PYTHAGOREAN THEOREMS  Contact Us At Contra Costa College  Math Dept. Home Page 

What's this? See below. 
Pythagorean TheoremSSome "Not So Familiar" Implications 
The Pythagorean Theorem states that for a right triangle
with legs a and b and hypotenuse c, we have
the Pythagorean Equation:
a^{2} + b^{2} = c^{2}
If we think of this geometrically, we are saying that the sum
of the areas of the squares drawn on the two legs, as shown to the left,
is equal to the area of the square drawn on the hypotenuse.
This is the traditional Pythagorean result, both algebraically and
geometrically. 
True for Regular Hexagons. 
However, still thinking in geometric terms, we can construct any set of 3 regular
polygons (triangles, pentagons, hexagons, etc.) using a and
b and hypotenuse c, respectively, as the common sides of the three polygons.
And still, the sum of the areas of the figures drawn on the two legs is equal to
the area of the figure drawn on the hypotenuse. 
True for Pythagorean Similar Triangles. 
Further, the Pythagorean result is valid for similar figures so
constructed on the legs and the hypotenuse. They don't have to be
only regular polygons. However, the two controlling dimensions of
of the figure that determine its area, must satisfy what we will call
the Pythagorean Similarity Condition. That is, if we consider the
leg a, then the two controlling dimensions of the figure
constructed on leg a might be xa and ya, some
real multiples of a. The Similarity Condition is that these
same multiples of b and c must also exist in the
controlling dimensions of those figures, xb, yb, xc
and yc. Thus, the areas of the three figures are kxaya,
kxbyb and kxcyc. k is just another factor, it could
be 1 or pi or any other real number. The result is that the
Pythagorean Equation is just multiplied by kxy:
(kxy)a^{2} + (kxy)b^{2} = (kxy)c^{2}

True for similar Pentagrams. 
The three pentagrams at the top of this page are Pythagorean Similar. 
True for Circles & Semicircles. 
The sum of the areas of the circles drawn on the two legs is equal to
the area of the circle drawn on the hypotenuse.
(Can you write the appropriate equation?) Also, related to these Pythagorean circles, see Problem 4 below in the Problem Section. 
What about Buffalos? 
These buffalos are of the Buffalo Nickel fame. However, these three are also
Pythagorean Buffalos in that they are Pythagorean Similar. Click below to see the Pythagorean relationship and a gallery of other familiar figures presented as Pythagorean Similar. Pythagorean Gallery 
True for Boxes. 
Suppose we extend the Pythagorean figure into the third dimension. Then we have ma^{2} + mb^{2} = mc^{2}
and that is neat! 
PROBLEM SECTION:
Problem 1: Binomial Expansions &
The Pythagorean Theorem Try Problem 1. (for more about this topic see the link : Pentagons & The Pythagorean Theorem near the bottom of this page ) 
Problem 2: Constructing a Square Equal To The Sum of Two Given Squares Try Problem 2. 
The Following Problems Are Related To Problem 2: Makes use of the geometric mean of two numbers. Try Problem 2a. Problem 2b: Constructing a Square Equal To The Area of a Given Triangle Try Problem 2b. Problem 2c: Constructing a Square Equal To The Area of a Given Pentagon Try Problem 2c. 
Problem 3: Sectional Pythagorean Theorem Try Problem 3. 
Problem 4: Pythagorean Similar Rectangles Try Problem 4. 
Problem 5: Circle Pythagorean Theorem (Lunes) Try Problem 5. 
Problem 6: A Trapezoid Pythagorean Theorem Try Problem 6. 
Problem 7: A Pythagoragram
Try Problem 7.

When we examine the unit circle, the x and y coordinates of points on that circle
are equal to the sine and cosine of the angle the radius makes with the positive
xaxis.
x^{2} + y^{2} = 1
®
COS^{2}( q ) + SIN^{2}( q )
= 1
The latter equation is known as the Pythagorean Identity in trigonometry.
Also, by substitution,
COS^{2}( q ) + COS^{2}( b )
= 1
(Notice that b is also the angle that the radius line makes
with the yaxis. For future reference, these angles are called the direction angles
of the radius line and COS( q ) and COS( b )
are called the direction cosines of the radius line.)
Now we can multiply the above equations by r^{2}
(r COS(q ))^{2} + (r SIN(q ))^{2}
= r^{2}
and
x^{2} + y^{2} = r^{2}
the equation for a circle in Cartesian coordinates as a direct result
of the Pythagorean Theorem.
(See below, the discussion of the
Unit Sphere for more on direction angles and direction cosines. )
In special circumstances, when the triangle is not a right triangle, we can talk about parallelograms and rhombuses instead of squares and create a special 'Pythagorean Relationship'. See the figures below:


