Assignment 3

# Assignment 3 - MATH 348: Assignment 3 (ﬁnal version)...

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Unformatted text preview: MATH 348: Assignment 3 (ﬁnal version) Leonid Chindelevitch July 22, 2008 1. Show that if a regular n-gon can be constructed with a ruler and compass, then a regular 2n-gon can also be constructed with a ruler and compass. Hint: Use trigonometry and half-angle formulas. 2. Using the doubling method of Archimedes sketched in class, compute a sequence of approximations to π from above and below by computing the perimeters of regular polygons inscribed and circumscribed in a circle of unit diameter starting with a square. Stop when the diﬀerence between your upper and lower bound is ≤ 10−5 . How many digits of accuracy did you obtain? 3. Determine for which integers n an angle of n◦ can be constructed with a ruler and compass. 4. Verify the following geometric solution of the quadratic equation x2 − gx + h = 0 (with h = 1 if g = 0). The real roots are given by the intersection of the x-axis and the circle with diameter having endpoints (0, 1) and (g, h). 5. Given compass points that are the vertices of an equilateral triangle, give a compass construction for the center of the triangle. 6. Suppose 0 < x < π/2. Prove that an angle of x can be constructed with a ruler iﬀ tan(x) ∈ Q. Hence or otherwise, show that it is impossible to construct an equilateral triangle with a ruler alone. BONUS. Construct a square each of whose extended sides passes through one of four given points. BONUS. Construct the circles that are tangent to three given circles. [Very hard!] 1 ...
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## This note was uploaded on 12/10/2011 for the course MATH 348 taught by Professor Karigiannis during the Summer '06 term at McGill.

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