My right angled triangle has sides measuring n, n^2, n^3.
Evaluate its area, perimeter and centre of gravity.
n^2 + n^4 = n^6 a Golden equation
If t = n^2, t(t^2 - t - 1) = 0
t = (1 ± √5)/2 or phi
Since this is a triangle, negative and complex solutions are rejected
n^2 = (1 + √5)/2 = 1.6180339887499
Trying to simplify by setting n = a + b√5 leads to complex answers
n = 1.27201964951407
n^2 = 1.6180339887499
n^3 = 2.05817102727149
If the triangle's vertices are at (0,0), (n,0), (0,n^2)
Area = 1.02908551363575
Perimeter = 4.94822466553546
Centroid: (n/3, (n^2)/3)
(0.424006549838023, 0.539344662916632)
(I'm fairly sure the CG of an xyz right triangle is (x/3, y/3) from the right angle.)
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Posted by Larry
on 2024-01-15 18:43:45 |
Answer
