STRAIGHT-LINE AND COMPASS CONSTRUCTIONS.

Problem 11

Suppose you're shown a line segment and told that it is (exactly) one inch long.

11. Show how you can Construct a line segment sqrt(n) inches long, for any integer n > 2, as well.

Solution

1. Construct a right-angled triangle with sides adjacent to the right-angle of length Sqrt(2) and 1.
2. The hypotenuse of this will be of length l; from Pythagoras, l squared = 2 + 1 = 3, so l = Sqrt(3).
3. Continue the process, using side of length Sqrt(3) and 1; Sqrt(4) and 1; Sqrt(5) and one, ... This will allow you to produce a line segment of length Sqrt(n) for any natural number n.