8.7 Consider the real Cartesian plane R2, with lines and betweenness as before (

Question

8.7 Consider the real Cartesian plane R2, with lines and betweenness as before (Exam- ple 7.3.1), but define a different notion of congruence of line segments using the distance function given by the sum of the absolute values: where A=(al ,a2) and B = (bi, b). Some people call this "taxicab geometry" be- cause it is similar to the distance by taxi from one point to another in a city where all streets run east-west or north-south. Show that the axioms (C1), (C2), (C3) hold, so that this is another model of the axioms introduced so far. What does the circle with center (0,0) and radius 1 look like in this model?

Explanation / Answer

Let's take a point (x,y) on the circle and distance of this point from circle is always one

so d (A,B) = |a1 - b1| + | a2 - b2 |

Distance is 1 and A = (x,y) and B = (0,0)

1 = |x - 0| + |y - 0|

This is the model of circle

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