5.3: Areas, lengths and angles
Problem 174 A rectangular piece of fruitcake has a layer of icing on top and down one side to form a larger rectangular slab of cake (as shown in Figure 3 ).
Figure 3: Icing on the cake
Describe how to make a single straight cut so as to divide both the fruitcake and the icing exactly in half. (The thickness of the icing on top is not necessarily the same as the thickness down the side.)
Problem 175
(a) What is the angle between the two hands of a clock at 1:35? Can you find another time when the angle between the two hands is the same as this?
(b) How many times each day do the two hands of a clock ‘coincide’? And at what times do they coincide?
(c) If we add a second hand, how many times each day do the three hands coincide?
Problem 176 The twelve hour marks for a clock are marked on the circumference of a unit circle to form the vertices of a regular dodecagon ABCDEFGHIJKL. Calculate exactly (i.e. using Pythagoras’ Theorem rather than trigonometry) the lengths of all the possible line segments joining two vertices of the dodecagon.
Problem 177 Consider the lattice of all points (k,m,n) in 3-dimensions with integer coordinates k , m , n . Which of the following distances can be realised between lattice points?
Problem 178
(a) Five vertices A, B, C, D , E are arranged in cyclic order. However instead of joining each vertex to its two immediate neighbours to form a convex pentagon, we join each vertex to the next but one vertex to form a pentagonal star, or pentagram ACEBD. Calculate the sum of the five “angles” in any such pentagonal star.
(b) There are two types of 7-gonal stars. Calculate the sum of the angles at the seven vertices for each type.
(c) Try to extend the previous two results (and the proofs) to arbitrary n -gonal stars.
Problem 179
(a) A regular pentagon ABCDE with edges of length 1 is surrounded in the plane by five new regular pentagons - ABLMN joined to AB , BCOPQ joined to BC , and so on.
(i) Prove that M, N , X , Y lie on a line.
(ii) Prove that MPSVY is a regular pentagon.
(iii) Find the edge length of this larger surrounding regular pentagon.
(b) Given a regular pentagon MPSVY , with edge length 1, draw the five diagonals to form the pentagram MSYPV . Let PY meet MV at A, and MS at B ; let PV meet MS at C and SY at D; and let SY meet VM at E .
(i) Prove that ABCDE is a regular pentagon.
(ii) Prove that A , B , and M are three vertices of a regular pentagon ABLMN , where L lies on MP and N lies on MY .
(iii) Find the edge length of the regular pentagon ABCDE .