In the second figure above (stretched), we no longer have squares on the sides
of the triangle. We have parallelograms (rectangles, in this case) and
a rhombus. Still, the Pythagorean relationship holds, the sum
of the areas of the rectangles drawn on the two legs
is equal to the area of the rhombus drawn on the hypotenuse of the
right triangle. But the triangle may not be a right triangle under
other transformations. See the next two figures:


In these figures the Pythagorean relationship still holds, the sum of the areas of the parallelograms drawn on the two sides is equal to the area of the parallelogram drawn on the third side of the triangle.
In these special circumstances the area of the "red" figure is equal to twice
the area of one of the "yellow" figures (the yellow figures have
equal areas) because of the 3 by 3 grid we produced in the original
set up.
Try this: Draw an equilateral triangle, set up a 3 by 3 grid work around it, and show that the areas of rhombuses drawn on two of the sides of the triangle is equal to the area of the larger parallelogram drawn on the third side. 
If the triangle is not a right triangle, then the Pythagorean Equation no longer holds.
But, it can be brought back into balance by including one more term in the equation,
2ab*Cos( g) ,
a^{2} + b^{2} = c^{2} + 2ab*Cos( g )
where g is the angle in the triangle opposite side c.
This equation is known as (one of the) Law of Cosines. The other equations in the Law of Cosines
are variations wherein the side and the angle opposite it are chosen from any of the three
possibilities for the triangle.
So for any triangle, the sum of the areas of the squares on two of the sides of the triangle is equal to the area of the square on the third side plus (or minus) some percentage of twice the area of a rectangle, that is 2ab, with sides equal to the first two sides of the triangle!(The percentage mentioned above being the value given by COS(g) which varies from 0 to ±100%
Extended To Boxes. 
Pythagorean Theorem Extended to 3 Dimensions
In the same way we might cut off the corner of a rectangle, making a right triangle, we might cut off the corner of a box, making a right tetrahedron . And just as
a^{2} + b^{2} = c^{2}
A^{2} + B^{2} + C^{2} = D^{2}
and that is really neat!


A Pythagorean Quadruple Problem. 
A Pythagorean Quadruple Problem 
Diagonal of Box. 
Pythagorean Theorem For Diagonal of a Box
Given the dimensions of a box are a, b, and c. Since a^{2} + b^{2} = e^{2} and e^{2} + c^{2} = d^{2} we have by substitution a^{2} + b^{2} + c^{2} = d^{2} Thus, the square of the diagonal of a box is equal to the sum of the squares of
the dimensions of the box. 
Check the problem section above for "Sum Of The Squares Of The Sides Of The Trapezoid ... Problem" as it relates to this figure. Problem Section 
Note on the Unit Sphere:
Just as in the case of the Unit Circle
(See above, the discussion of the
Unit Circle )
a radius line in 3 dimensions to the surface of a Unit Sphere, makes angles with
each of the positive coordinate axes, x, y, and z.
Call these angles a, b, g,
respectively. Then the Pythagorean relationship is
COS^{2}( a ) + COS^{2}( b )
+ COS^{2}( g )
= 1
The angles are called the direction angles of the radius line and
COS( a ), COS( b ), and
COS( g ) are called the direction cosines
of the radius line.
If you draw a picture of Unit Sphere and the radius to the point
(x, y, z) on the surface, the direction cosines are equal to x, y and z, respectively. Furthermore,
they are the edges of a box inside the sphere and the radius line is its diagonal.
And consequently,
x^{2} + y^{2} + z^{2} = r^{2}
is the equation for a sphere in Cartesian coordinates, a direct result
of the Pythagorean Theorem.
Can you find the length of the diagonal of a unit cube? a unit square?
http://mathworld.wolfram.com/PythagoreanTriple.html http://mathworld.wolfram.com/PythagoreanQuadruple.html 
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This plot shows points with integer coordinates (a, b), plotted in the xyplane where [a, b, Ö(a^{2} + b^{2})] is a Pythagorean Triple. 

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Bill Casselman's feature article http://plus.maths.org/issue16/features/perigal/ 
send comments to Thomas M. Green c/o CCC Math Dept.
hwalters@contracosta.cc.ca.us
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