One of the angles of a right triangle is always equal to 90 degrees. This angle is the right angle. The two sides next to the right angle are called the legs and the other side is called the hypotenuse. The hypotenuse is the side opposite to the right angle, and it is always the longest side.
The Pythagorean theorem says that the area of a square on the hypotenuse is equal to the sum of the areas of the squares on the legs. In this picture, the area of the blue square added to the area of the red square makes the area of the purple square. It was named after the Greek mathematician Pythagoras:
If the lengths of the legs are a and b, and the length of the hypotenuse is c, then, .
There are many different proofs of this theorem. They fall into four categories:
- Those based on linear relations: the algebraic proofs.
- Those based upon comparison of areas: the geometric proofs.
- Those based upon the vector operation.
- Those based on mass and velocity: the dynamic proofs.
Proof[change | edit source]
The proof uses three lemmas:
- Triangles with the same base and height have the same area.
- A triangle which has the same base and height as a side of a square has the same area as a half of the square.
- Triangles with two congruent sides and one congruent angle are congruent and have the same area.
The proof is:
- Blue triangle has the same area as the green triangle, because it has the same base and height (lemma 1).
- Green and red triangles both have two sides equal to sides of the same squares, and an angle equal to a straight angle (an angle of 90 degrees) plus an angle of a triangle, so they are congruent and have the same area (lemma 3).
- Red and yellow triangles' areas are equal because they have the same heights and bases (lemma 1).
- Blue triangle's area equals area of yellow triangle's area, because
- Brown triangles have the same area for the same reasons.
- Blue and brown each have a half of the area of a smaller square, sum of their areas equals half of the area of the bigger square, so halves of the areas of small squares are the same as a half of the area of the bigger square, so their area is the same as the area of the bigger square.
Proof using similar triangles[change | edit source]
From the image . And by replacing equations (1) and (2):
Multiplying by c:
Pythagorean Triplets[change | edit source]
Pythagorean Triples are three whole numbers which meet the equation .
One well known example is the 3-4-5 triangle: if a=3 and b=4, or . This can also be shown as
The three-four-five triangle works for any multiples of 3, 4, and 5; such as 6, 8, and 10; or 30, 40 and 50. Another, less-often used example is the 12-5-13 triangle.
References[change | edit source]
- Loomis, Elisha S. 1927. The Pythagorean proposition: its proofs analysed and classified and bibliography of sources. Cleveland, Ohio.