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Q19E

Expert-verified
Found in: Page 470

### Abstract Algebra: An Introduction

Book edition 3rd
Author(s) Thomas W Hungerford, David Leep
Pages 608 pages
ISBN 9781111569624

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

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

See the step by step solution

## Step 1: Conceptual Introduction

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

## Step 2: 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.

## Step 3: 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 1st ,a line perpendicular to$L$ and passing through$A$ is constructible, say $N$ .

## Step 4: 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.

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