Taxicab Geometry Practice Problems (part 2) Some more problems to get you familiar with taxicab geometry In light of problem 7 on the previous page, it appears that when we investigate something involving two points, it would be worth our while to consider three possibilities: (i) the points are on the same horizontal (or vertical) line, (ii) the points are on a non-diagonal line from each other, and (iii) the points are on the same 45-degree line. 1) Given two points, In Euclidean geometry, an ellipse is the set of points where the sum of the distances to the two given points is a constant. Let's see what a taxi-ellipse looks like. For each part below, given A and B, draw the set of points P such that t(P; A)+t(P; B) = 12. (a) A = (0; 0);B = (6; 0) (b) A = (0; 0);B = (4; 2) (c) A = (0; 0);B = (3; 3) 1 2) Given two points, in Euclidean geometry, a hyperbola is the set of points where the difference between the distances to the two given points is a constant. For each part below, draw the set of points such that jt(P; A) − t(P; B)j = 2. (a) A = (0; 0);B = (6; 0) (b) A = (0; 0);B = (4; 2) (c) A = (0; 0);B = (3; 3) 2 3) For parabolas, we need to know the distance from a point to a line. So, experiment and decide how to determine the taxicab distance from a points to a line. Make sure to consider horizontal/vertical lines, slanted lines with slopes other than ±1, and diagonal lines with slope exactly ±1. 4) In Euclidean geometry, a parabola is the set of points equidistant from a given fixed point and fixed line. Draw the taxicab parabolas below: (a) A = (0; 4), line is x-axis (b) A = (0; 4), line is y = 2x (c) A = (0; 4), line is y = x 3 5) It's not one-hundred percent clear how angles should be measured in taxicab geometry. Should we measure as a portion of the taxi unit circle? Should we use the same angle measure as we do in Euclidean geometry? Here is why some of these questions are so hard to answer. (a) Consider the points A(4; 0);B(0; 4) and the origin O(0; 0). No matter how we measure angles, it seems like \AOB should be a right angle. What is the distance t(A; B)? 0 0 0 0 (b) What if the points move to A (3; 1) and B (−1; 3). Should \A OB be a right angle? What is t(A0;B0)? 00 00 00 00 (c) Move the points once more to A (2; 2) and B (−2; 2). What about \A OB and distnace t(A00;B00) now? (d) What do the three triangles 4AOB; 4A0OB0; 4A00OB00 tell you about SAS? The Pythagorean Theorem? 6) Can you find two equliateral triangles (all sides the same length, in taxicab measure) that are not congruent? Talk about SSS not being true! 7) Some things do stay the same, though. Show that the tringle inequality for three points in the plane is still true. 4.