Pythagoras' Theorem - A Proof
What is Pythagoras' theorem?
Pythagoras' Theorem states that for any right-angled triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
Put algebraically, using our diagram above where a and b are our two perpendicular sides and c is the hypotenuse, we get a^2 + b^2 = c^2.
Pythagoras' Theorem has many important applications across mathematics from simple geometry through to trigonometry and can even be used with n-dimensional solids!
It was known by the ancient Babylonians and Egyptians as far back as 1900 BC and Pythagorean triples (whole numbers that satisfy the equation such as 32 + 42 = 52 and 52 + 122 = 132) can be found on the Plimpton 322 tablet, a Babylonian clay tablet dating from approximately 1800 BC. It is also believed that ancient Egyptian builders may have used rope with equally spaced knots and the knowledge that a triangle with sides of 3, 4 and 5 is right-angled to ensure the accuracy of right-angles when constructing the pyramids.
Pythagoras' theorem was also discovered independently in other cultures around the world such as Mesopotamia, China and India, however it is the ancient Greeks who are most commonly associated with this theorem.
Put algebraically, using our diagram above where a and b are our two perpendicular sides and c is the hypotenuse, we get a^2 + b^2 = c^2.
Pythagoras' Theorem has many important applications across mathematics from simple geometry through to trigonometry and can even be used with n-dimensional solids!
It was known by the ancient Babylonians and Egyptians as far back as 1900 BC and Pythagorean triples (whole numbers that satisfy the equation such as 32 + 42 = 52 and 52 + 122 = 132) can be found on the Plimpton 322 tablet, a Babylonian clay tablet dating from approximately 1800 BC. It is also believed that ancient Egyptian builders may have used rope with equally spaced knots and the knowledge that a triangle with sides of 3, 4 and 5 is right-angled to ensure the accuracy of right-angles when constructing the pyramids.
Pythagoras' theorem was also discovered independently in other cultures around the world such as Mesopotamia, China and India, however it is the ancient Greeks who are most commonly associated with this theorem.
A bust of Pythagoras in the Vatican Museum, Rome
Pythagoras of Samos
The theorem is named after the ancient Greek mathematician and philosopher Pythagoras (c 569 - 495 BC). Although earlier civilizations where aware of aspects of the theorem, it is Pythagoras who is credited with the first proof of the theorem, although no evidence of this proof remains. Interestingly, as the Greeks were much more adept at geometry than they were at algebra, Pythagoras would not have thought of the theorem as an algebraic one, but instead as a triangle with a square attached to each side, where the areas of the two smaller squares added up to the area of the larger square on the hypotenuse.
Little reliable evidence about Pythagoras' life exists today and when researching him, it is very difficult to separate fact from fiction. Many mathematical and scientific discoveries are attributed to him such as irrational numbers, the regular solids, and of course the theorem that bears his name, but again, it is difficult to know what was actually Pythagoras and what was discovered by his followers and pupils.
Little reliable evidence about Pythagoras' life exists today and when researching him, it is very difficult to separate fact from fiction. Many mathematical and scientific discoveries are attributed to him such as irrational numbers, the regular solids, and of course the theorem that bears his name, but again, it is difficult to know what was actually Pythagoras and what was discovered by his followers and pupils.
Proving Pythagoras' theorem
There are many ways to prove Pythagoras' Theorem and in this article we are going to use a quick, concise one which uses geometry and some simple algebra.
To begin with look at the diagram below which consists of a large square with a smaller square inside it, angled to create four right-angled triangles around its edges.
To begin with look at the diagram below which consists of a large square with a smaller square inside it, angled to create four right-angled triangles around its edges.
Two squares, one inside the other
If we label one of the triangles so that the perpendicular sides are a and b, and the hypotenuse is c, we can quickly see that the remaining lengths in the diagram must also all be a, b and c.
This can be demonstrated by using the fact that angles in a triangle and angles on a straight line both add up to 180°. Using these facts we can quickly see that the angles in the triangles are all the same, hence the four triangles are similar. Furthermore, as each triangle has a side of the smaller square as a hypotenuse, the hypotenuses are all the same length, hence the four triangles must all be congruent (identical in size).
Using this fact, we can now label all of the lengths with a, b or c.
This can be demonstrated by using the fact that angles in a triangle and angles on a straight line both add up to 180°. Using these facts we can quickly see that the angles in the triangles are all the same, hence the four triangles are similar. Furthermore, as each triangle has a side of the smaller square as a hypotenuse, the hypotenuses are all the same length, hence the four triangles must all be congruent (identical in size).
Using this fact, we can now label all of the lengths with a, b or c.
Square diagram with labelled edges
Pythagoras' theorem proof - The algebra bit
We are now going to calculate the area of the large square (you will see why soon).
Each side is of length a + b, so we get:
Area of large square = (a + b)^2
= a^2 + b^2 + 2ab
We can also express the area of the large square as the areas of the small square (c^2) and the four triangles (1/2 × ab).
Area of large square = c^2 + 4 ×1/2 × ab
= c^2 + 2ab
As these two expressions are both the area of the large square they must be equal to each other so:
a^2 + b^2 + 2ab = c^2 + 2ab
and cancelling 2ab from each side leaves us with:
a^2 + b^2 = c^2
which is Pythagoras' theorem. Proof complete.
Each side is of length a + b, so we get:
Area of large square = (a + b)^2
= a^2 + b^2 + 2ab
We can also express the area of the large square as the areas of the small square (c^2) and the four triangles (1/2 × ab).
Area of large square = c^2 + 4 ×1/2 × ab
= c^2 + 2ab
As these two expressions are both the area of the large square they must be equal to each other so:
a^2 + b^2 + 2ab = c^2 + 2ab
and cancelling 2ab from each side leaves us with:
a^2 + b^2 = c^2
which is Pythagoras' theorem. Proof complete.
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