SOLUTION: Use Pythagoras’ theorem to construct a line segment with the given length, {{{sqrt(10)}}} Please explain. Thanks

We will use this identity  = .

1. Open the compass for 1 unit of length.
   (This operation is of the kind "always can be made").

2. Draw a straight line in the plane, using the ruler.
   (This operation is of the same kind "always can be made").

3. Using the compass, mark the segment of the length 3 in the straight line.
   (see the lesson HOW TO construct a congruent segment and a congruent angle using a compass and a ruler in this site).

   Let A be the starting point of this segment, and B be its ending point.

4. At the end of this segment, at the point B, erect the perpendicular line to the base straight line, using the compass and the ruler.
   (see the lesson HOW TO bisect a segment using a compass and a ruler on how to do it).

5. In this perpendicular, mark the segment of the length 1 with starting point B using the compass. 
   It will give you the point C in the perpendicular.

6. Connect the points A and C using the ruler.
   The segment AC has the length , by the Pythagorean theorem.

   It is the segment you need.

The problem is solved.