The side lengths of a right-angled triangle are in geometric progression and the shortest side has length 2. What is the length of the hypotenuse?
We have that the sides are
2 , 2r , 2r^2 so.....by the Pythagorean Theorem
2^2 + (2r)^2 = (2r^2)^2
4 + 4r^2 = 4r^4 divide through by 4
1 + r^2 = r^4 rearrange
r^4 - r^2 - 1 = 0 let r^2 = m and we have
m^2 - m - 1 = 0 complete the square on m
m^2 - m + 1/4 = 1 + 1/4
(m - 1/2)^2 = 5/4 take the positive root
m - 1/2 = √5 / 2
m = [ 1 + √5 ] / 2 = r^2 ⇒ this is known as the irrational number "Phi"
So
m = r^2 = Phi
√m = r = √Phi
So.....the side lengths are
2 , 2 √Phi , 2Phi
So....the hypotenuse is 2Phi units in length
Proof :
2^2 + (2√Phi)^2 = (2Phi)^2
4 + 4Phi = 4Phi^2 divide through by 4
1 + Phi = Phi^2 which is an identity