the taxicab ellipse with foci ( 3, 2) and (3, 2), so that the total distance from each point to the foci is 14. a. b.

10 10 7. Investigating SSS postulate in Taxicab Geometry: Consider the following triangles: Does d T (A,B) = d T (A',B')? Does d T (A,C) = d T (A',C ')? Does d T (B,C) = d T (B',C ')? Is triangle ABC congruent to A' B'C '? In Euclidean geometry the SSS postulate say that if three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent. Does SSS hold in Taxicab geometry? 8. Investigating SAS postulate in Taxicab Geometry: Consider triangles ABC and DEF. Does d T (A,B) = d T (D,E)? Does ABC DEF? Does d T (B,C) = d T (E,F)? Is triangle ABC congruent to DEF? Does SAS hold in Taxicab geometry?

11 11 When would I teach this? Taxicab geometry would be suitable for middle school and secondary students just after covering coordinate geometry and the distance formula. References: The problems in this worksheet are from the Taxicab Geometry Worksheets created by Dr. Anders Hendrickson at Concordia College: Other material on Taxicab Geometry may be found at: column/fcarc- taxi Taxicab geometry via geogebra:

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