Question

Prove that (r,s) is a constructible point if and only if r and s are constructible numbers.

Short answer

(r,s) is a constructible point if and only if r and s are constructible numbers.

Step-by-step solution

Recalling the properties of constructible.

We know that If C is constructible point and L is constructible line , then line perpendicular to Land passing through C is constructible.

Proving s is constructible.

Suppose $\left(r,s\right)$be a constructible point. X-axis is constructible.

Hence by the above statement, draw a line perpendicular from the point $\left(r,s\right)$to the X-axis which gives us a point $\left(r,0\right)$ and hence this implies that L, r is constructible.

Similarly , in the same way, draw a line perpendicular from the point $\left(r,s\right)$to the Y-axis which gives us a point $\left(0,s\right)$ and hence this implies that is constructible.

Proving is  constructible.

Assume that r and s are constructible numbers.

Then $\left(r,0\right)$and $\left(0,s\right)$are constructible. Now draw a line perpendicular to the X-axis passing through $\left(r,0\right)$and a line perpendicular to the Y-axis passing through $\left(0,s\right)$.

Hence , this implies that $\left(r,s\right)$ be a constructible point.