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
I worked out the nth term for the length of the sides, area and perimeter in ODD right-angled triangles.
I then looked at the multiples of odd triples and found out their nth terms using the sequence formula.
TRIPLE: 3,4,5
TRIPLE: 5,12,13
TRIPLE: 7,24,25
I put the nth terms in a table against the number triple it was (t). I was then able to work out formulas for the side lengths areas and perimeters of any ODD TRIPLE.
Using the formula : length of shortest side = (2t +1)n
I can work out the length of the shortest side for the 6th multiple of the 4th odd triple.
6th multiple : n = 6
4th odd triple : t = 4
(2t + 1)n
(2 x 4 + 1) x 6 = 54
When n and m are any positive integers these general formulas can be used to generate Pythagorean triples.
eg: n = 2
n² + m² m = 1
n² - m²
2mn
shortest side = 3
middle side = 4
longest side = 5
Therefore:
(n² - m²)² + (2mn)² = (n² + m²)²
(2² - 1²)² + (2 x 1 x 2)² = (2² + 1²)²
(4 – 1)² + 4² = (4 + 1)²
3² + 4² = 5²
9 + 16 = 25
a + b = c therefore this must be a Pythagorean triple.
Using these general formulas I can work out the Pythagorean in which area = perimeter.
Formula for Perimeter :
(n²-m²) + 2mn + n² + m²
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 = 5 9 + 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
