Consider the real Cartesian plane R^2, with lines and betweenness as before (Example 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: d(A,B) = |a1-b1| + |a2-b2|, where A = (a1,a2) and B = (b1,b2). Some people call this "taxicab geometry" because 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 in this model?
All parts, please. I will post the 8.7 and 9.3 exercises below. The parts are labeled in the image
8.7 says
Consider the real Cartesian plane R^2, with lines and betweenness as before (Example 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: d(A,B) = |a1-b1| + |a2-b2|, where A = (a1,a2) and B = (b1,b2). Some people call this "taxicab geometry" because 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 in this model?
9.3 says
Consider the real Cartesian plane where congruence of line segments is given by the absolute value distance function (Exercise 8.7). Using the usual congruence of angles that you know from analytic geometry (Section 16), show that (C4) and (C5) hold in this model, but that (C6) fails. (Give a counterexample.)
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