Two given line segments have the lengths a and b.
Using only a compass and an unmarked straightedge construct a line segment whose length is equal to the root-mean-square of a and b.

First
construct a rectangle with sides of length a and b.
Then construct a circle whose diameter is a diagonal, PQ, of the rectangle.
Now draw the perpendicular bisector of that diagonal to cross the circle at R.
The distance |PR| is the RMS value of a and b.

Proof

|PQ| = sqrt(a^{2} + b^{2}) is the hypotenuse of the right
isosceles triangle PQR,