Thursday, January 29
Puzzles and Polygons [1.2]
- Dissections, cut and paste methods of measurement.
- Cutting and reassembling polygons.
- Tangrams.
- Tangram Activities last class
Tangoes: a commercial game based on tangrams
Tapestry Project from previous Math 103 students.
Cutting and reassembling polygons.
Convex: Any two points in the figure have
a line segment connecting them. If that line segment is always inside the figure, then the figure
is called "convex".
Making Dissection Puzzles:
- Dissections
(Junkyard)
- Equidecomposable
polygons (translation from Portugese)
Where we are going:
* Scissors congruence: A sc= B means figure A can be
cut into pieces that can be reassembled to form figure B.
This is also described using the word "equidecomposable".
"A and B are equdecomposable to B." - SC
= is a reflexive, symmetric, and transitive relation. [like congruence
and similarity in geometry and equality in arithmetic]
- Theorem I : A sc= B implies Area(A) = Area(B)
- Theorem II [The converse of Theoerm I!]: Area(A) = Area(B)
implies A sc= B !!
- The presidential puzzles: Washington, ..., Jefferson,...,Lincoln,
... ,Clinton, Bush II.
- Simple cases as evidence
and a foundation for building toward the proof of Theorem II.:
- A triangle is SC to
a rectangle. Activity:
- A rectangle is SC to a square.
radius = r = (a+b)/2 #
- length of DC = a- r
= r - b
- x^{2} + (CD)^{2} = r^{2}
- x^{2} = r^{2} - (CD)^{2} [Revised! 2-3-04]
=r^{2}
- (r-b)^{2}
= r^{2} - (r^{2} -2rb+ b^{2})
= 2rb - b^{2}
= (2r-b)*b [Now notice that from # 2r-b = a!]
= ab
- Two squares are SC to a single square.
- A polygon is SC to
a square.
- Triangulation
.: Any polygom can be decomposed into triangles!
- If two polygons have equal area, then they are
SC to the same square!