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Pythagoras Theorem
Introduction to Pythagoras
Most people have heard of Pythagoras and know that is something to do with geometry but not much else? Many of the things that we take for granted such as satellite navigation, mobile telephones and air travel all use the principles that Pythagoras proved over two thousand year ago. Once you understand the principles of this simple theory then you can use it to calculate things such as measuring triangles, calculating distances and many other things

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Pythagoras Theorem
Introduction to Pythagoras
Most people have heard of Pythagoras and know that is something to do with geometrybut not much else? Many of the things that we take for granted such as satellitenavigation, mobile telephones and air travel all use the principles that Pythagorasproved over two thousand year ago.Once you understand the principles of this simple theory then you can use it tocalculate things such as measuring triangles, calculating distances and many otherthings that you haven’t even thought about yet?
What is Pythagoras?
Pythagoras was a mathematician who lived over 2000 years ago in Greece. His exactdate of birth is not known but it’s thought that he was born around 580 BC and diedaround 500 BC.He also founded a religious movement called Pythagoreanism which had manyfollowers. Pythagoreans believed that many things in life could be explained bymathematics.Pythagoras is best known for proving the Pythagorean Theorem, as it is known in theUS, or Pythagoras Theorem in the UK, which has been used by scientists, students,mathematicians and engineers in their daily lives ever since. Many things which wetake for granted nowadays would not be possible if it was not for PythagorasTheorem.
Definition
The longest side of the triangle is called the hypotenuse , so the formal definition isthat in a right angled triangle the square of the hypotenuse is equal to the sum of thesquares of the other two sides
What's Pythagoras Theorem?
Pythagoras' theorem states that in any right angled triangle, the square of thehypotenuse is equal to the sum of the squares of the other two sides.Unless you understand the theory then this statement can seem a little baffling?Like many things that are difficult to understand, it’s often easier to break thetheory down into easily understandable pieces?
What’s a Right Angled triangle?
A right angled triangle has a right angle of 90° as one of itsinterior angles. In the image here the right angle is markedwith the white square.Any triangle like this is a right angled triangle, it doesn'tmatter which side the right angle is on or what size thetriangle is - the rule applies to all of them. This is what makesthe theory as useful as it can be applied to shapes as small asthis one or those which are many miles wide
.
What’s the Hypotenuse?
According to some sources the word hypotenuse derives from the Greek words forhypo (under) and teinein (stretch) or tenuse (side).The hypotenuse is the longest side of a right angled triangle. As you can see, thehypotenuse 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 sidesmarked 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 calculatelots of things withouthaving to know the exact exact length of each side?For example, if you knew that side x was 3 metres andside y was 4 metres long then you would be able to workout the length of the hypotenuse quite easily?On the image here the hypotenuse is z and the other 2sides are x and y .Firstly you need to work out the sum of the squares ofboth 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 25The length of the hypotenuse therefore is the square root of 25√25 = 5 (5 x 5 = 25)
So you have now calculated that the length of the hypotenuse is 5 metre
s
Its Problems
T
he Pythagorean Theorem must work in any 90 degree triangle. This means that if youknow two of the sides, you can always find the third one.
I
n the right triangle at the left, we know that:h
2
= 6
2
+ 8
2
Simplifying the squares gives:h
2
= 36 + 64and then:h
2
= 100h = 10
(by doing the square root of 100)
Here's another one:
I
n this example, the missing side is not the long one. But thetheorem still works,as long as you start with the hypotenuse:15
2
= x
2
+ 9
2
Simplifying the squares gives:225 = x
2
+ 81and then:225 - 81 = x
2
144 = x
2
12 = x
(Notice that we had to rearrange the equation)
Now something different will happen:
I
n the right triangle at the left, we know that:h
2
= 7
2
+ 10
2
Simplifying the squares gives:h
2
= 49 + 100h
2
= 149
This square root is not perfect. A calculator gives:
h = 12.2
(rounded to one decimal place)
A
gain, the missing side is not the long one. But we start with thehypotenuse:18
2
= x
2
+ 11
2
Simplifying the squares gives:324 = x
2
+ 121and then:324 - 121 = x
2
203 = x
2
14.2 = x
(We had to rearrange the equation, and round the answer.)
Here's a real problem that uses the theorem:
H
ow far up a wall will an 11m ladder reach, if the foot of the ladder must be 4m from the base of the wall?11
2
= x
2
+ 4
2
121 = x
2
+ 16121 - 16 = x
2
105 = x
2
10.2 = x
The ladder will reach 10.2 metres up the wall.

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