Constructing the reciprocal of a segment

Question

How can one construct the reciprocal length of a line segment? For example, given any line segment a, how can $\frac{1}{a}$ be constructed?

I was told that it can be solved by creating similar triangles, but I do not get it.

Answer 1

There are several answers to your question. However, the question is meaningless unless the length $1$ is also given, since $\frac 1a$ has different units than $a$ does.

In this diagram, $AD$ is your given length $a$, and $AE$ is the unit length, $1$. Ray $\overrightarrow{AE}$ is an arbitrary ray that starts at point $A$. By construction, $AB=AE=1$ and segment $\overline{BG}$ is parallel to segment $\overline{DE}$.

Then triangles $ABG$ and $ADE$ are similar, so we have the proportion

$$\frac{AG}{AE}=\frac{AB}{AD}$$

or

$$\frac{AG}{1}=\frac{1}{a}$$

So $AG$ is your desired reciprocal $\frac 1a$.