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13.1 Given AB = 1, construct segments of length √2, √√3, √5, √6, √7, √10 in 5 steps or fewer each, making the constructions independent of each other.

13.1 Given AB = 1, construct segments of length √2, √√3, √5, √6, √7, √10 in 5 steps or fewer each, making the constructions independent of each other.

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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[Geometry Over Fields] How do you solve this question? Consider Hilbert's axioms

As a practical application of this result, we will find expressions using nested square roots for some lengths that are constructible with ruler and compass, such as the sides of regular polygons inscribed in a circle. Note that if a particu- lar angle x is constructible, then its trigonometric functions, in particular sin a and cos x, can be expressed using square roots. For example, from the right isosceles triangle with sides 1, 1, √2 we obtain 1 sin 45º = cos 45º = -√2. From the 30° -60°-90° triangle with sides 1, √3, and 2 we obtain cos 60° = sin 30° = 1/1, 2 sin 60° = cos 30° = 1/√√3. 30° 2 √2 450 1 √3 1
Transcribed Image Text:As a practical application of this result, we will find expressions using nested square roots for some lengths that are constructible with ruler and compass, such as the sides of regular polygons inscribed in a circle. Note that if a particu- lar angle x is constructible, then its trigonometric functions, in particular sin a and cos x, can be expressed using square roots. For example, from the right isosceles triangle with sides 1, 1, √2 we obtain 1 sin 45º = cos 45º = -√2. From the 30° -60°-90° triangle with sides 1, √3, and 2 we obtain cos 60° = sin 30° = 1/1, 2 sin 60° = cos 30° = 1/√√3. 30° 2 √2 450 1 √3 1
13.1 Given AB = 1, construct segments of length √√2, √3, √5, √6, √7, √10 in 5 steps or fewer each, making the constructions independent of each other.
Transcribed Image Text:13.1 Given AB = 1, construct segments of length √√2, √3, √5, √6, √7, √10 in 5 steps or fewer each, making the constructions independent of each other.
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