### Taxicab Activities in Sketchpad

**Tools**
Arrowhead |
Point |
Compass |
Straight Edge |
Alphabet |
Script View |

**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
**Construct**.
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
Aqua.
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
highlights everything.
Use the **Script tool** to **Create New Tool** and check the
**Show Script View**
box.

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.

**Square Circles**
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
Taxicab Metric**
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.

**Precinct Problem**
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?