This preview shows page 1. Sign up to view the full content.
Unformatted text preview: MATH 348: Assignment 3 (ﬁnal version)
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
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