Thursday, February 21, 2013

Common Pythagorean Triples


Introduction to Pythagoras online study:

The Pythagoras Theorem is a statement relating the lengths of the sides of any right triangle.

 The theorem states that:

For any right triangle, the square of the hypotenuse
is equal to the sum of the squares of the other two sides.

Mathematically, this is written:

c^2 = a^2 + b^2

We define the side of the triangle opposite from the right angle to be the hypotenuse, c. It is the longest side of the three sides of the right triangle. The other two sides are labelled as a and b.

pythagoras theorem



Pythagoras generalized the result to any right triangle. There are many different algebraic and geometric proofs of the theorem. Most of these begin with a construction of squares on a sketch of a basic right triangle. We show squares drawn on the three sides of the triangle. For a square with a side equal to a, the area is given by:

A = a * a = a2

So the Pythagorean theorem states the area c2 of the square drawn on the hypotenuse is equal to the area a2 of the square drawn on side a plus the area b2 of the square drawn on side b.

pythagoras online study-Pythagorean triplets


A knowledge of Pythagorean triplets will also help the student in working the problems at a faster pace.

 The study of these Pythagorean triples began long before the time of Pythagoras.

There are Babylonian tablets that contain lists of such triples, including quite large ones.

There are many Pythagorean triangles all of whose sides are natural numbers. The most famous has sides 3, 4,

and 5. Here are the first few examples:

32 + 42 = 52;

52 + 122 = 132;

82 + 152 = 172;

282 + 452 = 532

There are infinitely many Pythagorean triples,that is triples of natural numbers (a; b; c) satisfying the equation a2 + b2 = c2.

If we take a Pythagorean triple (a; b; c),and multiply it by some other number d, then we obtain a new Pythagorean triple

(da; db; dc). This is true because,

(da)2 + (db)2 = d2(a2 + b2) = d2c2 = (dc)2 :

Clearly these new Pythagorean triples are not very interesting. So we will concentrate our attention on triples with no common factors.They are primitive Pythagorean triples

A primitive Pythagorean triple (or PPT for short) is a triple of numbers

(a; b; c) so that a, b, and c have no common factors1 and satisfy

a2 + b2 = c2:

There are 16 primitive Pythagorean triples with c ≤ 100:

( 3 , 4 , 5 )

( 5, 12, 13)

( 7, 24, 25)

( 8, 15, 17)

( 9, 40, 41

(11, 60, 61)

(12, 35, 37)

(13, 84, 85)

(16, 63, 65)

(20, 21, 29)

(28, 45, 53)

(33, 56, 65)

(36, 77, 85)

(39, 80, 89)

(48, 55, 73)

(65, 72, 97)

 One interesting observation in a primitive  Pythagoras triple is  either a or b must be a multiple of 3.


pythagoras online study-Solved examples


The Pythagorean Theorem must work in any 90 degree triangle. This means that if you know two of the sides, you can always find the third one.

 pythagoras solution1



In the right triangle, we know that:

c^2 = 6^2 + 8^2

Simplifying the squares gives:

                                                   c2= 36 + 64

                                                  c2 = 100    

                                                   c = 10       

                                      (taking the square root of 100)



In this example, the missing side is not the long one. But the theorem still works, as long as you start with the hypotenuse:

pythagoras solution2

                                                15^2 = a^2 + 9^2

Simplifying the squares gives:

                                                225 = a2 + 81

                                         225 - 81 = a2                 

                                                144 = a2        

                                                  12 = a  

                                                  a   = 12

                              (Notice that we had to rearrange the equation)

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