Assignment 8 (Due 11/17/04)

8.1 Another cake problem

Little Euclid has again brought a cake to a party this week. It has icing on the top and the sides. But this time he dropped his cake pan, so the cake came out in the kite shape below. 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? Carefully draw the five pieces of cake in this figure, explaining how any special points are located. (Each piece of cake should be one connected piece, not a bunch of separate little pieces.)

8.2 Circumcenter of an Isosceles Triangle

Given an isosceles triangle ABC, with |AB| = |AC| = b and |BC| = a, what is the radius R of the circumcircle of ABC? (The answer should be in terms of a and b, if possible.)

8.3 Properties of Dilations

Suppose that D is a dilation with center O and ratio r (either r > 0 or r <0).

a) Prove that for any points A and B, that |D(A)D(B)| = |r| |AB|. Hint: Draw a figure.

b) If m is a line, prove that its image D(m) is either the line m itself (if O is on m) or is a line m' parallel to m.

c) If ABC is a triangle, prove that the triangle D(A)D(B)D(C) is similar to triangle ABC.

8.4 Composition of isometries

a) Suppose that S and T are isometries. Prove that the composition ST is also an isometry.

b) Suppose that M and N are line reflections, prove that MN is the inverse of NM.

8.5 Double line reflection (parallels)

Let m and n be parallel lines, distance d apart. Let M and N be the isometries that are reflection in m and n. Given points A and B, with A' = M(A) and B' = M(B) the reflections of these points in m and A'' = N(A') and B'' =N(B') be the reflections of A' and B' in n.

a) Prove that |AA''| = |BB''| = 2d. (You should check more than one case for A or B: the case points between m and n and the cases of points not between m and n.)

b) Show that AA''B''B is a parallelogram, provided that AB is not perpendicular to m and n.

c) Let p and q be two more lines parallel to m and n, with distance d from p to q. Denote by P and Q the line reflection transformation in p and in q. For any A, prove that either PQ(A) = MN(A) or QP(A) = MN(A).

d) From (b) and (c), prove that either PQ = MN or QP = MN. (Note: The hardest part of this proof is to figure out what the statements mean and then how one can check equality.)

8.6 Point Symmetry in the Coordinate Plane

a) Draw a picture and write a convincing explanation that on the number line, for two numbers a and b, the point m = (1/2) (a + b) is the midpoint of a and b.

b) Then for two points A = (a1, a2) and B = (b1, b2), find a formula for the midpoint M and explain why the formula is correct.

c) Then solve this equation for B to find the formula for the point reflection of A with center M.

d) If E = (e1, e2) and F = (f1, f2), then if H_E is point reflection with center E and H_F is point reflection with center F, then use (c) to find the formula for H_EH_F(x, y) for any point P = (x, y).