Brain Teaser:
Show that the red and blue areas in the first picture are equal:
Answer:
Use the second picture above to follow the proof. BD divides parallelogram
ABCD in two equal halves: triangles ABD and BCD. The smaller, similar
parallelograms DiEh and EfBg are also bisects by BD. This tells us that
EfB and EgB are equal in area and that DiE and DhE are also equal in area.
From this information, we can determine the area of AfEi and EgCh in terms
of the above triangles: AfEi = ABD  EfB  DiE, and EgCh = BCD  EgB  EhD.
Using substitution we can see that the area of AfEi and EgCh are equal.
180 Degrees in a Triangle
Let students draw triangles of any form on pieces of paper. Have
them tear of the corners of their triangle and line them up side
by side as shown below. Because the sum of the three angles is always
180 degrees, the bottom side will always be a 180 degree angle, or a
straight line.
Pythagorean Theorem Proof
Notice that the area of the square is (a+b)^2, and that the area of the
central square is c^2. Using the results from the brain teaser above it can
be shown that the two colored areas in the second picture are equal.
Choose the dimensions of the second square such that the area is
also (a+b)^2. If you remove the same four triangles from the
second square, you get two squares, one of size a^2 and one of size b^2.
Since the areas of the squares are the same and the areas of the four triangles
are the same, there is an equal area uncolored in each square.
So, the area a^2 plus b^2 is equal to the area of c^2.
