18.1 Taxicab Geometry Homework
1. Consider Taxicab and Euclidean distances.
a. Which is usually greater, Taxicab distance or Euclidean distance?
b. Is the reverse ever true?
c. Are they ever equal?
2. Does the triangle inequality hold in Taxicab geometry?
3. Prove betweenness in taxicab geometry. That is, if A-B-C then AB* + BC* = AC*.
4. If three points A, B, and C are plotted so that no two of the taxicab distances AB, BC, and AC are equal, the points are said to form a taxi-scalene triangle.
a. Sketch a taxi-scalene triangle that is scalene in Euclidean geometry.
b. Sketch a taxi-scalene triangle that is isosceles in Euclidean geometry.
c. Sketch a taxi-scalene triangle that is equilateral in Euclidean geometry
5. If three points A, B, and C are plotted so that exactly two of the taxicab distances AB, BC, and AC are equal, the points are said to form a taxi-isosceles triangle.
a. Sketch a taxi-isosceles triangle that is scalene in Euclidean geometry.
b. Sketch a taxi-isosceles triangle that is isosceles in Euclidean geometry.
c. Sketch a taxi-isosceles triangle that is equilateral in Euclidean geometry
6. If three points A, B, and C are plotted so that, in taxicab distances AB = BC = AC, the points are said to form a taxi-equilateral triangle.
a. Sketch a taxi-equilateral triangle that is scalene in Euclidean geometry.
b. Sketch a taxi-equilateral triangle that is isosceles in Euclidean geometry.
c. Sketch a taxi-equilateral triangle that is equilateral in Euclidean geometry
7. Does the Pythagorean Theorem hold for taxicab geometry? Its converse?
8. Consider triangle ABC with A (0, 0), B (3, 1) and C (1, 3) and triangle XYZ with X (5, 0), Y (7, 2) and Z (5, 4). Find the lengths of the sides of each triangle. Is AB = XY? Is AC = XZ? Is BC = YZ? Is triangle ABC congruent to triangle XYZ?
9. Ideal City has perfect square blocks and equally spaced streets running north and south, east and west. The usual grid coordinates of a Cartesian plane is imposed over Ideal City with the origin at the town square. The dispatcher for the city Police Department receives a report of an accident at X (-1, 4). There are two police cars in the area, car C at (2, 1) and car D at (-1, -1). Which car should be sent to the scene of the accident?
10. There are three high schools in Ideal City: Fillmore at (-4, 3), Grant at (2, 1) and Harding at (1, 6). Draw in school boundary lines so that each student in the city attends the high school nearest home.
11. If Veggie Value wants to open a restaurant equally distant from each of the three high schools, where should it be located?