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Q17E

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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 $C$ be a constructible point and $L$ a constructible line. Prove that the linethrough $C$ perpendicular to $L$ is constructible. [Hint: The case when $C$ is on was done in Example 1. If is not on $L$ and $D$ is a constructible point on $L$ , the circle with center $C$ and radius $CD$ is constructible and meets $L$ at the
constructible points $D$ and $E$ . The circles with center $D$ , radius $CD$ and center
$E$ , radius $CE$ intersect at constructive points $C$ and $Q$ . Show that line $CQ$ is
perpendicular to $L$ .]

The line through a constructible point $C$ and perpendicular to $L$ is constructible.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Introduction

To coordinate the plane, consider any two points and treat the distance between them as a unit length. Any point in the coordinate plane that may be built with straightedge and compass constructions is referred to as constructible. If a line goes through at least two constructible points, it is said to be constructible.

Step 2: Assuming the claim to be true

Let’s suppose that a line through a constructible point and perpendicular to a constructible line is constructible. Now, consider the point $C$ and the line $L$ , which are both constructible.

Assume that $C$ is a constructible point on the line $L$ , and that $D$ is also a constructible point on the line $L$ . Draw a circle with radius $CD$ that meets the line at points $D$ and $E$ using the compass at constructible point $C$ (as in figure ).

Figure 1

Step 3: Constructing circles

Make two circles, one with the center at $E$ and the radius $ED$ , and the other with the center at $D$ and the radius $ED$ . Assume that the circles overlap at A and B (as in figure $2$ ).

Figure 2

Step 4: Considering triangles

Let consider triangles $AEC$ and $ACD$ (as in figure $3$ ).

Figure 3

$AE$ and $AD$ are both equivalent to $ED$ on both sides. The triangles' sides $EC$ and $CD$ are equal, while the side $AC$ is shared by both. As a result of the side-side-side congruence rule, the triangles $AEC$ and $ACD$ are congruent.

As a result, the angles $AEC$ and $ACD$ are complementary angles. As a result, the line $AC$ is parallel to the line $L$ .

Step 5: Assuming C is not on line L

Let's assume that point $C$ is not on the constructible line $L$ . Consider the point $D$ on the line L that is constructible.

Draw a circle of radius $CD$ that meets the line $L$ at the points $C$ and $D$ by placing the compass point at $C$ (as in figure $4$ ).

Figure 4

Step 6: Constructing circles and considering its intersection

Make two circles, one with a center at $D$ and a radius of , $CD$ and the other with a center at $E$ and a radius of $CE$ . Consider the intersection of the two circles at points $C$ and $Q$ (as in figure $5$ ).

Figure 5

Step 7: Considering the triangles

Take a closer look at the $DOQ$ and $DOC$ triangles (as in figure $6$ ).

Figure 6

Step 8: Claim conclusion

$DQ$ , being the radius of the circle with the center at $D$ is equal to the side $CD$ $CD$ .

Both triangles have the same side $DO$ , and the side $CO$ equals $OQ$ (since the circle of the circle bisects the chord).

As a result of the side-side-side congruence rule, the triangles $DOC$ and $DOQ$ are congruent.

As both the angles $DOC$ and $DOQ$ are equivalent and complementary, so the angle $DOC$ is measured at 90 degrees. As a result, the line $CQ$ is parallel to the line $L$ .

The claim's proof is now complete.

Hence,Proved.

Most popular questions for Math Textbooks

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

Prove that r is a constructible number if and only if a line segment of length $| r |$ can be constructed by straightedge and compass, beginning with a segment oflength $1$ .

Use straightedge and compass to construct a line segment of length $1 + \sqrt{3}$ , beginning with the unit segment.

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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