How to use the Pythagorean theorem?

right triangles

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:

pythagorean theorem

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:

egyptian triangle

$\ c^2 = a^2 + b^2$

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.

egyptian triangle task

$ 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.

pythagorean theorem 2023

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

pythagorean theorem 2023  Integers (181.1 KiB, 1,119 hits)

pythagorean theorem 2023  Decimals (190.9 KiB, 691 hits)

pythagorean theorem 2023  Radicals (467.3 KiB, 899 hits)

pythagorean theorem 2023  Triangle area (139.6 KiB, 1,038 hits)

Specials

pythagorean theorem 2023  Special right triangles (389.8 KiB, 1,163 hits)

pythagorean theorem 2023  Multi-step triangle problems (175.1 KiB, 1,109 hits)