Pythagorian Triples

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Introduction

I am going to investigate Pythagorean Triples.  In this investigation, I shall explore right-angled triangles as used in Pythagoras’s Theorem.  Pythagoras’s Theorem is ‘A²+B²=C².  Some examples of Pythagorean Triples are (3,4,5), (5,12,13) & (7,24,25).  These could be used in the triangle below.

        

In right-angled triangles like this one, the area can be found by using this equation -         0.5 x Base x Height.

        

The perimeter in these triangles is always found using this equation – Side A + Side B + Side C.

                                                         

To begin my investigation, I am going to investigate the family of Pythagorean Triples where all three sides are positive integers and the shortest side is an odd number.  I will start with the shortest side being 3cm and progress until I have 9 readings.

‘B + 1’

 3² + 4² = 5² (9+16=25)
Area = 0.5 x 3 x 4 =6cm²

Perimeter = 3cm + 4cm + 5cm = 12cm

        

5² + 12² = 13² (25+144=169)

Area = 0.5 x 5 x 12 = 30cm²

Perimeter= 5cm+12cm+13cm=30cm²

7²+24² = 25² (49+576 = 625)

Area = 0.5 x 7 x 24 = 84cm²

Perimeter= 7cm+24cm+25cm=56cm²

        9²+40² = 41² (81+1600 = 1681)

Area = 0.5 x 9 x 40 = 180cm²

Perimeter = 9cm+40cm+41cm = 90cm

        

11²+60² = 61² (121+3600 = 3721)

Area = 0.5 x 11 x 60 = 330cm²

Perimeter = 11cm+60cm+61cm = 132cm

        13²+84² = 85² (169+7056 = 7225)

Area = 0.5 x 13 x 84 = 546cm²

Perimeter = 13cm+84cm+85cm = 182cm

        15²+112² = 113² (225+12544 = 12769)

Area = 0.5 x 15 x 112 = 840cm²

Perimeter = 15cm+112cm+113cm = 240cm

        17²+144² = 145² (289+20736 = 21025)

Area = 0.5 x 17 x 144 = 1224cm²

Perimeter = 17cm+144cm+145cm = 306cm

19²+180² = 181² (360+32400 = 32760)

        Area = 0.5 x 19 x 180 = 1710cm²

Perimeter = 19cm+180cm+181cm = 380cm

        

I will now add all of my results into a table so I can see them more clearly and see if there are any trends and rules.

        From this table, I have found a rule to find Side A, a rule to find Side B and rule to find Side C.

Rules For ‘B+1’

A = Side A

B = Side B

C = Side C

N = Term Number

Side A -        2n+1.  This is explained by two multiplied by the term number plus one.  For example, if the term is 5 then 2 x 5+1=11.

Side B -        an+n.  This is explained by Side A multiplied by the term number plus the term number.  For example, if the term number is 4 and Side A is 9, 4x9+4 =40.

Side C -        B+1.  This is explained by Side B plus 1.  For example, if Side B is 5, Side C would be six.

Join now!

To further extend my investigation, I am going to investigate the family of Pythagorean Triples where all three sides are positive integers and the shortest side is an even number.  I will start with the shortest side being 6cm and progress until I have 10 readings.

‘B+2’

I am not going to draw any triangles from now on because I find it much easier and quicker to use trial and improvement to find the results for my table than it would drawing 10 triangles.  Here is my results table for ‘B+2’.

From this table, I ...

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