MTH 250 Graded Assignment 4 Measurement Material from Kay, sections 2.4, 3.2, 2.5, 2.6 Q1: Suppose that in a certain metric geometry* satisfying axioms D1 through D3 [Kay, p78], points A, B, C and D are collinear, and AB 13, BC 9, AC 8, CD 5, BD 11, AD 2 *clearly not the Euclidean metric, because if it were, these points couldn t be collinear! a) Construct a table that shows all the potential betweenness relations between any 3 points (see below where I ve gotten it started, and note that if you have an ordering such as A B C, you don t also need C B A, because that s really the same thing). b) Then, for each, state whether XY YZ XZ, XY YZ XZ, or XY YZ XZ (where X, Y, and Z are any of the combinations of A, B, C and D on the table). c) Indicate for each combination whether or not betweenness holds, and whether or not the triangle inequality is satisfied. There are 12 possible combinations here to check, and the systematic way to list is to note that out of the four point set { A, BCD,, } there are four distinct combinations of three points, and within a 3 point combination, there are three different orderings (permutations), not counting reversals. Table on next page, since it won t all fit here. I filled in a couple bits to show what I m looking for; you fill in the rest.
AB 13, BC 9, AC 8, CD 5, BD 11, AD 2 A B C (or C B A) A C B (or B C A) AB BC AC (13 9 8 ) Not between ( A B C is false) Triangle inequality satisfied A B D (or D B A)
Q2: Sketch a triangle ABC on a grid with vertices A (4,3), B (7,1), C (5,7). (a) Calculate the side lengths AB, BC, and AC in the Euclidean metric. (b) Is the triangle equilateral, isosceles, or scalene in this metric? (c) Calculate the side lengths AB, BC, and AC in the Taxicab metric. (d) Is the triangle equilateral, isosceles, or scalene in this metric? (e) While the Euclidean and Taxicab metrics measure distance in different ways, the text notes [p.129] that the measures of angles are the same in each. Whatever the measures of A, B and C are in the figure above, they re the same no matter what metric we re considering. 1. In Euclidean geometry (go back and draw on your knowledge of high school geometry), do any of the angles in triangle ABC have the same measure (e.g, is it possible that m A m Band so on)? Explain briefly. 2. All the above calculations (assuming they re done correctly!) are a reminder about why it s important to keep straight what s part of the definition of a thing (which would be
true in all geometries), and what is a theorem about a thing (which would only be true in specific geometries where it s been proven). Is has congruent base angles part of the definition of an isosceles triangle? Explain based on the above. Q3: Points with their coordinates are shown in the coordinate plane to the right. Note the points are given as A (1, 3), B (6,10), and C (11,18) it s a little hard to read. Assume the Euclidean metric. Is A B C? Explain why or why not (be sure to refer to precise definition of betweenness).
Q4: Identify each of the following sets appearing in the figure (each one should resolve to a single thing; e.g. a point, a segment, a ray, a line, an angle): a) CE CA b) CD AG c) DF CBF d) FD CBF e) CEG AE
Q5: In the diagram, the coordinates of BC, BE, and BD are given as shown. Also given is that BA is opposite BC, and that BG is opposite BE. Find the measures of angles ABG and GBD. Justify each step in your calculations. [Kay, section 2.5, #6, p100] Anything you use to do to a calculation, you need to find the definition of that thing, or a theorem related to that thing, and support that it meets the definition (or hypothesis of the theorem) from the givens above. Prove every little thing you do, even if it seems obvious! (See suggested problems for examples.)
Q6: We are given that MN has coordinate 20, and MQ has coordinate 60. Also given that QMN and QMP are a linear pair of angles, and rays MX and MY bisect QMN and QMP respectively. Find m QMX and m QMY, and the coordinates of rays MX (labeled (a) in the diagram) and MY (labeled (b) in the diagram). Justify your calculations. For this one, you can assume betweenness relations are as apparent in the figure. That s because this question is related to problem #13, which is the more general case, and it s discussed in the notes and in that problem, it s shown that the betweenness relations can be formally established from those givens anyway. You re also welcome to use the result of problem #13 as a justification in this one!
Q7: Prove that if two angles have a side in common that passes through the interior point of an angle formed by the other two sides, then the other two sides are perpendicular if and only if the given angles are complementary. [Kay, section 2.5, #16, p102] Given: angle ABC with interior point D, forming angles ABD and DBC with common ray BD. Be sure to prove both directions of the if and only if in addition to the above, assume first that the two sides (rays) are perpendicular and show the angles are complementary, then assume the angles are complementary, and show the sides (rays) are perpendicular. This is a short proof; just make sure again you re working within Kay s axiomatic system and using results as they appear in the text!