Pythagorean Theorem Inequality
Used to classify triangles by angles
Longest side² < short side² + short side² - ACUTE triangle
Longest side² = short side² + short side² - RIGHT triangle
Longest side² > short side² + short side² - OBTUSE triangle
The Pythagorean Theorem describes the relationship
between the sides of a right triangle.
leg² + leg² = hypotenuse²
or
short side² + short side² = long side²
A Pythagorean triple is a set of integers, a, b, and c,
that could be the sides of a right triangle if a² + b² = c².
If not, then what kind of triangle is it? Or, is it not a triangle at all?
7, 24, 25
8, 9, 10
60, 11, 61
73, 19, 18
81, 1, 82
33, 42, 9
Many mathematicians over the centuries have
developed formulas for generating side lengths for
right triangles. Some generate Pythagorean triples,
others just generate the side lengths for a right
triangle.
Masères
n , n² - 1 , n² + 1
2
2
Of course today we particularly remember
Pythagoras for his famous geometry
theorem. Although the theorem, now
known as Pythagoras's theorem, was
known to the Babylonians 1000 years
earlier he may have been the first to prove
it.
Number rules the universe.
-Pythagoras
n , n² - 1 , n² + 1
2
2
Find the sides of a Pythagoras
triangle if n = 3.
3, 4, 5
Find the sides of a Pythagoras triangle
if n = 2.
2, 3/2, 5/2
Pythagoras,
contorniate
medallion engraved
between AD 395
and 410
Why might you want to restrict n to odd positive integers in
Pythagoras’s formula?
a² - 1 , a , a² + 1
4
4
It was claimed that Plato's real name was
Aristocles, and that 'Plato' was a nickname
(roughly 'the broad') derived either from the
width of his shoulders, the results of training
for wrestling, or from the size of his forehead.
Although Plato made no important
mathematical discoveries himself, his belief
that mathematics provides the finest training
for the mind was extremely important in the
development of the subject. Over the door of
the Academy was written:Let no one unversed in geometry enter here.
a² - 1 , a , a² + 1
4
4
Find the sides of a Plato
triangle if a = 4.
3, 4, 5
Find the sides of a Plato triangle if
a = 7.
11.25, 7, 13.25
Why might you want to restrict values of a to even positive
integers greater than 2 in Plato’s formula?
x - y , xy , x + y
2
2
Euclid's most famous work is his treatise on
mathematics The Elements. The book was a
compilation of knowledge that became the
centre of mathematical teaching for 2000 years.
Probably no results in The Elements were first
proved by Euclid but the organisation of the
material and its exposition are certainly due to
him.
x - y , xy , x + y
2
2
Find the sides of a Euclid triangle if
x = 3 and y = 1.
1, 3 , 2
Find the sides of a Euclid triangle if
x = 10 and y = 4.
3, 40 , 7
Find the sides of a Euclid triangle if x = 5 and y = 2.
3/2, 10 , 7/2
Why might you want to restrict values of x and y to either even
or odd numbers in Euclid’s formula?
2pq , p² - q² , p² + q²
Maseres wrote many mathematical works which
show a complete lack of creative ability. He rejected
negative numbers and that part of algebra which is
not arithmetic. It is probable that Maseres rejected
all mathematics which he could not understand.
Masères
2pq , p² - q² , p² + q²
Find the sides of a Masères triangle 2pq
if p = 4 and q = 1.
8, 15, 17
Find the sides of a Masères triangle
if p = 2.6 and q = 1.5.
7.8 , 4.51, 9.01
p² - q²
What restriction would you impose on values for p and q in
Masères’ formula?
COLORED NOTE CARD
Finding Pythagorean Triples
Pythagorean Triple - A set of three whole numbers such that a² + b² = c²
Pythagoras’ formula
Plato’s formula
n² + 1
a²
a²
n² - 1
n ,
-1 , a ,
+1
,
2
4
4
2
-use odd positive integers
-even positive integers greater
than 2
Euclid’s formula
Maseres’ formula
x - y , xy
2
, x + y
2
-both even or both odd, not always
a triple
2pq , p² - q² , p² + q²
-Whole numbers
. . . one number equal to 16.
. . . one number equal to 17.
. . . the numbers 9 and 7.
. . . the numbers 5 and 6.