Constructing the Square Root of x

The next construction that we will need is the construction of square roots. Suppose the length of AB is one, and the length of CD is x. We want to construct a segment of length Öx. First construct the line j though A and B and the segment m though C and D. Construct the circle c₁ having center B and radius of length x. Construct the intersection E of the line j and the circle c₁, so that E and A are not on the same side of B. Let l be the segment AE, and let F be its midpoint. Construct the circle c₂ with center F through the point E. Construct the line m perpendicular to l though B. Construct one of the intersection points of m and c₂, and label it G. The length of segment BG is Öx.

Click here for a GSP script for the construction of a segment of length Öx,

given AB = 1 and CD = x.

Click here for a GSP sketch of the construction.

Click here for a proof of why this construction works. (This is from Homework 2, problem 5a.)