PROJECT for Chapter 1 Taxicab

OBJECTIVE Compare in taxicab geometry to distances in . Materials: ruler, graph paper, colored pencils, poster paper

In taxicab geometry, distances are measured along paths that are made of horizontal and vertical segments. Diagonal paths are not allowed. This simulates the movement of taxicabs in a city, which can travel only on streets, never through buildings.

FINDING TAXICAB DISTANCES

Follow these steps to learn more about in taxicab geometry.

B B 14 B

10

A A A 12

1 Copy points A and 2 Trace paths from A 3 Calculate the B on a piece of to B using the grid distances covered graph paper. lines. You may move by your paths. horizontally and vertically but not diagonally.

INVESTIGATION

Repeat Steps 2 and 3 above several times, then answer the exercises. 1. Compare your paths and those of your classmates. Are the distances always the same? What is the length of the shortest possible path from A to B? 2. The length of the shortest path from A to B is called the taxicab distance from A to B. Can you find other paths from A to B that have the same distance? 3. Use the Distance Formula to find the from A to B. Which is greater, the taxicab distance from A to B or the Euclidean distance? 4. On another piece of graph paper, plot a new pair of points. Find the taxicab distance and Euclidean distance for the points. Repeat this for several pairs of points. Write a general statement that compares taxicab distance and Euclidean distance.

5. Write a general formula for the taxicab distance from A(x1, y1) to B(x2, y2).

66 Chapter 1 TAXICAB

Suppose you are in a city with a square grid and want to find all the places you can reach if you walk 4 blocks along the streets. This forms a figure in taxicab geometry in the same way that a is formed in Euclidean geometry.

INVESTIGATION

6. On a piece of graph paper, plot a point O(0, 0). Plot all the points that have a taxicab distance of 4 units from point O. How are the points arranged? 7. To find the circumference of your taxicab circle, imagine that you are walking a shortest taxicab path that passes through all the points you drew in Investigation Exercise 6. Draw one such path. What is its total length? (Remember that all distances in taxicab geometry are horizontal or vertical.) 8. To find the diameter of your taxicab circle, draw a shortest taxicab path that joins two points on the circumference and also passes through the center O. What is its length? 9. In Euclidean geometry, the constant π is defined as the ratio of a circle’s circumference to its diameter. If π is defined in the same way in taxicab geometry, does it have a constant value? If so, what is the value?

PRESENT YOUR RESULTS TAXICAB Gather your work and present it as a poster. GEOMETRY • Include your answers to the Investigation 8 Exercises. 8 • Include your calculations and drawings 4 you used to find the value of π in taxicab geometry. 4 8 • Summarize how taxicab geometry and 8 Euclidean geometry are different. C = 4 p 8 d = 2 p 4

EXTENSIONS

In Euclidean geometry, the points that lie between two points A and B form a segment. In taxicab geometry, what kind of region do the points that lie between two points A and B form? (Hint: A point C lies between A and B in taxicab geometry if this equation is true for the taxicab distances: AC + CB = AB.) Æ In Euclidean geometry, the midpoint of AB bisects the segment into two parts. All the points in one part are closer to A than to B, and all the points in the other part are closer to B than to A. Is there a set of points that bisects the set of points between A and B in taxicab geometry? What does it look like?

Project 67