Beyond Pythagoras

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".

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

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

a) 5, 12, 13

a2 + b2 = c2

52 + 122 = 132

25 + 144 = 169

69 = 169

b) (7, 24, 25)

a2 + b2 = c2

72 + 242 = 252

49 + 576 = 625

625 = 625

(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.

Let Term number = Tn

Let Shortest Side length = SL

Tn ------ 2n + 1 = SL

If term number = 1; 2(1)+1=3

If term number = 2; 2(2)+1=5

If term number = 3; 2(3)+1=7

This works for all Pythagorean triples that have an odd numbered SL.

I have investigated another three Pythagorean triples and they are stated below.

Serial No.

Shortest side a

Middle Side b

Longest side c

3

4

5

2

5

2

3

3

7

24

25

4

9

40

41

5

1

60

61

6

3

84

85

I will now test the Pythagorean triples I have just found with the aid of Pythagoras theorem and a diagram.

a) (9, 40, 41)

a2 + b2 = c2

92 + 402 = 412

81 + 1600 = 1681

Scale: 1cm=2cm

b) (11, 60, 61)

a2 + b2 = c2

12 + 602 = 612

21 + 3600 = 3721

Scale: 1cm=2cm

c) (13, 84, 85)

a2 + b2 = c2

32 + 842 = 852

69 + 7056 = 7225

Scale: 1cm=4cm

I want to predict two other Pythagorean triples outside the table, and I will do this with the help of the observations I have made.

We know that the shortest side and longest side is an odd number and the longest side is one more than the middle side.

The shortest side increases by two each time, the middle side's difference increases by four each time, and the longest side is one more than the middle side.

. Since the shortest side increases by two each time, the last number we investigated was 13, therefore 13 + 2 = 15,

Shortest side = 15

Since the middle side's difference increases by four each time, the last

number was 84 and the value before that was 64, the difference between 84

and 64 is 24, so we add 4 to 24,

which is 28, therefore 84 + 28 = 112,

Middle Side = 112

The longest side is 1 more than the middle side, therefore 112 + 1 = 113

I would now like to check if the values found are accurate and would check it with the help of the Pythagoras theorem.

a2 + b2 = c2

52 + 1122 = 1132

2769 = 12769

The results show that (15,112,113) is a Pythagorean triple.

I would like to investigate another Pythagorean triple using the same method.

2. We add 2 to 15 which is 17.

Shortest Side = 17

112 - 84 = 28, I will add 4 to 28 which is 32, therefore 112 + 32 = 144

Middle Side = 144

The longest side is 1 more than the middle side, therefore 144 + 1 = 145

Longest side = 145

I would like to check if the values found are accurate with the aid of a Pythagoras theorem.

a2 + b2 = c2

72 + 1442 = 142

21025 = 21025

The results show that (17,144,145) are Pythagorean triples.

With the help of the site, http://www2.math.vic.edu/~fields/puzzle/triples.html I am able to generate around 70 more Pythagoras theorem.

Serial No.

Shorter Side a

Middle Side b

Longest side d

3

4

5

2

5

2

3

3

7

24

25

4

9

40

41

5

1

60

61

6

3

84

85

7

5

12

13

8

7

44

45

9

9

80

81

0

21

220

221

1

23

264

265

2

25

312

313

3

27

364

365

4

29

420

421

5

31

480

481

6

33

544

545

7

35

612

613

8

37

684

685

9

39

760

761

20

41

840

841

21

43

924

925

22

45

012

013

23

47

104

105

24

49

200

201

25

51

300

301

26

53

404

405

27

55

512

513

28

57

624

625

29

59

740

741

30

61

860

861

31

63

984

985

32

65

2112

2113

33

67

2244

2245

34

69

2380

2381

35

71

2520

2521

36

73

2664

2665

37

75

2812

2813

38

77

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".

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

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

a) 5, 12, 13

a2 + b2 = c2

52 + 122 = 132

25 + 144 = 169

69 = 169

b) (7, 24, 25)

a2 + b2 = c2

72 + 242 = 252

49 + 576 = 625

625 = 625

(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.

Let Term number = Tn

Let Shortest Side length = SL

Tn ------ 2n + 1 = SL

If term number = 1; 2(1)+1=3

If term number = 2; 2(2)+1=5

If term number = 3; 2(3)+1=7

This works for all Pythagorean triples that have an odd numbered SL.

I have investigated another three Pythagorean triples and they are stated below.

Serial No.

Shortest side a

Middle Side b

Longest side c

3

4

5

2

5

2

3

3

7

24

25

4

9

40

41

5

1

60

61

6

3

84

85

I will now test the Pythagorean triples I have just found with the aid of Pythagoras theorem and a diagram.

a) (9, 40, 41)

a2 + b2 = c2

92 + 402 = 412

81 + 1600 = 1681

Scale: 1cm=2cm

b) (11, 60, 61)

a2 + b2 = c2

12 + 602 = 612

21 + 3600 = 3721

Scale: 1cm=2cm

c) (13, 84, 85)

a2 + b2 = c2

32 + 842 = 852

69 + 7056 = 7225

Scale: 1cm=4cm

I want to predict two other Pythagorean triples outside the table, and I will do this with the help of the observations I have made.

We know that the shortest side and longest side is an odd number and the longest side is one more than the middle side.

The shortest side increases by two each time, the middle side's difference increases by four each time, and the longest side is one more than the middle side.

. Since the shortest side increases by two each time, the last number we investigated was 13, therefore 13 + 2 = 15,

Shortest side = 15

Since the middle side's difference increases by four each time, the last

number was 84 and the value before that was 64, the difference between 84

and 64 is 24, so we add 4 to 24,

which is 28, therefore 84 + 28 = 112,

Middle Side = 112

The longest side is 1 more than the middle side, therefore 112 + 1 = 113

I would now like to check if the values found are accurate and would check it with the help of the Pythagoras theorem.

a2 + b2 = c2

52 + 1122 = 1132

2769 = 12769

The results show that (15,112,113) is a Pythagorean triple.

I would like to investigate another Pythagorean triple using the same method.

2. We add 2 to 15 which is 17.

Shortest Side = 17

112 - 84 = 28, I will add 4 to 28 which is 32, therefore 112 + 32 = 144

Middle Side = 144

The longest side is 1 more than the middle side, therefore 144 + 1 = 145

Longest side = 145

I would like to check if the values found are accurate with the aid of a Pythagoras theorem.

a2 + b2 = c2

72 + 1442 = 142

21025 = 21025

The results show that (17,144,145) are Pythagorean triples.

With the help of the site, http://www2.math.vic.edu/~fields/puzzle/triples.html I am able to generate around 70 more Pythagoras theorem.

Serial No.

Shorter Side a

Middle Side b

Longest side d

3

4

5

2

5

2

3

3

7

24

25

4

9

40

41

5

1

60

61

6

3

84

85

7

5

12

13

8

7

44

45

9

9

80

81

0

21

220

221

1

23

264

265

2

25

312

313

3

27

364

365

4

29

420

421

5

31

480

481

6

33

544

545

7

35

612

613

8

37

684

685

9

39

760

761

20

41

840

841

21

43

924

925

22

45

012

013

23

47

104

105

24

49

200

201

25

51

300

301

26

53

404

405

27

55

512

513

28

57

624

625

29

59

740

741

30

61

860

861

31

63

984

985

32

65

2112

2113

33

67

2244

2245

34

69

2380

2381

35

71

2520

2521

36

73

2664

2665

37

75

2812

2813

38

77