Assignment 3

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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ...
View Full Document

This note was uploaded on 12/10/2011 for the course MATH 348 taught by Professor Karigiannis during the Summer '06 term at McGill.

Ask a homework question - tutors are online