ANSWERS AND COMMENTS

Assignment 1B - Due Friday 10/3

1.3 - The Fundamental Property of a Perpendicular Bisector

(a) Write the definition of the perpendicular bisector of a segment.

Comment:

Definition: The perpendicular bisector of a segment is the line through the midpoint of the segment that is perpendicular to the segment.

There were 3 common errors in answering this problem. They are not just mistakes, they are interesting illustrations of difficulties in understanding the role of definitions in mathematics.

1. Some people said that the perpendicular bisector is a line or a segment or a ray. This is not correct. There is only one perpendicular bisector of a segment, so the correct definition can't make it several things. It is a line. There are a couple of simple reasons: if it is a segment or ray, which one? More seriously, in (c) below you prove that the perpendicualar bisector is the set of points equidistant from the endpoints. This set is a line, not a segment or a ray. If you PROVE it, it should be true.
2. Some people defined the the perpendicular bisector as the set of points equidistant from the endpoints. It would be possible to write a geometry book with this as the definition. But then you would have to prove that the perpendicular bisector is a line perpendicular to the endpoints. It would not be part of the definition. The standard approach, and the one we have adopted in this course, is to define the perpendicular bisector as a special line and then prove special properties about it, as in (c). Notice that if you do take this non-standard definition, you should not prove anything to prove (c), since (c) is then true by definition. For our definition, it must be proved.
3. A few people gave the correct definition and then added some extra information about distances, etc. The drawback to this idea of putting lots of stuff into the definition (besides not agreeing with the definition that we are using in the course) is that you then have to prove that all the stuff in the definition does not lead to a contradiction. Also, every time you assert something is a perpendicular bisector, you have to check off all the properties.

(b) If AB is a segment and P is any point on the perpendicular bisector of AB, prove that the triangle PAB is isosceles (with PA = PB).

If M is the midpoint of AB, you can see that triangle PMA = triangle PMB by SAS. See B&B for detailed proof.

(c) If a triangle PAB is isosceles (with PA = PB), prove that P is a point on the perpendicular bisector of AB.

This is one of the properties of isosceles triangles proved on the first day. You can quote it.

1.4 - Kite proofs using what you have recently proved.

Given the kite ABCD in problem 1.1, use 1.3 and the properties of isosceles triangles to give very short (one or two line) proofs of these statements:

These is a second opportunity to prove properties of kites. You can use the isosceles properties and and also 1.3 above. You can't really use what we proved about kites, unless you just want to dispose of the proofs by saying that we have already proved all 3. Then stop. You should either prove them or quote them, but it makes no sense to prove them using the kite properties (including SSS).

(a) line AC is the perpendicular bisector of segment BD.

We are given AB = AD and CB = CD. By 1.3c this means both A and C are on the perpendicular bisector of BD. Since two points line on only one line, then AC is the perpendicular bisector of BD.

(b) ray AC bisects angle BAD and ray CA bisects angle DCB.

Since AC is the perpendicular bisector of BD, by (a) , in the isosceles triangle ABD, this line bisects the angle BAD, by properties of isosceles triangles. For the same reason in triangle CDB, this line bisects the angle BCD.

(c) triangle ABC is congruent to triangle ADC.

Since angle BAC = angle DAC by (b), AC = AC, and we are given AB = AD, the triangle ABC = triangle ADC by SAS.