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(a² + b²) is the hypotenuse of the right isosceles triangle PQR,

so |PR| = |PQ|/sqrt(2) = sqrt((a² + b²)/2).

Posted by Harry on 2014-06-18 13:36:48