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Abstract Algebra: An Introduction
Found in: Page 470
Abstract Algebra: An Introduction

Abstract Algebra: An Introduction

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 lineL . Prove that the line through A parallel to Lis constructible. [Hint: Use Exercise 17 to find a constructible line M through A , perpendicular to L. Then construct a line through A perpendicular toM .]

The line through A parallel to L is constructible.

See the step by step solution

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 C is constructible point and L is constructible line , then line perpendicular to L and passing throughC is constructible.

Step 3: Construct a line perpendicular to  L.

The line L is constructible, then the pointA is also constructible but not on lineL . From the step 1st ,a line perpendicular toL and passing throughA is constructible, say N .

Step 4: Construct a line perpendicular to N.

Similarly, draw a line perpendicular to Nand passing through AO , say O.

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

Hence, the line through Aparallel to L is constructible.

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