Pythagoras Theorem

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Pythagoras Theorem is a + b = c. a being the shortest side, b being the middle side and c being the longest side (hypotenuse) of a right angled triangle.

The numbers , 4 and 5 satisfy this condition

+ 4 = 5

because = x =

Help with essay on Pythagoras Theorem

4 = 4 x 4 = 16

5 = 5 x 5 = 5

and so + 4 = + 16 = 5 = 5

The numbers 5, 1, 1 and 7, 4, 5 also work for this theorem

5 + 1 = 1

because 5 = 5 x 5 = 5

1 = 1 x 1 = 144

1 = 1 x 1 = 16

and so 5 + 1 = 5 + 144 = 16 = 1

7 + 4 = 5

because 7 = 7 x 7 = 4

4 = 4 x 4 = 576

5 = 5 x 5 = 65

and so 7 + 4 = 4 + 576 = 65 = 5

, 4, 5

Perimeter = + 4 + 5 = 1

Area = ½ x x 4 = 6

5, 1, 1

Perimeter = 5 + 1 + 1 = 0

Area = ½ x 5 x 1 = 0

7, 4, 5

Perimeter = 7 + 4 + 5 = 56

Area = ½ x 7 x 4 = 84

From the first three terms I have noticed the following -

• a increases by + each term

• a is equal to the term number times then add 1

• the last digit of b is in a pattern 4, , 4

• the last digit of c is in a pattern 5, , 5

• the square root of (b + c) = a

• c is always +1 to b

• b increases by +4 each term

• (a x n) + n = b

From these observations I have worked out the next two terms.

I will now put the first five terms in a table format.

Term Number n Shortest Side a Middle Side b Longest Side c Perimeter Area

1 4 5 1 6

5 1 1 0 0

7 4 5 56 84

4 40 41 0 180

5 11 60 61 1 0

I have worked out formulas for

1. How to get a from n

. How to get b from n

. How to get c from n

4. How to get the perimeter from n

5. How to get the area from n

My formulas are

1. n + 1

. n + n

. n + n + 1

4. 4n + 6n +

5. n + n + n

To get these formulas I did the following

1. Take side a for the first five terms , 5, 7, , 11. From these numbers you can see that the formula is n + 1 because these are consecutive odd numbers (n + 1 is the general formula for consecutive odd numbers) You may be able to see the formula if you draw a graph

. From looking at my table of results, I noticed that an + n = b. So I took my formula for a (n + 1) multiplied it by n to get n + n. I then added my other n to get n + n. This is a parabola as you can see from the equation and also the graph

. Side c is just the formula for side b +1

4. The perimeter = a + b + c. Therefore I took my formula for a (n + 1), my formula for b (n + n) and my formula for c (n + n + 1). I then did the following -

n + 1 + n + n + n + n + 1 = perimeter

Rearranges to equal

4n + 6n + = perimeter

5. The area = (a x b) divided by . Therefore I took my formula for a (n + 1) and my formula for b (n + n). I then did the following -

(n + 1)(n + n) = area

Multiply this out to get

4n + 6n + n = area

Then divide 4n + 6n + n by to get

n + n + n

To prove my formulas for a, b and c are correct. I decided incorporate my formulas into a + b = c -

a + b= c

(n + 1)+ (n + n)= (n + n + 1)

(n + 1)(n + 1) + (n + n)(n + n) = (n + n + 1)(n + n + 1)

4n + n + n + 1 + 4n4 + 4n + 4n + 4n= 4n4 + 8n + 8n + 4n + 1

4n + 4n + 1 + 4n4 + 8n + 4n= 4n4 + 8n + 8n + 4n + 1

4n4 + 8n + 8n + 4n + 1 = 4n4 + 8n + 8n + 4n + 1

This proves that my a, b and c formulas are correct

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