Taxicab Activities in Sketchpad
Creating a Taxicab Script to Measure Taxicab Distance
Open up Sketchpad and use the point tool to create two points that are
not on the same vertical or horizontal line.
Use the arrowhead tool to select the point on the left
and under Transform, release on Translate.
Change the angle to 0 degrees.
You will see that the translation is in the horizontal direction 1 cm
and that this translated point is closer to the second point.
Construct a line passing through the point and its translate by
first selecting both points and releasing on Line under Construct
Construct the perpendicular to this line through the other point
by selecting the line and the other point and releasing on
Perpendicular Line under Construct
Select both lines and release on Intersection under
Using Display/Hide, hide the lines and the translated point.
You now have 3 points remaining (which form a right triangle).
Using the straight edge
tool, construct the horizontal part of the triangle.
Under Display, scroll down to Color and release on
Using the straight edge tool, construct the vertical part of the triangle.
Under Display, scroll down to Color and release on Orange.
Using Measure/Length, select and then
measure the horizontal and vertical distances.
Under Measure or Number, release on Calculate,
and compute the sum of these
distances (ie the taxicab distance) by clicking on measurement 1, then +, then
measurement 2. Hit Ok. You'll see a measurement in cm.
Under Display, Hide the first two measurements and the
horizontal and vertical distances. You now have
3 points, the aqua horizontal line, the orange vertical line, and the
taxicab distance, like in the following picture:
Use the arrowhead tool in order to create a big rectangle that
Use the Script tool to Create New Tool and check the
Show Script View
You have just created a script that will measure the taxicab distance
between two points.
Save your Sketchpad file as yourname.gsp and leave the script
open. Mail the file to yourself or use the personal storage space on
ASULearn so that you can use this later.
Under File release on New Sketch. Under Graph
release on Show Grid.
Using Graph, plot points, precisely plot the point (-6,4).
Create a taxicab circle of radius 2 about the point (-6,4) by using
the graph/plot point
feature to create the four vertices of your taxicab circle.
Use the point tool (not the graph/plot point feature) to
create an approximate point on the boundary. Use the arrowhead tool to
select the point (-6,4) and then
the boundary point you created, and then
click on All Steps from your taxicab script tool.
Notice that the script
will now run and it will calculate the taxicab distance for you.
Move the point approximately
on the boundary to see that the taxicab distance to the
center remains the same as the horizontal and vertical distances change.
Finding All the Points Equidistant from A=(0,0) and B=(3,3) in the
Under File release on New Sketch and
nder Graph release on Show Grid.
Carefully create the point (3,3) using the Graph/plot points feature.
Use the point tool to create a
point that is NOT on the axes and is not directly
above, directly below, or directly to the left/right of A or B.
Next use your Taxicab Script tool to measure the distances from this point
to both A and
B (click on 2 points and then All Steps in the script and then redo this).
Move the point around and note the distance to A and B in order
to find all of the points that are equidistant from A and B.
Sketch a diagram of these equidistant points on a sheet of paper and
show Dr. Sarah when you think that you have found them all.
In a town having perfect square blocks and equally spaced streets running
north and south, east and west, two police stations are to be located at
A=(0,0) and B=(3,3). The town officials want to divide the town into
two precincts - Precinct 1 served by Station A and
Precinct 2 served by Station B. What are the real-life issues that would
go into deciding how the boundary should be drawn?
Explain your ideas to Dr. Sarah when you are ready.
3 Noncollinear Points
In Euclidean Geometry, 3 noncollinear points determine a unique circle.
Is this true in taxicab geometry?