[unreality]

A rectangle exists with dimensions ab, where 'a' and 'b' are the values of the lengths of its two sides. Find a ratio for a/b so that the following is true:

When cut in half [cut so that the shorter side is left intact and the longer side is halved], the ratio of a:b or b:a (it will alternate a/b and b/a) is perserved and remains constant, no matter how many times you cut the rectangle into smaller and smaller rectangles.

Prove that this is the only ratio of sides that allows this property (excluding negative side lengths of course )

[Guest]

The ratio is...

square root of 2 : 1

[Guest]

As for the proof...

Well I was always bad at proofs, but I'll try to explain it the best that I can.

Assume side B is the longer side, and A is the shorter side, so the ratio is B:A.

Then, if we cut B in half, the ratio must be the same, the new ratio being A:(B/2).

Therefore, B/A must equal A/(B/2)

Which simplifies to B/A=2A/B or B^2 = 2A^2

Therefore, (B^2 / A^2) = 2, take the square root of both sides to obtain B/A = square root of 2.

So the only (positive) ratio that works is square root of 2 to 1.

[unreality]

Good job

I solved it as sqrt(2)/2 as the ratio, but then realized that was 1/sqrt(2) which is just the opposite of a/b, the other ratio of the alternating. The answer is √2

(hopefully the sqrt symbol shows)