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Consider the following right triangle with altitude CD. Figure 5.18a shows this with three triangular flaps folded over the ABC. The flaps are ABC, ADC, and BDC. Each student should be working along with you as you develop this demonstration.

D
B
C
A
Figure 5.18a

Notice that ADC ? CDB ? ACB. In Figure 5.18a, ADC and
CDB are folded over ACB. So clearly AreaADC + AreaCDB = AreaACB. If we unfold the triangles (including the ACB itself), we
get the following (see Figure 5.18b) that shows that the relationship of the similar polygons (here right triangles) is the extension of the Pythagorean theorem:

The sum of the areas of similar right triangles on the legs of a right
triangle is equal to the area of the similar right triangle on the
hypotenuse of the triangle.

This essentially “proves” the Pythagorean theorem by paper folding!

P’
Q’
Q’
Q
P’
P
C’
D’
D” D
C
B
A
Figure 5.18b

Taken From :Math Wonders to inspire teacher and student

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The following technique addresses those students who fall in between the two extremes of the good visualiser and the student who has no visual capacity at all.
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Notice that each time a square is taken from a Golden Rectangle the
resulting rectangle is also a Golden Rectangle.

Once they have drawn the above figure, students ought to be encouraged to draw quarter circular arcs, as shown in Figure 5.11c. The resulting figure approximates a logarithmic spiral.

P
S R
K
H L Q
T
N J
G
M F D
B
A
E
Figure 5.11c

We can locate the vanishing point of the spiral by drawing the diagonals of the two largest Golden Rectangles, as shown in Figure 5.11d.

P
S R
K
H L Q
T
N J
G
M F D
B
A
E
Figure 5.11d

Students who construct this figure accurately will see that BD and GF
contain the diagonals of the other Golden Rectangles as well. Moreover, BD?GF .

A similar spiral can be drawn by locating the centers of each of the squares in succession from largest to smallest and drawing (see Figure 5.11e).

S
R
L N M
K
J
G H
F
E
D
B C
A
Figure 5.11e
There is no end to the beauty of this rectangle!

Taken From :Math Wonders to inspire teacher and student

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One of the most celebrated relationships in mathematics is the Pythagorean theorem. Why is this so much in the minds of adults, who usually remember this above all else learned in school mathematics? Could this be because we usually refer to the theorem with the first three letters of the alphabet, and it is like learning your ABCs? Whatever makes it popular, it still requires a proof for us to be able to accept it as a theorem. This unit presents a very simple proof that you might want to use instead of the one usually provided in the textbook. Because it is rather simple, you may want to present it out of context a bit earlier than usual. You must be cautioned that it does depend on a theorem about the chord of a circle that must be presented first.

Clearly, the Pythagorean theorem is the basis for much of geometry and all of trigonometry. For this reason, be careful about discovering a new proof to make sure that it is not based on a relationship established by the Pythagorean theorem itself. Such is the case with trigonometry. No proof of the Pythagorean theorem can use trigonometric relationships because they are based on the Pythagorean theorem—a clear case of circular reasoning. Students should be clear on what is meant by circular reasoning.

Now to the proof. It is very simple, but it is based on a theorem that states that when two chords intersect in the circle, the product of the segments of one chord is equal to the product of the segments of the other chord. In the figure below, this would mean that for the two intersecting chords:

p
q = r
s.
s
q
r
p

Now consider the circle with diameter AB perpendicular to chord CD:
c+b

a
c
b
c-b
a
c
D C
A
O
B
E

From the theorem stated above, c ? b c + b = a2. Then c2 ? b2 = a2 and, therefore, a2 + b2 = c2. The Pythagorean theorem is proved again. Although there have been many proofs after its publication, a nice collection of 370 proofs of the Pythagorean theorem is by Elisha S. Loomis.? This is a lovely resource for all mathematics teachers.

? A classic source for 370 proofs of the Pythagorean theorem is E. S. Loomis, The Pythagorean Proposition (Reston, VA: NCTM, 1968).

Taken From :Math Wonders to inspire teacher and student

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