• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Beyond Pythagoras

Extracts from this document...

Introduction

BEYOND PYTHAGORAS

Pythagoras theorem states that in a right angled triangle, a² + b² = c², this can also be interpreted as (shortest side) ² + (middle side) ²

= (longest side) ².

Two sets of numbers that satisfy the theorem are 5, 12, 13 and

7, 24, 25.

5² + 12² = 13²                                 7² + 24² = 25²

25 + 144 = 169                                     49 + 576 = 625

The perimeter of a triangle is calculated using a + b + c. The area of a triangle is calculated using ½ x a x b. The perimeters and areas for the triangles above are:

PERIMETER –  

        5 + 12 + 13 = 30

        7 + 24 + 25 = 56

AREA -

        ½ x 5 x 12 = 30

        ½ x 7 x 24 = 84

Below is a table of side length, area and perimeter for Pythagorean triples that start with ODD numbers.

TERM

LENGTH OF

LENGTH OF

LENGTH OF

PERIMETER

AREA

SHORTEST SIDE

MIDDLE SIDE

LONGEST SIDE

1

3

4

5

12

6

2

5

12

13

30

30

3

7

24

25

56

84

4

9

40

41

90

180

5

11

60

61

132

330

Using the sequence formula :

nth term = a + (n-1)d + ½ (n-1) (n-2)c

a = first number

d = first difference

c = second difference

...read more.

Middle

20

25

60

150

nth

3n 

4n 

5n 

12n 

6n²

TRIPLE: 5,12,13

TERM

LENGTH OF

LENGTH OF

LENGTH OF

PERIMETER

AREA

SHORTEST SIDE

MIDDLE SIDE

LONGEST SIDE

PERIMETER

AREA

1

5

12

13

30

30

2

10

24

26

60

120

3

15

36

39

90

270

4

20

48

52

120

480

5

25

60

65

150

750

nth

5n 

12n

13n 

30n 

30n²

TRIPLE: 7,24,25

TERM

LENGTH OF

LENGTH OF

LENGTH OF

PERIMETER

AREA

SHORTEST SIDE

MIDDLE SIDE

LONGEST SIDE

1

7

24

25

56

84

2

14

48

50

112

336

3

21

72

75

168

756

4

28

96

100

224

1344

5

35

120

125

280

2100

nth

7n 

24n 

25n 

56n 

84n²

I put the nth terms in a table against the number triple it was (t). I was then able to work

...read more.

Conclusion

Formula for Area:

(n² - m²) 2mn

2

Perimeter = Area:

(n² - m²) + 2mn + n² + m² = (n² - m²) 2mn

        2

I simplified the formula to give so that I was able to work out values for n and m.

(n² - m²) + 2mn + n² + m² = (n² - m²) 2mn

                                                        2

2mn + 2n² = (n + m)(n -m)mn

                                  2n(m + n) = (n + m)(n – m)mn

                                           2 = m x (n - m)

m = 2    (n – m) = 1    n = 3

                                               or            m = 1    (n – m) = 2   n = 3

When these two solutions for n and m are inserted into the general formulas the two Pythagorean triples with perimeter = area are generated.

m = 2   n = 3

n² - m ²        2mn                n² + m²

3² - 2²         2x2x3 = 12        3² + 2²

9 – 4 = 59 + 4 = 13

Pythagorean triple in which area = perimeter : 5, 12, 13

m = 1   n = 3

n² - m²        2mn        n² + m²

3² - 1²        2x1x3 = 6        3² + 1²

9 - 1 = 8        9 + 1 = 10

Pythagorean triple in which area = perimeter: 6, 8 , 10

...read more.

This student written piece of work is one of many that can be found in our GCSE Fencing Problem section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related GCSE Fencing Problem essays

  1. t shape t toal

    1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

  2. Geography Investigation: Residential Areas

    I did this by using a Bi-Polar analysis amongst other things, they can all be found on my 'external questionnaire'. The results from my 'external questionnaire' could have not been improved by collecting more data because I can only analyse the street once with the same questionnaire when filled in by myself.

  1. Regeneration has had a positive impact on the Sutton Harbour area - its environment, ...

    The following gives reasons on why I asked each question and what I hoped it would tell me. Question 1- "Are you local to the area of Plymouth?" From this I could distinguish how popular the area was with both local people and also how many tourists were in the area.

  2. Beyond Pythagoras

    cm 144 cm 145 cm 306 cm 1224 cm� 19 cm 180 cm 181 cm 380 cm 1710 cm� 21 cm 220 cm 221 cm 462 cm 2310 cm� 23 cm 264 cm 265 cm 552 cm 3036 cm� 25 cm 312 cm 313 cm 650 cm 3900 cm� 27

  1. Beyond pythagoras - First Number is odd.

    Just in case, I will check it using the first 3 terms. 2n� + 2n +1 = 5 2 x 1� + 2 x 1 + 1 = 5 2 + 2 + 1 = 5 5 = 5 The formula works for the first term.

  2. Beyond Pythagoras

    Hence, the next lengths of the three longest sides would be: 25 + 16 = 41; then 41 + 20 = 61 and 61+ 24 = 85. ( Also, we can see that there is another way to find the longest sides besides as mentioned above.

  1. Beyond Pythagoras

    The longest side of the triangle (c) is always 2 more than b because I am investigating triangles where c = b + 2. Therefore, the formula must be n2 + 1. Area (A) The formula for the area (A)

  2. Beyond Pythagoras.

    = + 2 Now I will complete the square on '4n2 + 6n + 2' to see what the solution to this is. 4n2 + 6n + 2 4(n +3)2 - 9 + 2 4(n +3)2 - 7 4(n +3)2 = 7 (n +3)2 = 1.75 n + 3 =

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work