# Pythagorean theorem

The sum of the areas of the two squares on the legs ($a$ and $b$) equals the area of the square on the hypotenuse $c$.

We can imagine something like this: Image credit: Wikimedia Commons user AmericanXplorer13

### Egyptian triangle

The right Triangle whose length of sides is equal to $3, 4$ and $5$ is called the Egyptian triangle.

Let’s forget that we know that the length of hypotenuse is equal to $5$ and try to calculate it using Pythagorean theorem.

In a right angled triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides.

And it is valid in different order:

If the square of the hypotenuse is equal to the sum of the squares of the other two sides the triangle is a right angled.

We have: $\ c^2 = a^2 + b^2$

Where $a = 4$, and $b = 3$.

$c^2 = 4^2 + 3^2$
$c^2 = 16 + 9$
$c^2 = 25$
$c=5$

If we know lengths of any two sides of a right triangle, we can use this theorem to quickly find the third. $c^2 = a^2 + b^2$

Where $c = 10$, and $b = 83$.

$a^2 = c^2 – b^2$
$a^2 = 100 – 64$
$a^2 = 36$
$a = 6$

If we have any other triangle, we can also use Pythagorean theorem. If we have length of $b$ and $v_a$, and $a=12$ we can find out other lengths in these triangle:

$v_a=6cm$

$b=10cm$

_______

$c=?$

$b^2=v_a^2+x^2$

$x^2=b^2-v_a^2$

$x^2=100-36$

$x^2=64$

$x=8cm$

Now, we can calculate:

$a^2=v_a^2+y^2$

$y^2=a^2-v_a^2$

$y^2=144-36$

$y^2=108$

$y=\sqrt{108}$

$y=10.39cm$

$c=8+10.39$

$c=18.39 cm$

## Pythagorean theorem worksheets

Right triangle Integers (181.1 KiB, 1,051 hits) Decimals (190.9 KiB, 590 hits) Radicals (467.3 KiB, 767 hits) Triangle area (139.6 KiB, 975 hits)

Specials Special right triangles (389.8 KiB, 1,083 hits) Multi-step triangle problems (175.1 KiB, 1,048 hits)