Beyond Pythagoras

Pythagorean Triples:

Three integers a, b, and c that satisfy a2 + b2 = c2 are called Pythagorean Triples.

a2 + b2 = c2

The numbers 3, 4 and 5 satisfy the condition because:

32 + 42 = 52

32 = 3 x 3 = 9

42 = 4 x 4 =16

52 = 5 x 5 = 25

32 + 42 = 9 +16 = 25 = 52

. Each of the following sets of numbers satisfy a similar condition of

(smallest number)2 + (middle number)2 = (largest number)2

a) 5, 12, 13.

52 + 122 = 132

52 = 5 x 5 = 25

122 = 12 x 12 = 144

132 = 13 x 13 = 169

52 + 122 = 25 +144 = 169 = 132

b) 7, 24, 25

72 + 242 = 252

72 = 7 x 7 = 49

242 = 24 x 24 = 576

252 = 25 x 25 = 625

72 + 242 = 49 +576 = 625 = 252

The numbers 3, 4 and 5 can be the lengths - in appropriate units - of the sides of a right-angled triangle.

3 5

4

The perimeter and area of this triangle are :

Perimeter = 3 + 4 + 5 = 12 units

Area = 1/2 x 3 x 4 = 6 square units

The numbers 5, 12, and 13 can also be the lengths - in appropriate units - of a right-angled triangle :

5 13

12

The perimeter and area of this triangle are :

Perimeter = 5 + 12 + 13 = 30

Area = 1/2 x 5 x 12 = 30

(c) This is also true for the numbers 7, 24 and 25:

7 25

24

The perimeter and area for this triangle are :

Perimeter = 7 + 24 + 25 = 56

Area = 1/2 x 7 x 24 = 84

Below is a table showing many Pythagorean triples:

Length of shortest side

(a)

Length of middle side

(b)

Length of longest side

(c)

Perimeter

Area

3

4

5

2

6

5

2

3

30

30

7

24

25

56

84

9

40

41

90

80

1

60

61

32

330

3

84

85

82

546

5

12

13

240

840

7

44

45

306

224

9

80

81

380

710

21

220

221

462

2310

23

264

265

552

3036

I have noticed that the first column goes up by two every time, there is a constant difference of two. I have also noticed that the second column is a quadratic sequence, with it going up in fours and then a difference of four at the end. The third column is simply the number in the second column plus one.

Generalisation:

(a)=

Term no.

2

3

4

5

Sequence

3

5

7

9

1

Difference

2

2

2

2

I have found the nth term formula to be 2n + 1

(b)=

Term no.

2

3

4

5

Sequence

4

2

24

40

60

2n2

2

8

8

32

50

R2 - R3

2

4

6

8

0

Difference

2

2

2

2

I have found the nth term formula to be 2n2 + 2n

This can also be expressed in the form of triangle numbers because:

(b)

Multiples

4

4 x 1

2

4 x 3

24

4 x 6

40

4 x 10

The triangle formula will be simply be 4t

(c)=

To find the nth term formula for this column is very simple. You take the formula for column (b) and just add one at the end because that is always the difference between column (b) and (c).

The formula will be 2n2 + 2n +1.

Pythagorean Triples:

Three integers a, b, and c that satisfy a2 + b2 = c2 are called Pythagorean Triples.

a2 + b2 = c2

The numbers 3, 4 and 5 satisfy the condition because:

32 + 42 = 52

32 = 3 x 3 = 9

42 = 4 x 4 =16

52 = 5 x 5 = 25

32 + 42 = 9 +16 = 25 = 52

. Each of the following sets of numbers satisfy a similar condition of

(smallest number)2 + (middle number)2 = (largest number)2

a) 5, 12, 13.

52 + 122 = 132

52 = 5 x 5 = 25

122 = 12 x 12 = 144

132 = 13 x 13 = 169

52 + 122 = 25 +144 = 169 = 132

b) 7, 24, 25

72 + 242 = 252

72 = 7 x 7 = 49

242 = 24 x 24 = 576

252 = 25 x 25 = 625

72 + 242 = 49 +576 = 625 = 252

The numbers 3, 4 and 5 can be the lengths - in appropriate units - of the sides of a right-angled triangle.

3 5

4

The perimeter and area of this triangle are :

Perimeter = 3 + 4 + 5 = 12 units

Area = 1/2 x 3 x 4 = 6 square units

The numbers 5, 12, and 13 can also be the lengths - in appropriate units - of a right-angled triangle :

5 13

12

The perimeter and area of this triangle are :

Perimeter = 5 + 12 + 13 = 30

Area = 1/2 x 5 x 12 = 30

(c) This is also true for the numbers 7, 24 and 25:

7 25

24

The perimeter and area for this triangle are :

Perimeter = 7 + 24 + 25 = 56

Area = 1/2 x 7 x 24 = 84

Below is a table showing many Pythagorean triples:

Length of shortest side

(a)

Length of middle side

(b)

Length of longest side

(c)

Perimeter

Area

3

4

5

2

6

5

2

3

30

30

7

24

25

56

84

9

40

41

90

80

1

60

61

32

330

3

84

85

82

546

5

12

13

240

840

7

44

45

306

224

9

80

81

380

710

21

220

221

462

2310

23

264

265

552

3036

I have noticed that the first column goes up by two every time, there is a constant difference of two. I have also noticed that the second column is a quadratic sequence, with it going up in fours and then a difference of four at the end. The third column is simply the number in the second column plus one.

Generalisation:

(a)=

Term no.

2

3

4

5

Sequence

3

5

7

9

1

Difference

2

2

2

2

I have found the nth term formula to be 2n + 1

(b)=

Term no.

2

3

4

5

Sequence

4

2

24

40

60

2n2

2

8

8

32

50

R2 - R3

2

4

6

8

0

Difference

2

2

2

2

I have found the nth term formula to be 2n2 + 2n

This can also be expressed in the form of triangle numbers because:

(b)

Multiples

4

4 x 1

2

4 x 3

24

4 x 6

40

4 x 10

The triangle formula will be simply be 4t

(c)=

To find the nth term formula for this column is very simple. You take the formula for column (b) and just add one at the end because that is always the difference between column (b) and (c).

The formula will be 2n2 + 2n +1.