§ 18.1 1. Consider Taxicab and Euclidean distances. Which is usually greater? Taxicab distance or Euclidean distance? Is the reverse ever true? Are they ever equal? a. Taxicab is usually the greater because it is the two legs of the right triangle where the Euclidean distance is the hypotenuse. b. Euclidean distance is never greater. c. The two distances are equal when the two points have the same x – coordinates or y – coordinates. That is they are vertical or horizontal .

2. Does the triangle inequality hold in Taxicab geometry. No. the triangle inequality states that given three sides of a triangle it is always true that the sum of any two is greater than the third side. OR a + b > c. Consider the triangle A(1, 5), B(5, 2) and C(2, 2): a = 3, b = 4, and c = 7 and a + b = c.

3. Prove betweenness in taxicab geometry. That is, if A-B-C then AB 3. Prove betweenness in taxicab geometry. That is, if A-B-C then AB* + BC* = AC*. Given: A – B - C Prove: AB* + BC* = AC*. Statement Reason 1. AB* = |xa – x b | + |y a – y b | Definition taxicab distance 2. BC* = |xb – x c | + |y b – y c | Definition taxicab distance 3. x a < x b < x c and y a < y b < y c or x c < x b < x a and y c < y b < y a A – B – C Given 4. AB* + BC* = |xa – x b | + |y a – y b | + |xb – x c | + |y b – y c | Addition 1 + 2 5. AB* + BC* = |xa – x b | + |xb – x c | + |y a – y b | + |y b – y c | Associative Addition 6. AB* + BC* = |xa – x c | + |yb – y c | Addition 7. AB* + BC* = AC*. Definition taxicab distance

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-scale triangle. Sketch a taxi-scalene triangle that is scalene in Euclidean geometry. Sketch a taxi-scalene triangle that is isosceles in Euclidean geometry. Sketch a taxi-scalene triangle that is equilateral in Euclidean geometry 4 5 5 6 7 7 a b c Take a Euclidean equilateral ∆ with sides equal to 1 unit and with one vertex at the origin and rotate it 20°. The taxicab dimensions will be 1.2817, 1.1584, 1.4089 giving a scalene triangle.

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. Sketch a taxi-isosceles triangle that is scalene in Euclidean geometry. Sketch a taxi-isosceles triangle that is isosceles in Euclidean geometry. Sketch a taxi-isosceles triangle that is equilateral in Euclidean geometry 4 5 (1.41, 3.874) 5 6 6 6 (4, 1) (0, 0) a b c

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. Sketch a taxi-equilateral triangle that is scalene in Euclidean geometry. Sketch a taxi-equilateral triangle that is isosceles in Euclidean geometry. Sketch a taxi-equilateral triangle that is equilateral in Euclidean geometry (2, 8) (7, 3) (0, 0) a b c I can’t find one!

7. Does the Pythagorean Theorem hold for taxicab geometry? Its converse? The triangle to the right is a right triangle and it does not satisfy the Pythagorean Theorem. CONVERSE: The triangle to the left satisfies the Pythagorean Theorem and it is not a right triangle

AB = XY = 4, AC = XZ = 4 and BC = YZ = 4 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? AB = XY = 4, AC = XZ = 4 and BC = YZ = 4 The two triangles are not congruent even though SSS holds!!!!

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? X D C CX = 6 DX = 5 Send car D. +

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. H G F +

11. If Veggie Value wants to open a restaurant equally distant from each of the three high schools, where should it be located? H G F + Veggie Value