Math 119: Ideas in Geometry - Homework Assignments and Comments

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This webpage will include the homework assignments and responses to queries about the homework assignments.

Homework Assignment 4

  • Use compass and straightedge alone when doing constructions. Use no other tools.
  • Your constructions should be neat, clean, and clearly labelled (if appropriate).
  • A construction is a method for solving problems. It is not enought to guess an answer which "looks right." Your construction should be (at least in theory) 100% accurate.
  • For the same reason, your construction should include a list of the steps you used to solve the problem. This should be neat and in a logically correct order. Do not try and fit this list of steps into a tiny corner of the sheet of paper with the construction. If necessary, list the steps on a separate sheet of paper.
  • If you are using a basic construction from Section 4.1, just list which construction you are using and how (e.g., construct a line perpendicular to line L through point A). You do not have to repeat the list of steps necessary to do that basic construction
  • Use the lengths, angles, etc., from this pdf file when doing your constructions. Constructions using different lengths will not receive any credit. This sheet of information will also be handed out in class.
  • You may find it helpful to use the Geogebra or Euclidean Toolbox online tools that we have already met this semester. This will allow you to experiment with constructions without drawing a thousand pictures. Use these to help figure out what the correct construction is but the construction you hand in must be complete and done by hand using compass and straightedge alone.
  • You must show all your work. This includes drawing enough of each arc so that it is clear what the radius and centre of the circle is.

    • Section 4.1

    Problem 15. (See above pdf file.)

    Section 4.2

    Problems 7, 17, 24, 26. (See above pdf file.)

    Section 4.3

    Problems 3, 10, 14, 33.

    Section 4.4

    Problem 4 (using compass and straightedge only).
    Due Date: Monday, 26th of April.

    Homework Assignment 3

    Section 3.1

    To some extent mathematics is an experimental science. Mathematicians look at examples and try different things in order to figure out what is true. After gaining a feeling for what is true, they then try and prove that their hunch is correct. With this in mind, do some experimentation (perhaps using one of the two links below) to explore the questions below, decide what you think is true, and then provide an explanation for why your answer is correct.
  • Geogebra An online tool that you can use to help visualize and experiment with the ideas of Euclidean Geometry. Also keep this in mind for later in the semester when we will do some compass and straightedge constructions. You can use this online without installing any files by clicking on Applet Start.
  • The Euclidean Toolbox An online tool to help you explore the world of euclidean geometry. See for yourself what theorems hold. It will also be useful later when we consider compass and straightedge constructions. THIS IS PROBABLY THE MOST STRAIGHTFORWARD ONE TO USE. (It may also help you to keep things straight in your mind if you draw different things in different colours - the triangle in red, the perpendicular bisector in blue, etc.)
  • If you are in one of the University's computer labs, it should have at least one of the following computer programs: (The Geometer's) Sketchpad, Dr Geo. You could also use one of these to experiment.
  • All of the above work in pretty much the same way. Select "point" from the menu or toolbar. Click where you want the point to go. Repeat. Select "line" or "line segment". Click on the two points that you want to connect with a line. Repeat until you have a triangle. Select "perpendicular bisector". Now select the two endpoints of the line segment you want to bisect. Voila! It now draws the perpendicular bisector. Select "none" or the cursor arrow symbol to drag points around and see how your triangle (and everything that you built that depends on that triangle) changes.
  • Your answers to the following should include a sketch of the situation showing what you have learned from your experiments. It should include a statement of what you think holds. It should include an explanation of why that statement holds. You do NOT need to include the sketches or computer pictures that you used to reach these conclusions.
  • a) When is the median to a side equal to the perpendicular bisector of that side?
  • b) Can the altitude to a side lie entirely outside the triangle (except for one point)?
  • c) Can the orthocentre of a triangle lie outside the triangle?
  • d) Can the perpendicular bisector of a side lie entirely outside the triangle (except for one point)?
  • e) Can the circumcentre of a triangle lie outside the triangle?
  • f) Where is the circumcentre of a right triangle?
  • g) Does a Miquel point always lie inside the triangle?
    (Hint: Draw a triangle. Add 3 points on the sides of the triangle. Use the "circumscribed circle" or "circle through 3 ponits" command to get the circles appearing in the definition of the Miquel Points. What happens as you move these 3 points around?)
  • h) Problem 18, Section 3.1 (This overlaps with many of the above questions - just refer to your answers for the above question for those parts.)
  • A single counterexample is enough to show a statement is false - one picture shows that it is not always true. To show that a statement is true requires more explanation.
  • Please let me know if you have any questions.

    • Section 3.2

    Problems 8, 18, 22, 27, 29, 38, 43, 47

    Due Date: Monday, 29th of March.
    If you want, you can hand in Problems 27 and 29 on Wednesday, 31st of March. However all the other problems are still due Monday, 29th of March

    Homework Assignment 2

    If you are using the online course notes, please note that for Section 2.4 the numbering of the problems online is different than the numbering used in the print version. The problem numbers below are for the print version. Click here for a pdf file that agrees completely with the print version. In the copy of the notes on the separate Course Notes webpage, these would be Section 2.4 Problems 7, 27, 36.


