Area and perimeter of the triangle 3-4-5
Area = ½ * 3 * 4 Perimeter = 3 + 4 + 5
= 6 units2 = 12 units
Area and perimeter of the triangle 5-12-13
Area = ½ * 5 * 12 Perimeter = 5 + 12 + 13
= 30units2 = 30units
Area and perimeter of the triangle 7-24-25
Area = ½ * 7 * 24 Perimeter = 7 + 24 + 25
= 84 units2 = 56 units
I can continue the patterns so far seen in the following way. Side a (the shortest side) is obviously odd numbers: 3,5,7 => 9,11… but side b is a little more complicated.
The difference between each length can be seen going up in each multiple of 4. So this pattern will continue:
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) 2
=> 81 + 1600 = 1681
TRUE: 9-40-41 is a Pythagorean triple.
Following the last example, it can be seen that the theorem still holds. So far, any right angled triangle. a, b, c a2 + b2 = c2
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
(11) 2 + (60) 2 = (61) 2
=> 121 + 3600 = 3721
TRUE: 11-60-61 is a Pythagorean triple.
The 13-84-85 triangle
The 15-112-113 triangle
As with all sequences, I could continue with the table above by working with the columns, already mentioned. I will now look for general patterns in these sequences. I shall issue each triangle done so far with a number n. (As seen in the first column in the two graphs) So n => 1 => 3-4-5, n => 2 => 5-12-13 and so on.
Another pattern that I observed earlier was in the ‘b’ column. These numbers involve ‘gaps’ of multiples of 4. I have noticed that these numbers multiplied by 4 are in fact triangle numbers.
I was also asked to investigate any patterns between the perimeter and the area of the triangles I have looked at so far.
Using my general patterns (GP) for sides a, b and c. I can use a formula to obtain a GP for the perimeter.
As P = a + b + c
=> p = (2n + 1) + (2n2 + 2n) + (2n2 + 2n + 1)
=> 4n2 + 6n + 2
As A = ½ab
=> ½(2n +1) (2n2 + 2n)
=> ½(4n3 + 2n + 2n2 + 4n2)
=> ½(4n3 + 6n2 + 4n2)
=> 2n3 + 3n2 + n
=> n(2n2 + 3n + 1)
=> n(n+1) (2n +1)
I noticed from my table of results so far that the triangle/Pythagorean triple, 5-12-13, has both perimeter and area the same as each other. In this triangle therefore, 4n2 + 6n + 2 must equal n(n +1) (2n +1). I could find other triangles where P = A. If I solve this for n:
4n2 + 6n + 2 = 2n3 + 3n2 + n
=> 0 = 2n3 - n2 - 5n - 2
=> 0 = n(2n2 - n - 5) - 2
1 => 1 The first difference is +2, +3, +4, +5 and +6.
2 => 3 The second difference is +1
3 => 6
4 => 10
5 => 15
6 => 21
While investigating, I discovered that there were some triples that did not follow the normal pattern. These were the 6-8-10 triangle and the 9-12-15 triangle. The reasons why they didn’t follow the normal patterns were:
- 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 triangle
(6)2 + (8) 2 = (10) 2
=> 36 + 64 = 100 TRUE: 6-8-10 is a Pythagorean triple.
The 9-12-15 triangle
(9)2 + (12) 2 = (15) 2
=> 81 + 144 = 225 TRUE: 9-12-15 is a Pythagorean triple.
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!
