Assignment 3 (55 points, Due Wed. October 7)
Reading
B&B, Chapter 4.
Main ideas
Problems to Turn In
In these proofs you can use theorems in B&B, Chaps 1-3.
You may find that Sketchpad is very helpful in investigating 3.2 and 3.3.
3.1 Lines in a cube (20 points)
(a) B&B p. 101 #12, #14, #15, #16
3.2 Midpoint Quadrilaterals (20 points)
Given a quadrilateral ABCD, the midpoints of the sides are the vertices of a quadrilateral which we call the midpoint quadrilateral of ABCD.
(a) For a quadrilateral ABCD, prove that the midpoint quadrilateral of ABCD is a parallelogram.
(b) If ABCD is a kite, what kind of polygon is the midpoint quadrilateral? Prove your assertion.
(Note: In a problem of this kind you should give the best or strongest possible answer whether or not it is spelled out each time. For example, it is true that the polygon is a parallelogram, but this is not a satisfactory answer.)
(c) If ABCD is a rhombus, what kind of polygon is the midpoint quadrilateral? Prove your assertion.
1.3 Special Midpoint Quadrilaterals (15 points--or more for unusually good answers)
(a) Consider this statement.
If the midpoint quadrilateral of quadrilateral ABCD is a rectangle, then ABCD is a kite.
Prove it if it is true. If it is false, give a counterexample and, if you can, explain for which quadrilaterals the midpoint quadrilateral is a rectangle.
(b) Consider this statement.
If the midpoint quadrilateral of ABCD is a rhombus, then ABCD is a rectangle.
Prove it if it is true. If it is false, give a counterexample and, if you can explain for which quadrilaterals the midpoint quadrilateral is a rhombus.