InterMath | Workshop Support | Write Up Template
Title
Medial Polygons
Problem Statement
Take any triangle ABC. Construct a triangle connecting the three midpoints of the sides,
which is called the medial triangle. Investigate the relationships (perimeter and area)
between the medial triangle and the original triangle. What conjectures can you make? Can
you prove them?
Problem setup
What are the relationships between the perimeter and area of a medial triangle and the original
triangle?
Plans to Solve/Investigate the Problem
I will use Geometer Sketchpad (GSP) to create a triangle. I will construct the midpoints of each
side and connect them to create a medial triangle. I will calculate the perimeter and area of these
two triangles.
Investigation/Exploration of the Problem
Here is a copy of the GSP sketch:
Perimeter ABC = 9.30 cm
Area ABC = 3.98 cm2 A
Perimeter DEF = 4.65 cm
F
Area DEF = 1.00 cm2
E
Perimeter ABC C
= 2.00
Perimeter DEF D
Area ABC B
= 4.00
Area DEF
The perimeter of the original triangle is two times that of the medial triangle. The area is four
times as large, which is the same as 2^2. This may come into play with the extension.
Extensions of the Problem
Explore the above idea using various polygons. Include convex and nonconvex polygons in
your explorations.
Perimeter ABCD = 12.03 cm Perimeter ABCD = 11.77 cm
A A
Area ABCD = 3.45 cm2 Area ABCD = 7.31 cm2
H E E
Perimeter EFGH = 5.60 cm Perimeter EFGH = 8.04 cm H
B B
Area EFGH = 1.72 cm2 D Area EFGH = 3.66 cm2
D
Perimeter ABCD F Perimeter ABCD F
= 2.15 G = 1.46 G
Perimeter EFGH Perimeter EFGH
Area ABCD C Area ABCD C
= 2.00 = 2.00
Area EFGH Area EFGH
As shown, there is no distinct relationship with the perimeter of a quadrilateral, but the area
is twice as large as the medial for convex and concave quadrilaterals.
Perimeter ABCDE = 8.36 cm Perimeter ABCDEF = 9.11 cm
Area ABC DE = 2.52 cm2 A Area ABCD EF = 5.23 cm2 A
L
Perimeter FGHIJ = 5.21 cm H G Perimeter GHIJKL = 7.84 cm F G
K B
Area HGFJI = 1.66 cm2 F B Area GHIJKL = 3.93 cm2
E C E
Perimeter ABCDE Perimeter ABCDEF J
H
= 1.60 I J = 1.16
Perimeter FGHIJ Perimeter GHIJKL D I C
Area ABC DE D Area ABCD EF
= 1.52 = 1.33
Area HGFJI Area GHIJKL
There are no distinct relationships of pentagons or hexagons. Though, the ratio of
perimeters and areas gets smaller as the number of sides increase.
Author & Contact
Nicole McDowell
nmcdowell@rockdale.k12.ga.us
Link(s) to resources, references, lesson plans, and/or other materials
The Sierpinski Triangle -
This java applet constructs medial triangles inside an equilateral triangle in the number of
iterations you specify.
[ java applet ]
http://math.rice.edu/~lanius/fractals/sierjava.html
Fractals in Pascal's Triangle -
This lesson plan examines coloring patterns in Pascal's triangle that generate Sierpinski's
gasket.
[ acrobat pdf ]
http://explorer.scrtec.org/explorer/explorer-db/rsrc/820889816-81ED7D4C.2.PDF