Department of Mathematics

Winter 2009 MATH 330B/539 Geometry Homework 8 Hints

Assigned Homework Set #8

• due Friday, March 13, 2009 at the start of class
• Now Solve This 7.4 part 1 (Let z₁ = a + bi and z₂ = c + di.)
• Now Solve This 7.4 part 2 (use labeling described below)
• 7.2 # 4, 7

• Hint for Now Solve This 7.4 part 2
  • Do this problem carefully and write it up clearly, because it provides the bulk of the solution to one of the other homework problems!
  • Labeling instructions (Do this to make my grading easier and to allow yourself to recycle this work in a later problem.)
    • Starting at the lower left corner and going counterclockwise, label the four quadrilateral vertices A, B, C, and D. Let z₁, z₂, z₃ and z₄ be the four complex numbers that are the coordinates of the four vertices.
    • In the square that has that has A and B as two of its vertices, let E be the vertex opposite B. Let F be the vertex opposite C in the square that has B and C as two of its vertices. Similarly label vertices G and H.
    • Let C₁ be the center of the square that has A and B as two of its vertices. Let C₂ be the center of the square that has B and C as two of its vertices. Similarly label vertices C₁ and C₁.
  • Determine the complex number coordinate of vertex E in the following way.
    • Determine the complex number that represents the vector that starts at vertex A and ends at vertex B. This complex number should involve the complex numbers z₁ and z₂.
    • Determine the complex number that represents this vector rotated 90 degrees clockwise. It should involve z₁ and z₂.
    • Using the rotated vector, determine the complex number that represents vertex E. It should involve z₁ and z₂.
  • Use the fact that C₁ is the midpoint of segment BE to determine the complex number coordinate of center C₁. It should involve z₁ and z₂.
  • Similarly, determine the complex number coordinates of vertices F, G, and H and centers C₂, C₃, and C₄.
  • Determine the complex number that represents vector(C₁,C₃). Call the complex number z_1,3.
  • Determine the complex number that represents vector(C₂,C₄) Call the complex number z_2,4.
  • Show that iz_1,3 = z_2,4.
  • Explain what this tells you about the relationship between vector(C₁,C₃) and vector(C₂,C₄) and about the relationship between segment(C₁,C₃) and segment(C₂,C₄).
• Hint for problem 7.2 #4
  • Labeling instructions (Do this in order to be able to recycle work from an earlier problem.)
    • Starting at the lower left corner and going counterclockwise, label the four parallelogram vertices A, B, C, and D. Let z₁, z₂, z₃ and z₄ be the four complex numbers that are the coordinates of the four vertices.
    • In the square that has that has A and B as two of its vertices, let E be the vertex opposite B. Let F be the vertex opposite C in the square that has B and C as two of its vertices. Similarly label vertices G and H.
    • Let C₁ be the center of the square that has A and B as two of its vertices. Let C₂ be the center of the square that has B and C as two of its vertices. Similarly label vertices C₁ and C₁.
  • Observe that in Now Solve This 7.4 part 2, you proved that for any quadrilateral(ABCD), the segment(C₁,C₃) and segment(C₂,C₄) are perpendicular and have the same length. That result is still valid in the current problem, where quadrilateral(ABCD) is a parallelogram. That takes care of the bulk of the work for the current problem!
  • The only additional thing that we must show is that the two segments bisect each other. That is, they share a common midpoint.
  • Determine the complex number coordinate of the midpoint of segment(C₁,C₃) and the complex number coordinate of the midpoint of segment(C₂,C₄). Show that they are the same number.
  • You are probably frustrated now, because the complex numbers representing the two midpoints don't turn out to be the same number. But that makes sense because so far, we have not used the fact that the quadrilateral is a parallelogram. If the quadrilateral is to be a parallelogram, the four points cannot all be free. Rather, three of the points are free and the fourth one is dependent. Using the fact that the opposite sides are parallel and congruent, you should be able to express z₄ in terms of z₁, z₂, and z₃. When you substitute this expression into your earlier work, you should find that the midpoints do in fact have the same complex number.
• Hint for problem 7.2 #7
  • This problem uses techniques from the previous two assigned problems. It will help to to do those two problems first, because the techniques won't be explained as fully here.
  • Labeling instructions (Do this in order to make my grading easier and your work clearer.)
    • Going counterclockwise, label the four vertices of one square A, B, C, and D.
    • Going counterclockwise, label the four vertices of the other square E, F, G, and H.
    • Label the midpoints of segments AE, BF, CG, and DH as I, J, K, and L, respectively.
  • Let z_A and z_B be the complex numbers that are the coordinates of vertice A and B.
  • Determine the complex number coordinate of vertex C in the following way.
    • Determine the complex number that represents the vector that starts at vertex A and ends at vertex B. This complex number should involve the complex numbers z_A and z_B.
    • Determine the complex number that represents this vector rotated 90 degrees counterclockwise. It should involve z_A and z_B.
    • Using the rotated vector, determine the complex number that represents vertex C. It should involve z_A and z_B.
  • In a similar way, determine the complex number coordinate of vertex D. It should also involve z_A and z_B. (This is not a typo. You should be able to express the coordinate of vertex D in terms of z_A and z_B.)
  • Let z_E and z_F be the complex numbers that are the coordinates of vertices E and F.
  • Using the same technique just used, determine the complex number coordinates of vertices G and H. Those coordinates should involve z_E and z_F.
  • Determine the complex number coordinates of midpoints I, J, K, and L.
  • Using the technique that you used in Now Solve This 7.4 part 2 to show that the segments IK and JL are perpendicular and have the same length.
  • Using the technique that you used in problem 7.2#4 to show that the segments IK and JL bisect each other.

• 7.2 # 1, 2, 6, 8, 11, 12, 13, 14, 15, 17
• Obvious typo in book's hint for #6: Point A should have coordinates (0,1), not point C.

• Hint for #1:
  • Establish notation
    • In this problem, there are nine points to be labeled, and each point will have a corresponding representation by a complex number. It helps to abuse notation by using the complex number that represents a point as the name of the point.
    • Starting at the lower left corner and going counterclockwise, label the four vertices z₁, z₂, z₃, z₄.
    • Let M₁ be the midpoint of segment(z₁z₂), and so on going counterclockwise for M₂, M₃, M₄. These four symbols are the names of the points and they also stand for the complex numbers that represent the points.
  • (a) Let M₅ be the midpoint of segment(M₁M₃) and let M₆ be the midpoint of segment(M₂M₄). Show that M₅ = M₆. That is, show that the complex numbers are the same.
  • (b) Let M₇ be the midpoint of segment(z₁z₃), let M₈ be the midpoint of segment(z₂z₄), and let M₉ be the midpoint of segment(M₇M₈). Show that M₉ = M₅.

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Last updated March 8, 2009.