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Beyond Pythagoras.

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Introduction

Beyond Pythagoras

      Having done Pythagoras’ theorem in detail in and out of class, I am familiar with what his theorem states. I can show that his theorem works on the following triples using details and substitution.

The 3-4-5 triangle

                                         (3) 2 + (4) 2 = (5) 2

                                    => 9 + 16 = 25                        TRUE: 3-4-5 is a Pythagorean triple.

image02.png

 The 5-12-13 triangle

image26.png

                                         (5) 2 + (12) 2 = (13) 2

                                    => 25 + 122 = 169                TRUE: 5-12-13 is a Pythagorean triple.image34.pngimage15.png

image02.png

image36.png

The 7-24-25 triangle

                                          (7) 2 + (24) 2 = (25) 2image01.pngimage00.png

                                     => 49 + 576 = 625                 TRUE: 7-24-25 is Pythagorean triple.

image02.png

image03.png

      During the methods, I have observed the following about the numbers and triangles used so far:

  • The longest side (hypotenuse) is related to the middle length side. i.e. middle side + 1
  • The shortest length side is always an odd number follow the pattern 3-5-7…
  • The middle length side is even and a multiple of 4.

      I have also been asked to find the area and perimeter of the triangles done in the last exercise. The formula for finding the area of any triangle is ½base X height or base X height.                                                                                                                                                                     2                                                                                                                                         2                                           image04.png

Area and perimeter of the triangle 3-4-5

Area = ½ * 3 * 4      Perimeter = 3 + 4 + 5

        = 6 units2                          = 12 units

...read more.

Middle

Difference

24

+ 16

40

+20

60

+24

84

      Finally the side c (The longest length) always seems to be ‘+1’ onto length b giving 41,61,41 triple. I will start at looking at the 9-40-41 triple. This can be seen in the table above.

The 9-40-41 triangle

                                                  (9) 2 + (40) 2 = (41) 2image05.png

                                            => 81 + 1600 = 1681        

   TRUE: 9-40-41 is a Pythagorean triple.

image02.png

     Following the last example, it can be seen that the theorem still holds. So far, any right angled triangle. a, b, c   a2 + b2 = c2image07.pngimage06.png

image08.png

      As suggested, I shall now investigate further right angled triangles where the shortest side (a) is an odd number. Having looked at the short sides (side a) 3,5,7 and 9, I will now continue with 11,13, and 15. The length of the middle side (side b) will continue 60,84 and 112 respectively. The hypotenuse (side c) is side b + 1. For each triangle their Area and perimeter can be seen below.

The 11-60-61 triangle

image10.pngimage09.png

                                                  (11) 2 + (60) 2 = (61) 2image11.png

                                            => 121 + 3600 = 3721

                                                                                    TRUE: 11-60-61 is a Pythagorean triple.image02.png

image12.png

The 13-84-85 triangle

image10.png

image14.pngimage13.png

image15.png

image16.png

The 15-112-113 triangle

image17.png

image02.png

5

11

60

61

132

330

6

13

84

85

182

546

7

15

112

113

240

840

8

17

144

145

306

1224

      As with all sequences, I could continue with the table above by working with the columns, already mentioned.

...read more.

Conclusion

  • The longest side (hypotenuse) is not related to the middle length side. For the other triangles it followed a pattern, b + 1 = c
  • Previously the shortest length side (a) was always an odd number. The 6-8-10 triangle doesn’t follow this pattern.

      But the pattern, which I noticed that was the middle length side (b) is always a multiple of 4 still continues. The area, perimeter and details of the triangles can be seen further on.

The 6-8-10 triangleimage33.pngimage32.pngimage31.png

                                        (6)2 + (8) 2 = (10) 2

                                   => 36 + 64 = 100                TRUE: 6-8-10 is a Pythagorean triple.

The 9-12-15 triangleimage36.pngimage35.pngimage37.png

                                         (9)2 + (12) 2 = (15) 2

                                    => 81 + 144 = 225                TRUE: 9-12-15 is a Pythagorean triple.

Length of shortest side

Length of middle side

Length of longest side

Perimeter

Area

6

8

10

24

24

9

12

15

36

54

If I solve for ‘n’,’n’ will be all the numbers where the perimeter and area will be equal to each other. The solution by trial and error can be time consuming therefore another way would be more appropriate. Solution by graph!

n

-3

-2

-1

0

1

2

3

y

-50

-12

0

-2

-6

0

28

image38.png

image39.pngimage40.png

...read more.

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