Constructing Lines of Length sqrt(n) Using Proof by Induction

[cccic]

Proof by Induction--Sqrt(n)

Homework Statement

Prove that if a line of unit length is given, then a line of length sqrt(n) can be constructed for each n.

Homework Equations

N/A

The Attempt at a Solution

So I'm not really sure where to begin...I assumed that a unit length is the representation of the natural numbers (1, 2, 3...n). And then I drew a triangle with unit length 1 on the legs and then constructed the hypotenuse to be sqrt(2). And then I drew a triangle with unit length 1 on a leg and unit length 2 on a leg and then I constructed the hypotenuse to be sqrt(5). But I don't know how to:

i) Write this as a formal proof by induction, or
ii) How to find some sqrt(n) with unit lengths, like sqrt(3) or sqrt(4).

 

[Office_Shredder]

What if you use the sqrt(2) line that you just constructed to make the sqrt(3) line somehow?

 

[cccic]

What if you use the sqrt(2) line that you just constructed to make the sqrt(3) line somehow?

So then (sqrt(2))^2+1^2 = c^2
c=sqrt(3)

Ok, that makes sense. Is there a way to generalize this as a rule or equation to prove with induction?

So far I have:

(1^2)+(1^2)=c^2
(1+1) = c^2
c=sqrt(2)

(sqrt(2))^2+1^2 = c^2
2+1 = c^2
c=sqrt(3)

(sqrt(2))^2+(sqrt(2))^2=c^2
2+2 = c^2
c = 2

etc etc?

How would I prove that I can find sqrt(n) for all n, such that n is a natural number?

 

[Office_Shredder]

Well, it says to use induction. We can use the line with length sqrt(1) to make a line of length sqrt(2). We can use the line with length sqrt(2) to make a line with length sqrt(3). Given a line of length sqrt(n), can you make a line with length sqrt(n+1)? And how does that help you with a proof by induction?