Homework 4, due Tuesday, September 14
In
problems 1, 2, 3, and 5, use only the Greek
construction rules or the GSP rules which can be derived from
them, starting from two points A and B. In each case, briefly
explain your construction by listing the steps. Also explain why
your construction works, using theorems of geometry.
For problems 1, 2, 3, and 5, you may use any theorems of geometry you know, not just the axioms and theorems mentioned in problem 4.
1.(a) Divide the line segment AB into 3 segments
of equal length.
(b) Explain how to divide the line segment AB into n segments
of equal length (n > 1). Illustrate your construction for n
= 7.
2.(a) Construct the circle c with center A
which passes through the point B.
(b) Inscribe an equilateral triangle in the circle c.
(c) Inscribe a regular octagon in the circle c.
3. (a) Construct a square s with base AB.
(b) Construct a square with area twice the area of the square
s.
(c) Construct a square with area three times the area of the square
s.
4. Write a complete proof that the sum of the angles of a triangle is 180 degrees. (Hint: Connect the midpoints of the sides of the triangle, so that the triangle is divided into four little triangles. Prove that all four of the little triangles are similar to the original triangle.) For your proof, you may use the five axioms or any theorem in our list of basic theorems.
5. (Extra credit) Construct a golden rectangle starting with the longest side. (A rectangle with base b and height h is a golden rectangle if b < h and b/h = h/(b+h).)
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