1. Let point A be the origin. Find several points that are 3 units from A.

1. § 19.1 • 1. Let point A be the origin. • Find several points that are 3 units from A. • Find all the points that are 3 units from A. • What would you call this set of points A + c. A taxicab circle .

2. 2. Let point A be the origin with B (5, 3). Graph the following sets of points. . • The taxi circle with center A and radius 2. • The taxi circle with center B and radius 3 1/2. • The taxi circle with center B and radius 6. • Calculate the distance from A to B. B A + d. AB* = 8

3. 3. Consider a taxi circle with radius r and side length s. • Write an equation for the perimeter (circumference) of the circle with respect to its side length. • Write an equation for the perimeter of the circle with respect to its radius. • Write an equation for the area of the circle with respect to its side length. • Write an equation for the area of the circle with respect to its radius. • What is a reasonable value for  in taxicab geometry? s + r a. P = 8r b. K = s 2 / 2 c. K = 2 r 2 a. K = r 2 so  = K/r 2 = 2.

4. 4. Given the line L with equation y = 3x and the point P (3, 5), construct a taxi circle with radius 3 that passes through P and touches L at exactly one point. (There are two such circles.)

5. 5. Do three noncollinear points always determine a unique taxicab circle? No

6. 6. Explain how to find the center of a taxi circle that goes through two points A and B. If you construct the line that is equidistant from points A and B the center will lie on that line. You cannot definitively determine the location of the center from two points.

7. 7. Once you have the center located in problem 5 explain how you would sketch the circle. Give an example. If you construct the line that is equidistant from points A and B the center will lie on that line. You cannot definitively determine the location of the center from two points and thus you cannot sketch the circle without additional information such as a radius.

8. 8. Let point A be the origin with B (5, 3). Now on a single sheet of graph paper: • Graph all the points that are 3 units from A and 5 units from B. • Graph all the points that are 1 unit from A and 7 units from B. • Graph all the points that are 0 units from A and 8 units from B. • Graph all the points that are 1 1/2 units from A and 6 1/2 units from B. • Graph all the points that are 4 units from A and 4 units from B. • Graph all the points that are 5 units from A and 3 units from B. • Graph all the points P so that the sum of the distance from A and B is 8. B A +

9. 9. When one is trying to find the shortest distance from a point A to a line you may think of it as a slowly inflating circle with center A until it just touches the line L. the radius of the circle at that moment would be the distance in question. With this idea in mind on a sheet of graph paper draw the line L through (3, 0) and (1, -4). Now • Find the shortest distance from the origin to the line. • Find the shortest distance from (0, 3) to the line. a. r = 3 b. r = 4 1/2 +

10. L F + • 10. Consider the point F at the origin and draw the line L through (3, 0) and (1, -4). Now • Sketch all the points 2 units from L. • Sketch all the points 2 units from F. • Sketch all the points 2 units from L and 2 units from F. • Sketch all the points that are the same distance from L and from F.

11. 11. Graph a parabola with directrix at y = - x and focus at (4, 4). +

12. 12. Consider the line through the origin and (4, 4): • Find the set of points P so that the taxicab distance from the line is equal to PF where the point F is at (2, 0). • Find the set of points P so that the taxicab distance from the line is equal to PF where the point G is at (-1, 3). • Find the set of points P so that the taxicab distance from the line is equal to PF where the point H is at (-3, 5). H G F +