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Q19E

Expert-verified

Abstract Algebra: An Introduction

Found in: Page 470

Book edition 3rd

Author(s) Thomas W Hungerford, David Leep

Pages 608 pages

ISBN 9781111569624

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Short Answer

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

The line through $A$ parallel to $L$ is constructible.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

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 $C$ is constructible point and $L$ is constructible line , then line perpendicular to $L$ and passing through $C$ 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 1^st ,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 $AO$ , 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.

Most popular questions for Math Textbooks

Prove that the set of all constructible numbers is a field.

Let a,b be constructible numbers. Prove that $a + b$ and $a - b$ are constructible

Prove that the midpoint of the line segment between two constructible points is a constructible point. [Hint: Adapt the hint to Exercise $17$ .]

Prove that every integer is a constructible number.

Prove the converse of Theorem $15.6$ : If r is in some quadratic extension chain , then r is a constructible number.[Hint: Theorem $15.1$ and Corollary $15.2$ .)

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