2. Triangles
● When you make a
triangle with side a, b
and c it will be a right
angled triangle.
● C is the longest side of
the triangle called the
hypotenuse (hi-pot-en-
noose)
● A and B are the other
two sides
5. Euclid’s proof that there are
infinitely many Pythagorean Triples
The proof was based on the fact that the
difference of the square of any two consecutive
numbers is always an odd number.
Examples:
2² – 1²= 4 -1=3 (an odd number)
15² – 14² = 225 - 196 =29 (an odd number)
6. Properties
It can be observed that a Pythagorean triples always
consists of:
● All even numbers, or
● Two odd numbers and an even number.
A Pythagorean Triple can never be made up of all odd
numbers or two even numbers and one odd number. This
is true because:
● The square of an odd number is an odd number and the
square of an even number is an even number.
● The sum of two even numbers is an even number and the
sum of an odd number and an even number is in odd
number.
7. Constructing Pythagorean Triples
It is easy to construct
sets of Pythagorean
Triples.
When m and n are
any two positive
integers (m<n):
●
a=n²-m²
●
b=2nm
●
c=n²+m²
Then, a, b, and c form a
Pythagorean Triple.
Examples:m=1 and n=2
●
A=2²-1²=4-1=3
● B=2*2*1=4
●
C=2²+1²=5
Thus, we obtain the first
Pythagorean triples (3, 4, 5)
● Similarity, when m=2 and
n=3 we get the next
Pythagorean Triple (5, 12,
13)
8. List of Pythagorean triples less than
1,000
The list only contains the first set (a, b, c) which is
a Pythagorean triple (Primitive Pythagorean
Triples). The multiple of (a, b, c), (ie.(na, nb, nc)),
which also form a Pythagorean Triple are not
given in the list. For example, it has already been
seen that (3, 4, 5) is a Pythagorean Triple and so
is (6, 8, 10). However (6, 8, 10) is obtained by
multiplying (3, 4, 5) by 2. Hence, only (3, 4, 5)
would be shown.
