In this exploration,
I will look at the changes of the area and perimeter of various
triangles when the length of the sides is doubled and tripled.
I first start
out with a triangle of any size. Then form another triangle that
has sides twice as long as the first one.
By forming a script, I can
choose a triangle of any size (by choosing any three points) and
another triangle with sides twice as long will automatically be
You can try this by clicking here to open a script in
GSP. Open a new sketch, choose any three points and see what happens
when you play the script.
By choosing different types
of triangles, we can see that we appear to always be getting a
pair of triangles with a ratio of .25 from the first to the second
triangle, and a ratio of .5 for the perimeters.
What happens if
we triple the length of the sides?
Area(Polygon CFB) = 1.012 square inches
Perimeter(Polygon CFB) = 5.481 inches
Area(Polygon HGK) = 9.109 square inches
Perimeter(Polygon HGK) = 16.443 inches
Area(Polygon CFB)/Area(Polygon HGK) = 0.111
Perimeter(Polygon CFB)/Perimeter(Polygon HGK) = 0.333
You can try this yourself
by clicking here and choosing any
three points to create your first triangle. GSP will then use
a script to create a second triangle with triple the length of
As you can see, regardless
of the points you choose, the pair of triangles seems to have
a ratio of .111 for the area (from first to second triangle) and
.333 for the ratio of the perimeters.
I think this is an excellent
exploration for students. It allows them to become familiar with
GSP by determining the best way to "double" and "triple"
the length of the sides. It is also an excellent way to introduce
the scripts tool so students can try this out with several sets
of triangles. This exploration also opens the door for many further
investigations. As students begin to see a pattern emerge (the
ratio between the perimeters is an obvious one), they will form
new questions- "If I increased the lengths by five, will
the ratio of the perimeters be .20?" This directly correlates
with the NCTM's standard of students engaging in active problem