    Section 2.1 Problems 13, 17
    Section 2.2 Problems 5, 8, 10, plus the problem (*) below
    Section 2.3 Problems 4, 6
    Section 2.4 Problems 2, 22, 31
    Problem (*): Consider the statement ∼ (P ∨ Q) = (∼ P) ∧ (∼Q). Explain why this logical statement is true in the following ways:
  • Choose suitable "real-world" statements P and Q. What do ∼ (P ∨ Q) and (∼ P) ∧ (∼ Q) say in this situation? Explain why they are two ways of saying the same thing.
  • Use a truth table to show that ∼ (P ∨ Q) = (∼ P) ∧ (∼ Q).
  • Show that the Venn diagrams corresponding to these two equations are equal. What effect will ∼ have on Venn diagrams?

    • Due Date: Monday, 1st of March.

    Comments/Responses to queries

    Section 2.2 Problem 10: This problem is probably easiest to explain with a very concrete real world traffic law. Yes, you can use a "meta-law" like, "If I break a traffic law, then I will get a ticket", but that is probably just making things harder for yourself. Be more specific: if I change lanes or if I come to a red traffic light or ...
    Section 2.4 Problem 2: Careful! What is the size of the white square in these pictures? It is not a or b.
    Proofs by Picture: These are quite simple if you use the right approach. For example, let's look at Problem 22. Describe the first picture (hint: your answer should include 1+2+3+...+n - the question tells you that you should be looking to find this expression inside your picture). Describe the second picture (hint: your answer should include n2 and n - the question tells you that you are interested in these expressions.) Why are both pictures equal? Set them equal and tidy up until we have exactly what is in the statement of the problem.
  • Describe both pictures: the statement of the problem tells us what to look for. Show where these expressions come from in your picture.
  • A one-picture proof is basically the same: we describe the picture (its area or the number of dots or ...) in two different ways, looking for the expressions given in the statement of the problem.
  • Say why these two pictures (or two ways of describing one picture) are equal.
  • Set the two pictures equal and perhaps tidy up a little.

    • Homework Assignment 1

    Section 1.1 Problems 2, 8, 15, 21
    Section 1.2 Problems 10, 18, 20, 26, 27
    Section 1.3 Problems 8, 11, 20, 23, 30
    Due Date: Monday, 8th of February
    Homeworks will be due on the Monday, returned on the Wednesday, and the first Hour Exam (covering Chapter 1) will then be held on the Friday, the 12th of February.
    Come by my office with any questions you have or send me an email. I will post here (with names removed) my responses to any questions which I think other people in the class will also be interested in.
    As I have said in class, start early. I will tell you when we have covered enough material to do each problem. Do not wait until the night before to start these. This class does not have a lot of homework, but if you try to do three weeks of homework in one night, it will seem like an awful lot of work.

    Comments/Responses to queries

    Section 1.1 Problem 2: The assumptions that Doug and Deena make are ordinary, common sense statements about driving from downtown Columbus to downtown Urbana.
    Section 1.1 Problem 8: Draw the picture that goes with this statement and explain how your picture displays the meaning of this statement. You are not being asked to calculate anything or prove anything.
    Section 1.1 Problem 21 (c): What does "arbitrarily far apart" mean? It means as far apart as you want. In Euclidean Geometry, if I say I want two points in the plane 5 inches apart, I can find them. If instead I say I want two points 30 inches or 50 inches apart, I can find them. No matter how far apart I want the points to be, I can find points that far apart. What happens if the points have to be on the sphere?
    Also, when we are working with spherical geometry, we are only working with one sphere of fixed size. For example, we are not going to be working with a sphere of radius 1 m in one problem and then a sphere of radius 10 m later on in the same problem. We pick one sphere and everything happens only on that sphere.
    Section 1.2 Problem 27 (e): What do we care about when we solve for line(x)? We only care about the number of x's and the number of constants on each side. We do not care about the number of x2's or x3's, etc.. These will cancel out, so we need not worry about them. How many x's are on the left hand side? On the right hand side?
    Section 1.3 Problem 11: Just try it and see. Draw a triangle with three 60 degree angles (and with a horizontal base to make everything easier to figure out). Do all three sides have the same Taxicab length or not? Explain.
    Section 1.3 Problem 23: How do we draw a parabola? We draw a circle of radius r centred at the point. We draw a line r away from the original line (on the same side as the point). Where these intersect will be the same distance away from both the point and the line - the points of intersection will be on our parabola. Repeat for different values of r until you know what the parabola looks like. See the notes for more details.
    Section 1.3 Problem 30: Collinear means all on the same straight line.