Pythagoras Theorem

Topic: 
Pythagoras Theorem


Syllabus:

·       Identify a right-angled triangle and its hypotenuse.

·       Define the Pythagoras’ theorem and understand its symbols.

·       Find the unknown side of a right-angled triangle when the other two sides are given.

·       Solve problems involving right-angled triangles with Pythagoras’ theorem.

·       Determining whether a triangle is right-angled given the lengths of 3 sides.

Who is Pythagoras?

Pythagoras (569-500 B.C.E.) was born on the island of Samos in Greece, and did much traveling through Egypt, learning, among other things, mathematics. Not much more is known of his early years. Pythagoras gained his famous status by founding a group, the Brotherhood of Pythagoreans, which was devoted to the study of mathematics. The group was almost cult-like in that it had symbols, rituals and prayers. In addition, Pythagoras believed that "Number rules the universe,"and the Pythagoreans gave numerical values to many objects and ideas. These numerical values, in turn, were endowed with mystical and spiritual qualities.

What's Pythagoras Theorem?

Pythagoras' theorem states that in any right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

click here on proof and other facts on Pythagoras

What’s a Right Angled triangle?

A right angled triangle has a right angle of 90° as one of its interior angles. In the image here the right angle is marked with the white square.

Any triangle like this is a right angled triangle, it doesn't matter which side the right angle is on or what size the triangle is - the rule applies to all of them.

What’s the Hypotenuse?

According to some sources the word hypotenuse derives from the Greek words for hypo (under) and teinein (stretch) or tenuse (side).

The hypotenuse is the longest side of a right angled triangle. As you can see, the hypotenuse is the side marked in purple on the image above.

What are the other two sides?

Now you know which side the hypotenuse is then this is easy, it's the two other sides marked in orange on the picture above.

OK, so now we know which side is which, what use is it?

Well, understanding the rule means that you can calculate lots of things without having to know the exact exact length of each side?

For example, if you knew that side x was 3 metres and side y was 4 metres long then you would be able to work out the length of the hypotenuse quite easily?

On the image here the hypotenuse is z and the other 2 sides are x and y .

Firstly you need to work out the sum of the squares of both x and y:

X² is 3 x 3 = 9

Y² is 4 x 4 =16

Z² = X² + Y² (25)

The sum of the other 2 sides squared is 25

The length of the hypotenuse therefore is the square root of 25

√25 = 5 ( 5 x 5 = 25)

Watch the video for examples on the application of Pythagoras Theorem

source:
http://www.geom.uiuc.edu/~demo5337/Group3/hist.html
http://www.ies.co.jp/math/java/geo/pythagoras.html