Math Genius’s “The Pythagorean Theorem” by Egrimm Lyrics
The Pythagorean theorem is a relation rooted in Euclidean Geometry, which offers the following postulate given any right triangle:
a² + b² = c²
a and b refers to the legs of the right triangle and c refers to the hypotenuse. This equation is referred to as the Pythagorean equation.
The theorem is named after the Greek philosopher/mathematician, Pythagoras, though further research shows he was certainly not the first one to recognize the relationship between the sides. With the Pythagorean equation came proofs from various mathematicians to prove that the theorem worked. Below are three famous examples, each explained:
The Pythagoras Proof
Proof by rearrangement
Euclid's Proof
However, consider the original statement regarding the theorem by Pythagoras:
"The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides."
Visually, this means:The biggest consequence of this is that the length of any of the sides of a given triangle won't be rational unless c² is equal to a perfect square. When that happens, a Pythagorean triple is formed, such that a and b are integers, and c² is a perfect square. The following are examples of Pythagorean triples, all with values less than 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)
The Pythagorean triples can be multiplied and still be a Pythagorean triple. For instance, the double of the triple (3, 4, 5) is (6, 8, 10).
a² +b² =c² (Pythagorean theorem)
6² +8² =10² (Substitution property)
36+64=100 (Multiplication property)
100=100 (Addition property/Reflexive property)
a² + b² = c²
a and b refers to the legs of the right triangle and c refers to the hypotenuse. This equation is referred to as the Pythagorean equation.
The theorem is named after the Greek philosopher/mathematician, Pythagoras, though further research shows he was certainly not the first one to recognize the relationship between the sides. With the Pythagorean equation came proofs from various mathematicians to prove that the theorem worked. Below are three famous examples, each explained:
The Pythagoras Proof
Proof by rearrangement
Euclid's Proof
However, consider the original statement regarding the theorem by Pythagoras:
"The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides."
Visually, this means:The biggest consequence of this is that the length of any of the sides of a given triangle won't be rational unless c² is equal to a perfect square. When that happens, a Pythagorean triple is formed, such that a and b are integers, and c² is a perfect square. The following are examples of Pythagorean triples, all with values less than 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)
The Pythagorean triples can be multiplied and still be a Pythagorean triple. For instance, the double of the triple (3, 4, 5) is (6, 8, 10).
a² +b² =c² (Pythagorean theorem)
6² +8² =10² (Substitution property)
36+64=100 (Multiplication property)
100=100 (Addition property/Reflexive property)