# Mathematics Investigation

explanatory Essay
949 words
949 words

Mathematics Investigation

In this report, I am going to investigate the Pythagorean Triple. The

Pythagoras Theorem was invented by Pythagoras, a Greek mathematician

and philosopher who lived in the 6th centaury BC.

The Pythagoras Theorem only works in right-angled triangles, where

there are three different lengthed sides, one short, one medium, and

the other long. A Pythagorean Triple is when a set of numbers satisfy

the condition: Shortest side2 + Medium side2 =Longest side2. Also all

the sides have to have positive integers.

Here is an example of a Pythagorean Triple:

1)

The above triangle is a Pythagorean Triple because it satisfies the

condition with all its sides being a positive integer. I will now work

out the perimeter and area of the above Pythagorean Triple.

There are also other Pythagorean Triples. Here they are:

2)

3)

Both the triangles 2) and 3) are Pythagorean Triples because they

satisfy the condition and all their sides have a positive integer.

Here is a table showing the results of the 3 Pythagorean Triples:

Triangle No.

Shortest side

Medium side

Longest side

Perimeter

Area

1)

3

4

5

12

62

2)

5

12

13

30

302

3)

7

24

25

56

842

From the above table, I can see a few patterns emerging. Here they

are:

i. The shortest side is always an odd number.

ii. The medium side is always an even number.

iii. The medium side plus one equals to the Longest side.

iv. The Longest side is always odd.

To show that these patterns are correct and to see if there are any

more patterns, I am going to extend this investigation and draw two

more Pythagorean Triples. Here they are:

4)

5)

#### In this essay, the author

• Opines that the patterns go on, and not only in the table.
• Describes the quantities in the table.
• Explains that they are going to work with scale 2 because they want to prove that other scales are better.
• Opines that the patterns go on, and not only in the table.
• Explains that scale 3, 4, 5 by 2 gives 6, 8, 10 which fits the formula a2+b2=c2 in the same way.