Pythagorus maths assignment

QUESTION 1

The term Pythagorean triple is meant to explain that if three different positive integers, which each measure the distance of one side of a right angle triangle, (usually known as either a, b and c or side1, side2 and side3) fit the rule a2 + b2 = c2 then the combination of those numbers is a Pythagorean triple. The concept is only correct when the triangle used is a right angle triangle because there must be a hypotenuse across from the right angle. The demonstration used consists of three triangles each of them use positive integers, are right angle triangles and they all fit the rule a2 + b2 = c2, which means they are Pythagorean triples.

Rule

〖side〗_1^2+〖side〗_2^2=〖hyp〗^2

〖 a〗^2+b^2=c^2 or c = √(a^2+b^2 )

Triangle 1 a = 3cm scale b = 4cm 1cm = 1cm c = 5cm ∴1:1

〖 a〗^2+b^2=c^2

〖 3〗^2+4^2=5^2

9+16=25

〖 c〗^2=25 c=√25 c=5

Triangle 2 a = 8cm scale b = 15cm 1cm = 1cm c = 17cm ∴1:1

〖 a〗^2+b^2=c^2

〖 8〗^2+〖15〗^2=〖17〗^2

64+225=289

〖 c〗^2=289 c=√289 c=17

triangle 3 a = 6cm scale b = 8cm 1cm = 1cm c = 10cm ∴1:1

〖 a〗^2+b^2=c^2

〖 6〗^2+8^2=〖10〗^2

36+64=100

〖 c〗^2=100 c=√100 c=10

QUESTION 2

Pythagoras’s theorem proves that a2 + b2 = c2 but his theorem is only applicable for right-angled triangles. Although isosceles are capable of being right-angle triangles because two of the sides are equal length the formula does not work.

in most cases; if a^2+b^2=c^2 then the triangle is right if a^2+b^2 greate...

... middle of paper ...

... be able the prove the law with cube the formula would be a3 + b3 = c3 but this formula does not work when used in a real problem. This premise defective because it attempts to translate a theorem applicable in two dimensions (squares) to three dimensions (cubes). If this premise were possible it would imply a fixed ratio of area to volume.

For example

〖a 〗^3+b^3=c^3

〖 3〗^3+4^3=5^3

27+64=125

Which is incorrect because 27 + 64 actually equals 91

If the 3D premise were capable of proving Pythagoras’ theorem the premise would need to be refined. Therefor a possible way of refining the premise is the adjustment to the choice of the 3D shape used.

The 3D shape chosen to replace the cube is a triangular based pyramid.

This premise is also defective as it does not fit the Pythagorean law. Therefor it is proven that 3D shapes are not applicable for Pythagoras’ law.