Assignment 7 (Due Wed, Nov 10)

7.1 Midpoint Quadrilateral

Last week you proved in 6.4 that the midpoint quadrilateral MNOP of any quadrilateral ABCD is a parallelogram.  In this problem continue this work.

a)      What is the ratio of the area of MNOP to ABCD? Prove it.

b)      Prove that MNOP is a rectangle if ABCD is a kite. 

c)      Prove the converse of (b) if it is true.  Otherwise state and prove when MNOP is a rectangle.

d)      Prove that MNOP is a rhombus if ABCD is a rectangle.

e)      Prove the converse of (d) if it is true.  Otherwise state and prove when MNOP is a rhombus.

7.2 Quadrilateral in a circle

Let ABCD be a quadrilateral inscribed in a circle.  If |AB| = |CD|, prove that BC is parallel to DA.  What is the name of this kind of quadrilateral?

7.3 Cake problem

Euclid has brought a square cake to a potluck dinner.  It has icing on the top and the sides.  There are five people at the party.  How can Euclid cut the cake so that each person gets an equal amount of cake and also an equal amount of icing?  (Each piece of cake should be one connected piece, not a bunch of separate little pieces.)