Beyond Pythagoras

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Mathematics Coursework – Beyond Pythagoras         May 2006

BEYOND PYTHAGORAS

MATHS COURSEWORK

INTRODUCTION

Pythagoras was a Greek mathematician and philosopher. He lived in 400 BC and was one of the first great mathematical thinkers. He spent most of his life in Sicily and southern Italy. He had a group of follows who went around and thought other people what he had taught them who were called the Pythagoreans.

Pythagoras himself is best known for proving that the Pythagorean Theorem was true. The Sumerians, two thousand years earlier, already knew that it was generally true, and they used it in their measurements, but Pythagoras proved that it would always be true. The Pythagorean Theorem says that in a right triangle, the sum of the squares of the two right-angle sides will always be the same as the square of the hypotenuse (the long side). A2 + B2 = C2

Pythagoras theorem can also help in real life. Here is an example:

Say you were walking though a park and wanted to take a short cut. With Pythagoras’s theorem you could work out exactly how long you would have to walk though the grass, rather then talking the long route by walking on the paths.

PLAN


I am going to investigate the three triangles I have been given. They are all right-angled triangles, with 3 sides, all different lengths.

The three triangles satisfy the Pythagoras theorem. The theorem states that the hypotenuse side (longest side) must equal the 2 shorter sides squared.

Here is the Pythagoras theorem:



PYTHAGORAS = a2 + b2 = c2

Here are the three triangles I have been given:

a)                                b)                                c)

          


I am now going to test if three triangles I have been given:

Triangle A  =    32 + 42 = C2
                          9 + 16 = C
2
                                25 = C
2
                                 5  = C


Triangle A is a Pythagorean triplet because when I put in the shortest and middle side into the formula, it gave me the answer that the hypotenuse was 5. Checking this against what I have been given, I can verify that it was correct. I will now test the other two triangle I have been given.


Triangle B =    52 + 122 = C2
                       25 + 144 = C
2
                               169 = C
2
                                 13 = C


This is also correct and matches with that I have been given.


Triangle C =     72 + 242 = C2
                        49 + 276 = C
2
                                625 = C
2
                                  25 = C


All these triangles are Pythagorean triplets.

ACTION


I am going to put all the numbers I have been given into a table. I want to investigate how I cold work out what the next numbers in the next triangle would be. I am putting it into a table because it is easier to see if there is any pattern.



Everything that I add to my table after this point will be in
blue.

I will now work out what the next three triangles in the family will be.

Sequence for shortest side

3   5   7   9   11   13
\   / \   / \   / \   / \   /
2    2     2    2    2

Every time 2 is being added to the previous number. I can work out that the next numbers will be 9, 11, and then 13.

Sequence for middle side

4     12     24     40     60     84
\      / \      / \      / \      / \      /
   8     12     16       20    24  
     \      / \      / \      / \      /
        4       4       4        4  

Here there is a continuous second difference of 4. I can tell that the number after 24 will be 40. I worked this out by finding the difference between 12 and 24 (12), adding 4 to it (16) then adding it on to 24.

Sequence for hypotenuse side

5     13     25     41     61     85
\      / \      / \      / \      / \      /
   8     12     16       20    24  
     \      / \      / \      / \      /
        4       4       4        4  

On this sequence the second difference is 4, and by adding 4 every time to the first non continuous difference I can tell that the next numbers will be 41, 61 then 85.

Test that triangles are Pythagoras

I will now test to see if the three new triangles I have got numbers for do comply with Pythagoras theorem. I will to this by adding side a
2 and side b2 and see if I get the answer I get matches what I got for side C in my table.

Triangle 4

A2 + B2      = C2
9
2 + 402     = C2
81 + 1,600 = C
2
1681          = C
2
√1681        = C
2
41               = Side C

This answer matches with what I predicted with my table. Therefore it is defiantly a Pythagoras triangle.

Join now!


Triangle 5



A
2 + B2       = C2
11
2 + 602    = C2
121 + 3600 = C
2
3721           = C
2
√3721         = C
2
61               = Side C

Again this is a Pythagoras triangle.

Triangle 6

A2 + B2      = C2
13
2 + 842    = C2
169 + 7056 = C
2
7225           = C
2
√3721         = C
2
85               = Side C

I now can be sure that all the ...

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