3.2 Pythagorean Theorem

 The Pythagorean Brotherhood was created 530 BC by the Greek mathematician and philosopher Pythagoras (580 – 495 BC).  The Pythagoreans led by Pythagoras himself constructed the first known proof for a known relation between sides in a right triangle.  If you know two sides of a right-angled triangle, you can calculate the third side.  This relationship has since been called the Pythagorean Theorem.
What many people do not know is that it was indeed not Pythagoras who discovered this relationship.  It was well-known long before he lived, but he was the first who discovered its proof.

 In a right-angled triangle, just like the name describes, one of the angles is right, in other words 90o. The two sides that extend from the right angle are called legs.  The side which is opposite the right angle, the longest side, is called the hypotenuse.

Pythagorean Theorem
The formula for the Pythagorean Theorem looks like this:

If we illustrate the relationship, it might look like this:

Try to count the panes in the squares which are drawn on each leg and compare that to the number of panes in the square which is drawn along the hypotenuse.

9 + 16 = 25
32+42 = 52
leg2 + leg 2 = hypotenuse2

The Inversion of the Pythagorean Theorem
The Egyptians also had triangles where each of the three sides were known and could in this way create a right angle because the sum of the squares of the legs were equal to the square of the hypotenuse. They use the inverse of the Pythagorean Theorem.

By using the inverse of the Pythagorean Theorem, we can find out of a triangle is right-angled.  If we know each of the three sides in a triangle, then it is right-angled if the sum of the squares of the legs are equal to the square of the hypotenuse.

Here are a few examples of how the Pythagorean theorem can be used:

Example 1

How long is the hypotenuse of the triangle in the figure?

According to Pythagorean Theorem:

(leg1)2 + (leg2)2 = hypotenuse2

(leg1)2 = 32 = 9
(leg2)2 = 52 = 25

(leg1)2 + (leg2)2 = 9 + 25 = 34

So:

Hypotenuse2 = 9 + 25 = 34
Hypotenuse = ≈ 5.8 cm

Example 2

Is the triangle in the figure right-angled?

In order for the triangle to be right-angled, the following needs to apply:

leg2 + leg 2 = hypotenuse2
(leg1)2 = 82 = 64
(leg2)2 = 52 = 25

(leg1)2 + (leg2)2 = 64 + 25 = 89

Hypotenuse2 = 92 = 81

We see that 81 ≠ 89.

The sum of the square of the legs is not equal to the square of the hypotenuse, and therefore, this triangle is not a right-angled.

Example 3

Is the triangle in the figure right-angled?

In order for the triangle to be right-angled, the following needs to apply:

leg2 + leg2 = hypotenuse2
(leg1)2 = 62 = 36
(leg2)2 = 8 2 = 64

(leg1)2 + (leg2)2 = 36 + 64 = 100

Hypotenuse2 = 102 = 100

The sums of the square of the legs are equal to the square of the hypotenuse, and this triangle is right-angled.