Two-Column Proof: Parallel Tangents
Date: 03/08/2002 at 00:44:37 From: Andrea Subject: Two-column proof I have to prove that tangents to a circle at the endpoints of a diameter are parallel. I'm not sure how to do a two-column proof, or I should say I don't understand them. Any advice would help. Thank you.
Date: 03/08/2002 at 13:03:24 From: Doctor Peterson Subject: Re: Two-column proof Hi, Andrea. Let's work through this slowly. What are you supposed to prove? tangents to a circle at the endpoints of a diameter are parallel The first thing to do is to extract from this problem a list of the "givens" (the things you are told are true). Here's the situation: We've drawn a circle, and then drawn a diameter of it (a chord going through the center). Then we've drawn the tangent to the circle at each end of the diameter. I'll make a picture and label the parts: A C -----------------ooooo+ooooo----------------+ ooooo | ooooo oooo | oooo oo | oo o | o oo | oo o | o o | o o | o o | o o O+ o o | o o | o o | o o | o oo | oo o | o oo | oo oooo | oooo ooooo | ooooo -----------------ooooo+ooooo----------------+ B D So here's what we are given: Given: Segment AB is the diameter of a circle with center O Line AC is tangent to circle O at A Line BD is tangent to circle O at B Now, what are we to prove? To prove: Lines AC and BD are parallel Now we have to construct a complete line of reasoning that starts with the given facts, and ends with the thing we want to prove. How can we do that? You can read a number of discussions of how to approach a proof in our FAQ: About Proofs http://mathforum.org/dr.math/faq/faq.proof.html The basic idea is to look at the facts you have and think of ways you can use them; and look at the goal and think of ways to approach it. What do you know about tangents to a circle? Perhaps you have seen a theorem that says they are perpendicular to the radius. If not, what theorems do you know about tangents? What do you know about parallel lines? If you have a transversal (such as the diameter in this case), what facts about it and the angles it makes can you use to show that the lines are parallel? I'll let you go dig in your book for the answers to those questions (because every book is a little different, so I don't know just what theorems you can use). Write back and tell me what you've found, and then we can see how to put them together into a proof. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 03/08/2002 at 20:44:18 From: Andrea Subject: Re: Two-column proof GIVEN - AC=BD TO PROVE - AC and BD are parallel STATEMENTS REASONS 1. Point A (outside circle O) Given 2. Point B (outside circle O) Given 3. Line AC tangent to O Given 4. Line BD tangent to O Given 5. Midpoint of AB is O Defining a point 6. Connect AB through midpoint Constructing a line 7. AC is perpendicular to AB 90-degree angle 8. BD is perpendicular to BA 90-degree angle 9. AC is parallel to BD AC=bd a c oooooooooooooooooooooooooo oooo ooo : ooo oo : oo oo : oo oo : oo oo : oo oo : oo oo O oo oo : oo oo : oo oo : oo oo : oo oo : oo oo : oo ooo : ooo ooooo d ooooooooooooooooooooooooooo B Is there anything else that I would need to do to this? I appreciate your help. Thanks.
