Assignment 9 (due Wednesday, 12/1)

Problem 9.1 Rectangular Cake

Little Euclid has made another cake. This one is a rectangular cake 4 inches by 6 inches. Again there is icing on the top and the sides. How can he cut this cake into 5 pieces as before, so that each person gets the same amount of cake and the same amount of icing? Explain briefly why your method works.

Problem 9.2 Altitudes

We have already proved that for a triangle ABC the perpendicular bisectors of the sides are concurrent. This problem shows a trick for proving that altitudes are concurrent by using this fact. Let A'B'C' be the midpoint triangle of ABC, whose vertices are the midpoints of the sides of ABC.

(a) Explain why the perpendicular bisectors of the sides of ABC are the altitudes of A'B'C'. Show that you can conclude that the altitudes of A'B'C' are concurrent.

(b) Prove that for any triangle ABC, there is a triangle DEF so that ABC is the midpoint triangle of DEF. Therefore, the altitudes of ABC are the perpendicular bisectors of the sides of DEF.

(c) Summarize your work to explain why the altitudes of any triangle are concurrent. The point of concurrency is called the orthocenter of ABC. Is this orthocenter always inside the triangle?

Problem 9.3 More Altitudes

Given a triangle ABC, let a be the circle with diameter BC, b the circle with diameter CA, and c the circle with diameter AB.

(a) Construct a figure showing an acute-angled triangle ABC (not a special triangle) and these 3 circles.

(b) Construct points DEF on the sides of the triangle so that AD, BE and CF are altitudes of ABC.

(c) State and explain the relationship between the feet of the altitudes of ABC and the circles a, b, c.

(d) Find some angles in the figure that are equal because they are inscribed angles with the same arcs.

Problem 9.4 Half-turns and Line Reflections

Let A be a point and m be a line. Let H be the half-turn with center A and let R be the line reflection in m.

(a) Prove that if A is on m, then HR = RH = a line reflection S. Tell which line is the mirror line for S.

(b) Prove that if A is not on m, the HR = a glide reflection G and RH = the glide reflection G^-1. Tell which line is the special line in the definition of G and which translation is the special translation in the definition of G.

Hint: There are many ways to do this. Any one is OK. You can use a convenient coordinate system. You can factor H into line reflections in a convenient way. You can compute the image of a triangle EFG and use the first Fundamental Theorem.

Problem 9.5 Dihedral Angle

Consider a pyramid ABCP with base an equilateral triangle ABC with side length s and a vertex P so that the other 3 triangular faces such as ABP are isosceles right triangles with right angles at P. 

(a) Construct a triangle that is a cross-section of the pyramid chosen so that one angle of the triangle is equal to the dihedral angle between the triangular faces ABP and triangle ABC. Find the side lengths of this triangle. Show your work.

(b) Calculate the exact dihedral angle (possibly as an inverse trig function) between the triangular faces ABP and ABC.

(c) Calculate the dihedral angle between faces ABP and BCP. (This one you should be able to do in your head when you visualize the shape!)