leg² + leg² = hypotenuse²
Acute angles are
complementary.
Two acute angles
leg
One right angle
leg
The Pythagorean Theorem describes the relationship
between the sides of a right triangle.
leg² + leg² = hypotenuse²
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.
3, 4 and 5 are a Pythagorean triple because 3² + 4² = 5² and all three
numbers are whole numbers.
7, 8 and 12 are NOT a Pythagorean triple because 7² + 8² = 12² even though
they are all whole numbers.
5, 9.5 and √115.25 are NOT a Pythagorean triple - 5² + 9.5² = √115.25 ²
BUT the three numbers are not whole numbers.
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
Use Pythagoras’ formula to find a
Pythagorean triple when n is an
odd number.
Find a Pythagorean triple using
Pythagoras’ formula when n = 7.
a² - 1 , a , a² + 1
4
4
Use Plato’s formula to find a Pythagorean
triple when a is an even positive
integer greater than 2.
Find a Pythagorean triple using Plato’s
formula when a = 6.
x - y , xy , x + y
2
2
Euclid’s formula won’t always give
you a Pythagorean triple. If you restrict
values of x and y to either two even
or two odd numbers in Euclid’s formula
you will at least have two whole numbers.
Find the lengths of the sides of a right
triangle if x = 7 and y = 9.
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
Use Maseres’ formula to find a Pythagorean
triple when you are given two whole numbers.
Find a Pythagorean triple using the numbers
9 and 10.
2pq
p² - q²
This is a special right triangle called a 45°-45°-90° triangle.
Why is it given this name?
This is a special right triangle because there
is a special relationship between the sides
of the triangle.
45°
10√2
Find the length of the hypotenuse
of this triangle. Simplify the radical.
10
45°
10
Check out these other 45°-45°-90° triangles. Find the length
of the missing side – simplify radicals. Can you figure
out the special relationship between the sides of a
45°-45°-90° triangle?
45°
4√2
4
7√2
7
1√2
1
1
45°
4
7
45°
9√2
9
12√2
2√2
12
45°
9
12
2
2
What is the special relationship between the lengths of the
sides of a 45°-45°-90° triangle?
45°
What is the length of each missing side of this
triangle?
leg√2
leg
45°
leg
This is a special right triangle called a 30°-60°-90° triangle.
Why is it given this name?
This is a special right triangle because there
is a special relationship between the sides
of the triangle.
Find the length of the missing leg
of this triangle. Simplify the radical.
60°
20
10
The legs of this type of triangle are
given special names.
30°
10√3
Check out these other 30°-60°-90° triangles. Find the length
of the missing side – simplify radicals. Can you figure
out the special relationship between the sides of a
30°-60°-90° triangle?
4 60°
8
60°
30°
60°
14
1
7
4√3
2
30°
1√3
30°
7√3
60°
60°
18
9
12
24
4
30°
30°
9√3
12√3
30°
2√3
60°
2
What is the special relationship between the lengths of the
sides of a 30°-60°-90° triangle?
What is the length of each missing side of this
triangle?
60°
2 (short leg)
short
leg
30°
short leg√3
COLORED NOTE CARD
Pythagorean Triples and Special Right Triangles
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
Euclid’s formula
x - y , xy
2
45°
leg√2
leg
45°
leg
, x + y
2
Maseres’ formula
2pq , p² - q² , p² + q²
short 60°
leg
2 (short leg)
30°
short leg√3