Date: 03/08/2002 at 22:50:19 From: Doctor Peterson Subject: Re: Two-column proof Hi, Andrea. I'll make some comments to help you refine your thinking and writing. >GIVEN - AC=BD >TO PROVE - AC and BD are parallel The given is wrong. You don't know anything about the lengths of AC and BD, if that's what you mean. The "givens" I gave you last time better reflect what you really know. >STATEMENTS REASONS > >1. Point A (outside circle O) Given >2. Point B (outside circle O) Given It would be better to say "_on_ circle O". >3. Line AC tangent to O Given >4. Line BD tangent to O Given I would say "tangent to O _at A_" (or B) to make it clear. >5. Midpoint of AB is O Defining a point This isn't really needed; O doesn't show up again. Well, I take that back: you need to mention somewhere that OA and OB are radii, so although it is not relevant that O is the midpoint, it is important that O is on AB. Do you have a theorem that says that a diameter passes through the center? >6. Connect AB through midpoint Constructing a line This reads as if you were giving instructions for a construction, rather than a proof. Constructions are often part of a proof (you may have to introduce a new line that was not mentioned in the "givens," for example; but in this case AB is the diameter you are given, so there is no need to say this. >7. AC is perpendicular to AB 90 degree angle >8. BD is perpendicular to BA 90 degree angle You need a better reason here. No one told you that this is a 90-degree angle, so that can't be the reason it's true. Do you have a theorem in your text that says that a tangent is perpendicular to the circle? That's what you need here. >9. AC is parallel to BD AC=bd Again, you haven't really given a reason. You need some theorem from which you can deduce this. AB is a transversal; do the angles BAC and ABD relate to any theorem you know? Some books prove a theorem that "two lines perpendicular to the same line are parallel," but without that, one of the standard transversal theorems will do. Looking over the proof as a whole, I see that both your "givens" and your "reasons" are lacking. Each "given" should appear among the statements somewhere; and each reason should be either "given," a definition, or a brief statement of a theorem. Try looking through some sample two-column proofs in our archives to see what they should look like: Geometry Proofs http://mathforum.org/dr.math/problems/victoria.11.07.01.html Two-Column Proof About Kites http://mathforum.org/dr.math/problems/dement.11.9.99.html Building Two Column Proofs http://mathforum.org/dr.math/problems/crystal9.12.98.html Parallel Lines: Two Column Proof http://mathforum.org/dr.math/problems/turtle9.9.98.html Building a Geometric Proof http://mathforum.org/dr.math/problems/karen.6.03.99.html Then write back and we can see how you've improved it. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 03/09/2002 at 09:06:48 From: Andrea Subject: Re: Two-column proof I think I'm getting better, or at least I hope I am. Here we go. GIVEN - Segment AB is the diameter of a circle with center O Line AC is tangent to circle O at A Line BD is tangent to O at B TO PROVE - AC and BD are parallel STATEMENTS REASONS 1. Point A (on circle O) Given 2. Point B (on circle O) Given 3. Line AC tangent to O at A Given 4. Line BD tangent to O at B Given 5. Intersect of AB is O Definition of Intersects 6. AC is perpendicualr to AB If a line in the plane of a circle is perpendi- cular to a radius at its outer endpoint, then the line is tangent to the circle (Theorem 9-2) 7. BD is perpendicular to BA Theorem 9-2 stated in 6 8. AC is parallel to BD Two lines parallel to a third line are parallel to each other (Theorem 3-10) I looked at some of the two-column proofs and I hope I am getting closer. These things are not that easy without the help of someone who understands them. Thank you for all your help; I would probably fail this course if it weren't for your site.
Date: 03/09/2002 at 22:24:21 From: Doctor Peterson Subject: Re: Two-column proof Hi, Andrea. I certainly understand that this can be hard to figure out on your own; I wish I could just stand there with you and guide you more interactively. Maybe there is someone you can find - a student who took geometry last year and wants to improve her knowledge by helping you? - so you can have someone to share ideas with more quickly. Let's go through your latest version: >GIVEN - Segment AB is the diameter of a circle with center O > Line AC is tangent to circle O at A > Line BD is tangent to O at B > >TO PROVE - AC and BD are parallel > >STATEMENTS REASONS > >1. Point A (on circle O) Given >2. Point B (on circle O) Given Since it isn't the mere fact that these points are on the circle that you will be using later on, it's probably more useful just to restate the "given": Segment AB is a diameter of the circle O. >3. Line AC tangent to O at A Given >4. Line BD tangent to O at B Given You may have to be more careful than I am being about symbolism. Where I've said "line AC" or "segment AC" there are symbols I can't type, with a line or segment over the "AC". I'm not sure how best to indicate "circle O" as opposed to "point