Construction of $\sqrt{ab}$ user ruler and compass

Question

Q. Given two line segments of length a and b. Draw a line segment of length $\sqrt{ab}$ using a ruler and compass.

I didn't get any idea how to approach to the solution.

Answer 1

Draw a line of length $a+b$. Construct the perpendicular line in the point they joint. The semicircle over $a+b$ cuts that perpendicular. Now the distance between that point and the joining point is $\sqrt{ab}$ due to Euclid.

Answer 2

Let the base of the large triangle be $a+b$, and the height $h$. By similarity of the small triangles,

$$\frac ha=\frac bh$$ so that $$h=\sqrt{ab}.$$

Answer 3

Join segments of length a and length b together on the same line. Call where they join Point J. Call their Midpoint M. Construct a circle centered at M through either end point of a+b. Construct a line perpendicular to a+b through point J. Call where it intersects the circle point Q. The length of QJ is $\sqrt{ab}$ as proven elsewhere. These above constructions follow from Euclid's postulates I, III, SAS, and ASA. So the constructions should be valid in neutral geometry.

A related approach. Construct a rectangle having one sidelength a and one side length b. Find a square having the same area as the starting rectangle. The side length of thes square will be $\sqrt{ab}$.