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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.

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

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