Mathematical Investigation

Mathematical Investigation

Pythagoras was a great mathematician who created theorems and one of

his famous theorems was the "Pythagoras Theorem".

You start with a right-angled triangle. The hypotenuse is labeled "c".

The bottom of the triangle is "b" and the side of the triangle is

labeled "a".

[IMAGE]

Pythagoras Theorem says that in any right angled triangle, the lengths

of the hypotenuse and the other two sides are related by a simple

formula. So, if you know the lengths of any two sides of a right

angled triangle, you can use Pythagoras Theorem to find the length of

the third side:

Algebraically: a2 + b2 = c2

The numbers 3, 4 and 5 satisfy the condition

9 + 16 = 25

Because 3x3=9

4x4=16

5x5=25

And so 9 + 16 = 25

[IMAGE]

I now have to find out if the following sets of numbers satisfy a

similar condition of:

(Shortest Side) 2 + (middle Side) 2 = (Longest side) 2

[IMAGE]

a) 5, 12, 13

a2 + b2 = c2

52 + 122 = 132

25 + 144 = 169

169 = 169

b) (7, 24, 25)

a2 + b2 = c2

72 + 242 = 252

49 + 576 = 625

625 = 625

[IMAGE]

(3, 4, 5), (5, 12, 13) and (7, 24, 25) are called Pythagorean triples

because they satisfy the condition, (Shortest side)2 + (Middle side)2

= (Longest Side)2

We know from the Pythagorean triples the shortest side is always an

odd number.

So far I have observed the following patterns:

The shortest side length advances by two each time.

Both the shortest and longest side lengths are always odd.

The longest length is always one unit more than the middle length.

I can immediately see the formula to get the shortest side length from

the term number.

The terms multiplied by two add one equals the shortest side length.