In this tutorial, we use GeoGebra
to explore the minimum sum of the distances of a point on the interior of a
triangle to its sides. We learn how to use the Point on
Object tool and the Perpendicular Line tool. We also learn how to compute using
the Input bar. The final output of this tutorial is shown in the following
applet.
Steps in Creating the Applet
No

Instruction

1

Using the Polygon
tool, construct triangle ABC.

2

Select the Point
on Object tool,
and then click the interior of the triangle to create point D.

3

Move point D and
observe what happens. Can you move point D outside
the triangle?

4

Next, we create a line perpendicular to AB passing
through D.
To do this, select the Perpendicular Line tool,
select segment AB,
and then select point D.

5

Create two more lines passing through D,
one perpendicular to AC and
the other perpendicular to BC.

6

Using the Intersect Two Objects tool, intersect the lines and the perpendicular
segments. Your drawing should look like the figure below.

7

Hide the three lines by right
clicking each line and clicking them and selecting Show Objects.

8

Using the Segment
between Two Points tool, construct segments DE, DG,
and DF.

9

To find the total distance t of
points D from E, F,
and G,
type t = g + h + i,
and then press the ENTER key.

10

Move the points to explore and observe the value
of t.

Application
A building is going to be
constructed with a triangle bounded by the roads connecting Buildings A, B, and
C. A connecting pathway is to be constructed from the three roads to the
building. Find the location of the building that will minimize the cost of
pathway construction.
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