Question

Prove that r is a constructible number if and only if a line segment of length $\left|r\right|$can be constructed by straightedge and compass, beginning with a segment oflength $\text{1}$.

Short answer

A number $\text{r}$ is constructible if and only if the line segment of length$\left|\text{r}\right|$ can be constructed by straightedge and compass.

Step-by-step solution

Conceptual Introduction

Drawing precise lines, line segments, shapes, circles, and other figures with a ruler, compass, or protractor is known as geometric construction.

Definition of constructible point

Constructible points are any points in the plane that can be created using straightedge and compass constructions. If the point $\left(\text{r,0}\right)$ is a constructible point, the real number r is said to be constructible.

If part

Assume a coordinate plane with the point $\text{P}\left(\text{1,0}\right)$ and the origin at $O\left(\text{0,0}\right)$

Assume that r is a constructible value with r>0.

The point $R\left(\text{r,0}\right)$is hence constructible (as in figure 1):

OR is an r-length segment that runs through two constructible points,$\text{P}\left(\text{1,0}\right)$and$R\left(\text{r,0}\right)$

As a result, the line segment OR can be constructed.

Only if part

Assume, on the other hand, that the line segment with length |r| is constructible. Let the line segmentis denoted by AB, with length$\left|r\right|$, and coordinate of point A as$(0,0)$.

Draw a circle with a radius equal to the length of the segment AB with the compass point at point B.

It's important to note that the intersection of a constructible circle and a constructible horizontal axis at point B is constructible.

Furthermore, because the length of the segment AB is r, the coordinate of the point B is$\left(\text{r,0}\right)$

As a result,$\left(\text{r,0}\right)$is a constructible point and so a constructible number.

Hence, proved.