# 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:

### 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, 805 hits)

**Decimals** (190.9 KiB, 402 hits)

**Radicals** (467.3 KiB, 591 hits)

**Triangle area** (139.6 KiB, 760 hits)

Specials

**Special right triangles** (389.8 KiB, 818 hits)

**Multi-step triangle problems** (175.1 KiB, 802 hits)