# Geometry for Elementary School/Pythagorean theorem

 Geometry for Elementary School Some impossible constructions Pythagorean theorem A proof of irrationality

In this chapter, we will discuss the Pythagorean theorem. It is used the find the side lengths of right triangles. It says:

In any right triangle, the area of the square whose side is the hypotenuse (the side of a right triangle opposite the right angle) is equal to the sum of areas of the squares whose sides are the two legs (i.e. the two sides other than the hypotenuse).

This means that if $\triangle ABC$ is a right triangle, the length of the hypotenuse, c, squared eqauls the sum of a squared plus b squared. Or:

$a^{2}+b^{2}=c^{2}\,$ Here's an example:

In a right-angled triangle, a=5cm and b=12cm, so what is c?

$a^{2}+b^{2}=c^{2}\,$ $c={\sqrt {5^{2}+12^{2}}}\,$ $c={\sqrt {25+144}}\,$ $c={\sqrt {169}}\,$ $c=13\,$ If c is not larger than a or b, your answer is incorrect. There may be a number of reasons that your answer is incorrect. The first is that you have calculated the sums wrong, the second is that the triangle you are trying to find the hypotenuse of is not a right angled triangle or the third is you have mixed up the measurements. There may be more finer points to having a wrong answer but the three stated are the most common

## Exercise

• Write about Pythagorean Theorum and its use to prove that ${\sqrt {0.5}}$ is an irrational number.