/sci/ - Science & Math » Thread #11374628

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Anonymous Mon 10 Feb 16:47:09 2020 No.11374628 View Reply Original Report

Quoted By: >>11374684

Does there exist a set of three right triangles A,B,C with natural side-lengths, on the same hypotenuse, such that

Area(A) + Area(B) = Area(C)?

Anonymous

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Anonymous Mon 10 Feb 2020 17:08:41 No.11374659 Report

Quoted By: >>11374678 >>11374699 >>11375412

Yes.

Anonymous

Anonymous Mon 10 Feb 2020 17:17:36 No.11374678 Report

Quoted By:

>>11374659
All side lengths need to be natural numbers.

Anonymous

Anonymous Mon 10 Feb 2020 17:19:41 No.11374684 Report

Quoted By:

>>11374628
No.

Area of right triangle with area A, side b and hypotenuse c is

A = b*sqrt(c^2-b^2)
A^2 = b^2 c^2 - b^4
b = sqrt((c^2+/-sqrt(c^4-4a^2))/2)

So b can only take two values. It would need to take at least three values for there to be three different triangles.

Anonymous

Anonymous Mon 10 Feb 2020 17:26:38 No.11374699 Report

Quoted By:

>>11374659
It said natural side lengths, I am assuming that means from the natural numbers. There are infinite solutions using the reals.

Anonymous

Anonymous Mon 10 Feb 2020 21:14:43 No.11375214 Report

Quoted By:

i dont think so tried to prove it but failed. shouldn't be too hard to prove or disprove it anyway im just dumb

Anonymous

Anonymous Mon 10 Feb 2020 22:30:56 No.11375412 Report

Quoted By:

>>11374659
Good start. Just express areas in terms of the radius r of that circle. I guess there's a number theoretical way to see if there are integral solutions to the equation coming up.

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