# Pythagorian Triples

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Introduction

## 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.

Term Number | Side A | Side B | Side C | Area (cm²) | Perimeter (cm) | |

1 | 3 | 4 | 5 | 6 | 12 | |

2 | 5 | 12 | 13 | 30 | 30 | |

3 | 7 | 24 | 25 | 84 | 56 | |

4 | 9 | 40 | 41 | 180 | 90 | |

5 | 11 | 60 | 61 | 330 | 132 | |

6 | 13 | 84 | 85 | 546 | 182 | |

7 | 15 | 112 | 113 | 840 | 240 | |

8 | 17 | 144 | 145 | 1224 | 306 | |

9 | 19 | 180 | 181 | 1710 | 380 |

Middle

64

510

514

16320

1088

15

68

576

580

19584

1224

16

72

646

650

23256

1368

17

76

720

724

27360

1520

18

80

798

802

31920

1680

19

84

880

884

36960

1848

20

88

966

970

42504

2024

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+4

Side A - 4n+8. This is explained by four multiplied by the term number then plus eight. For example, if the term number is four, then Side A would be twenty-four.

Side B - 2(n²+4n+3). This is explained by squaring the term number, adding 4 multiplied by the term number, adding 3 and then multiplying that number by 2.

Side C - B+4. This is explained by Side B plus 4. For example, if Side B is 4, Side C would be 8.

‘B+5’

Here is my results table for ‘B+5’.

Term Number | Side A | Side B | Side C | Area (cm²) | Perimeter (cm) |

1 | 15 | 20 | 25 | 150 | 60 |

2 | 25 | 60 | 65 | 750 | 150 |

3 | 35 | 120 | 125 | 2100 | 280 |

4 | 45 | 200 | 205 | 4500 | 450 |

5 | 55 | 300 | 305 | 8250 | 660 |

6 | 65 | 420 | 425 | 13650 | 910 |

7 | 75 | 560 | 565 | 21000 | 1200 |

8 | 85 | 720 | 725 | 30600 | 1530 |

9 | 95 | 900 | 905 | 42750 | 1900 |

10 | 105 | 1100 | 1105 | 57750 | 2310 |

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+5

Side A - 10n+5. This is explained by ten multiplied by the term number then plus five. For example, if the term number is one, then Side A would be fifteen.

Side B - an+5n. This is explained by Side A multiplied by the term number then plus five times the term number. For example, if the term number was 2 and side A was 25, Side B would be 60.

Side C - B+5. This is explained by Side B plus 5. For example, if Side B is 10, Side C would be 15.

‘B+6’

Here is my results table for ‘B+6’.

Term Number | Side A | Side B | Side C | Area (cm²) | Perimeter (cm) |

1 | 18 | 24 | 30 | 216 | 72 |

2 | 24 | 32 | 38 | 384 | 94 |

3 | 30 | 72 | 78 | 1080 | 180 |

4 | 36 | 105 | 111 | 1890 | 252 |

5 | 42 | 142 | 148 | 2982 | 332 |

6 | 48 | 189 | 195 | 4536 | 432 |

7 | 54 | 240 | 246 | 6480 | 540 |

8 | 60 | 297 | 303 | 8910 | 660 |

9 | 66 | 360 | 366 | 11880 | 792 |

10 | 72 | 429 | 435 | 15444 | 936 |

11 | 78 | 504 | 510 | 19656 | 1092 |

12 | 84 | 585 | 591 | 24570 | 1260 |

13 | 90 | 672 | 678 | 30240 | 1440 |

14 | 96 | 765 | 771 | 36720 | 1632 |

15 | 102 | 864 | 870 | 44064 | 1836 |

16 | 108 | 969 | 975 | 52326 | 2052 |

17 | 114 | 1080 | 1086 | 61560 | 2280 |

18 | 120 | 1197 | 1203 | 71820 | 2520 |

19 | 126 | 1320 | 1326 | 83160 | 2772 |

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+6

Side A - 6n+6. This is explained by six multiplied by the term number then plus six. For example, if the term number is one, then Side A would be fifteen.

Side B - 3(n²+4n+3) This is explained by squaring the term number, adding 4 multiplied by the term number, adding 3 and then multiplying that number by 3.

Side C - B+6. This is explained by Side B plus 6. For example, if Side B is 10, Side C would be 16.

‘B+7’

Here is my results table for ‘B+7’.

Term Number | Side A | Side B | Side C | Area (cm²) | Perimeter (cm) |

1 | 21 | 28 | 35 | 294 | 84 |

2 | 35 | 84 | 91 | 1470 | 210 |

3 | 49 | 168 | 175 | 4116 | 392 |

4 | 63 | 280 | 287 | 8820 | 630 |

5 | 77 | 420 | 427 | 16170 | 924 |

6 | 91 | 588 | 595 | 26754 | 1274 |

7 | 105 | 784 | 791 | 41160 | 1680 |

8 | 119 | 1008 | 1015 | 59976 | 2142 |

9 | 133 | 1260 | 1267 | 83790 | 2660 |

10 | 147 | 1540 | 1547 | 113190 | 3234 |

Conclusion

8

8n+16

4(n²+4n+3)

B+8

0.5xAxB

A+B+C

9

18n+9

an+9n

B+9

0.5xAxB

A+B+C

10

10n+20

5(n²+4n+3)

B+10

0.5xAxB

A+B+C

I have noticed that in the above table there are two different patterns. The two patterns are one when ‘B+…’ is even and when ‘B+…’ is an odd number. Because of this, I will now draw two separate tables.

Odds

B+… | Side A | Side B | Side C | Area (cm²) | Perimeter (cm) |

1 | 2n+1 | an+n | B+10 | 0.5xAxB | A+B+C |

3 | 6n+3 | an+3n | B+3 | 0.5xAxB | A+B+C |

5 | 10n+5 | an+5n | B+5 | 0.5xAxB | A+B+C |

7 | 14n+7 | an+7n | B+7 | 0.5xAxB | A+B+C |

9 | 18n+9 | an+9n | B+9 | 0.5xAxB | A+B+C |

Evens

B+… | Side A | Side B | Side C | Area (cm²) | Perimeter (cm) |

2 | 2n+4 | (n²+4n+3) | B+2 | 0.5xAxB | A+B+C |

4 | 4n+8 | 2(n²+4n+3) | B+4 | 0.5xAxB | A+B+C |

6 | 6n+6 | 3(n²+4n+3) | B+6 | 0.5xAxB | A+B+C |

8 | 8n+16 | 4(n²+4n+3) | B+8 | 0.5xAxB | A+B+C |

10 | 10n+20 | 5(n²+4n+3) | B+10 | 0.5xAxB | A+B+C |

Overall Rules

Key – X = B+… (whatever side c is)

Odds

Side A – 2X + X. This is explained by 2 times by Side C plus Side C.

Side B – AN + XN. This is explained by Side A times the term number, plus Side C times by the term number.

Area – ½ A X B. This is explained by 0.5 times Side A times Side B.

Perimeter – A+B+C. This is explained by Side A plus Side B plus Side C.

Evens

Side A – XN + 2X. This is explained by Side C times the term number, plus two time Side C.

Side B – (0.5 X)(n²+4n+3). This is explained by 0.5 times Side C, times by the term number squared plus four times the term number plus 3.

Area – ½ A X B. This is explained by 0.5 times Side A times Side B.

Perimeter - A+B+C. This is explained by Side A plus Side B plus Side C.

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