O"; sometimes a symbol for a circle is used, and sometimes the circle is given a separate name, like "circle c". I don't think I'd just say "tangent to O", though I have no trouble understanding it. You'll want to check your book's notation and copy it. >5. Intersect of AB is O Definition of Intersects I don't know what you mean by this. There is no such thing as "the intersect of a segment"; you need two things to intersect. I presume you mean midpoint, but I don't see how that comes into the rest of the proof, unless you mean this to suggest that therefore OA is a radius. >6. AC is perpendicular to AB If a line in the plane > of a circle is perpendi- > cular to a radius at its > outer endpoint, then the > line is tangent to the > circle (Theorem 9-2) You don't quite have the right theorem, though I expect that the right one is near 9-2. Do you see what is wrong? It says "IF it's perpendicular, THEN it's tangent." You want something that says "If it's tangent, THEN it's perpendicular," since you know you have a tangent. Usually you will just put a brief summary of the theorem ("tangent perp. to radius"), or its number in your book; but I'm glad to see it written out here for my own sake. Notice also that this is why you should have a statement that AB is a diameter; and technically you would probably need another statement that OA is a radius. I'm not sure whether you would find a theorem that says half of a diameter is a radius; that might be considered obvious just from the definitions. >7. BD is perpendicular to BA Theorem 9-2 stated in 6 When two statements have the same reason, just applied to different parts of the diagram, you can put them into one statement, or say "same," or do just what you did here. >8. AC is parallel to BD Two lines parallel to a > third line are parallel > to each other (Theorem > 3-10) Did you mean "perpendicular to a third line"? You have no third parallel line here, so the theorem you have stated isn't what you need. Again, I'm commenting on every statement just to give you as much help as I can, though some comments are trivial and others are important. I don't want you to be worried about details; the main thing at this stage is to get a feel for the big idea of a proof. I would be happy to see a quick summary of the proof: the lines are tangent at the ends of the diameter, therefore are perpendicular to it by theorem XXX; since they are perpendicular to the same line, by theorem YYY they are parallel. The details of the two-column proof sometimes can be overwhelming and hide what's really doing on; but it is still worthwhile to be able to write it out that way, in order to see for yourself that every detail can be covered, and there are no holes in your logic. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 03/10/2002 at 20:21:04 From: Andrea Subject: Re: Two-column proof Let me start out by saying thank you, I truly believe I'm learning from all this even though I'm a little slow at it. I have made corrections to two-column proof and here it is. GIVEN - Segment AB is the diameter of a circle with center O Line AC is tangent to circle O at A Line BD is tangent to O at B TO PROVE - AC and BD are parallel STATEMENTS REASONS 1. Segment AB is a diamter of circle O Given 2. Line AC is tangent to O at A Given 3. Line BD is tangent to O at B Given 4. AC is perpendicular to AB Tangent line is perpen- dicular to the radius 5. BD is perpendicular TO BA Theorem as stated in #4 (Theorem 9-1 - If a line is tangent to a circle, then the line is perpendicular to the radius drawn to the point of tangency) 6. AC is perpendicular to AB Two lines perpendicu- lar to same line are parallel (Theorem 3-7 In a plane two lines perpendicular to the same line are parallel Thanks so much for all your help, it is greatly appreciated.
Date: 03/10/2002 at 21:00:49 From: Doctor Peterson Subject: Re: Two-column proof Hi, Andrea. It does take time to get through this, doesn't it? That's probably because fixing a proof is more like editing an essay than other kinds of math, and that takes a lot of back-and-forth. I applaud your persistence, because not many have kept it up this well! I don't have to go through this line by line, because it's basically acceptable now. Any further stylistic changes I might suggest wouldn't benefit your understanding. Assuming this has to be turned in to your teacher, you may get feedback saying it doesn't fit expectations in some details - you book may have something more specific to say about why something perpendicular to the diameter is perpendicular to a radius as well, and you may have some preferred way to state theorems in the reasons column - but that's just details. You'll want to learn to fit the standards, because there's reason for doing so, but I can't tell you what they are. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 03/10/2002 at 23:32:16 From: Andrea Subject: Re: Two-column proof Thank you so much for all your help. It is good to know that you can find help and learn in the process. It has truly been a learning experience, thanks to you. Thanks again for all your support and guidance.
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