What kind of geometric constructions require marking of a unit length?

Question

Wondering if there is a rule to determine if a geometric construction or question requires marking a unit length. For example, constructing a square root length or product length (a*b) requires unit length while finding geometric mean doesn't. What determines this requirement?

Answer 1

The square root of $a$ is the geometric mean of $1$ and $a$. The product $ab$ is $a$ times the ratio $\frac{b}{1}$. All lengths in a diagram scale the same with a homothety; if we start with some collection of given lengths, the new lengths we get are all homogeneous of degree $1$ with respect to those lengths. In order to get something else, like a square root (degree $\frac12$) or a product (degree $2$), we need a standard reference length $1$ to pin down the scaling.