Engineering Calculus Notes 148

Engineering Calculus Notes 148 - AD cuts o² from BE a...

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136 CHAPTER 2. CURVES v. Given a rectangle with sides of length w and h , show that the side s of a square with the same area satisFes w : s = s : h . The construction of a segment of length s given segments of respective lengths w and h is given in Proposition 13, Book VI of Euclid’s Elements . 7. The Conchoid of Nicomedes: Nicomedes ( ca. 280-210 BC ) constructed the following curve: ±ix a point O and a line L not going through O , and Fx a length . Now, for each ray through O , let Q be its intersection with L and let P be further out along the ray so that QP has length a . (a) Show that if O is the origin and L is the horizontal line at height b , then the equation of the conchoid in polar coordinates is r = a + b csc θ. (b) Show that the equation of the same conchoid in rectangular coordinates is ( y b ) 2 ( x 2 + y 2 ) = a 2 y 2 . (c) Trisecting an angle with the conchoid: [ 25 , p. 148] Consider the following conFguration (see ±igure 2.13 ): Given a rectangle BCAF , suppose that the line FA is extended to E in such a way that the line
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Unformatted text preview: AD cuts o² from BE a segment DE of length precisely 2 AB . Now let AG bisect DE . Then AB = DG = GE ; show that these are also equal to AG . ( Hint: ∠ DAE is a right angle.) Conclude that ∠ ABG = ∠ AGB and ∠ GAE = ∠ GEA ; use this to show that ∠ GBA = 2 ∠ AEG . ( Hint: external angles.) ±inally, show that ∠ GBC = ∠ AEG , and use this to show that ∠ ABC = 3 ∠ GBC . How do we use this to trisect an angle? Given an angle, draw it as ∠ ABC where AC is perpendicular to BC . Now using B in place of ) and the line AC in place of L , with a = 2 AB , carry out the construction of the conchoid. Show that E is the intersection of the line through A parallel to BC with the conchoid. But then we have constructed the angle ∠ GBA to be one-third of the given angle. Challenge problems:...
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