
Cake geometry
Jill's Birthday cake is square, with side lengths of 20cm. She cuts it with 4 straight line cuts, all intersecting at point P and at 45 degrees to one another. with two of the cuts parallel to the sides of the cake. Thus the cake is cut into 8 pieces for the four boys and four girls at her party. Jill marks B and G in the icing of alternate pieces of cake about the point P for the boys and girls respectively. Of course, not everyone necessarily gets the same amount of cake.
A) Show that if P is any point on a diagonal of the cake, then the boys and girls get the same total of cake
B) Show that if P is any point on the cake then the boys and girls will get the same total amount of cake
C) If P is 6cm from one side and 8cm from the other side, show that the total amount of cake for the boys is the same as the total amount of cake for the girls.