Pythagoras Theorem.

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Chetan Mandlia        Mathematics        10 White

                                                         

Mathematics GCSE Coursework: Beyond Pythagoras

Pythagoras

Pythagoras was a Greek mathematician and philosopher who was born around 582BC and died around 500BC. The theorem that carries his name, Pythagoras’ Theorem, is perhaps the best-known theorem in the whole of mathematics. The theorem is about finding the lengths of sides in right-angled triangles.

Pythagoras Theorem

The theorem states that for any right-angled triangle, the area of the square formed on the hypotenuse is equal to the sum of the squares formed on the other two sides.

Or a2 + b2 = c2

Or 92 + 602 = 612

Pythagorean Triples

(3, 4, 5); (5, 12, 13); (7, 24, 25) and etc are all called Pythagorean triples because the satisfy the condition

a2 + b2 = c2 

The numbers 3, 4 and 5 satisfy the condition

The numbers 3, 4 and 5 satisfy the condition

               

   

               

The numbers 3, 4 and 5 satisfy the condition

               

Because  

               

 

The perimeter and area of this triangle are

Perimeter = 3 + 4 + 5 = 12 units

Area = ½ x 3 x 4 = 6units2

The perimeter and area of this triangle are

Perimeter = 5 + 12 + 13 = 30 units

Area = ½ x 5 x 12 = 30units2

The perimeter and area of this triangle are

Perimeter = 7 + 24 + 25 = 56 units

Area = ½ x 7 x 24 = 84units2

Results Table

Odd numbers for Length of Shortest Side (a).

The Length of the Shortest Side (a)

The length of the shortest side (a) can be calculated using n.

The relationship between a and n is that: -

a = 2n + 1

For example, if n = 6

a = 2n +1

a = (2 x 6) + 1

a = 12 + 1

a = 13

The Length of The Middle Side (b)

The length of the middle side (b) can be calculated using n and a.

The relationship between b, a and n is that: -

b = an + n

For example, if a = 9 and n = 4

b = an + n

b = (9 x 4) + 4

b = 36 + 4

b = 40

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The Length of The Longest Side (c)

The length of the longest side (c) can be calculated using a, n and b.

The relationship between c, n and a is that: -

c = an + n + 1

For example, if a = 9 and n = 4

c = an + a + 1

c = [(9 x 4) + 4] +1

c = [36 + 4] + 1

c = 40 + 1 = 41

The relationship between c and b is that: -

c = b + 1

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