Question

Let$\mathit{A}$ be a constructible point not on the constructible line$\mathit{L}$ . Prove that the line through $\mathit{A}$ parallel to $\mathit{L}$is constructible.[Hint: Use Exercise $\mathbf{17}$to find a constructible line M through $\mathit{A}$, perpendicular to $\mathit{L}$. Then construct a line through $\mathit{A}$ perpendicular to$\mathit{M}$ .]

Short answer

The line through $\text{A}$ parallel to $L$is constructible.

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.

Recalling the properties of constructible.

Consider that If $\text{C}$ is constructible point and $L$is constructible line , then line perpendicular to $L$and passing through$\text{C}$ is constructible.

Construct a line perpendicular to  L.

The line $L$ is constructible, then the point$A$ is also constructible but not on line$L$. From the step 1^st ,a line perpendicular to$L$ and passing through$A$ is constructible, say $N$ .

Construct a line perpendicular to N.

Similarly, draw a line perpendicular to$N$and passing through $\text{AO}$, say$O$.

The line $O$is perpendicular to $N$ which is perpendicular to$L$. Hence, $O$is perpendicular to $L$ and also passes through$A$.

Hence, the line through $A$parallel to $L$is constructible.