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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.
    
Associated Topics:
High School Conic Sections/Circles
High School Euclidean/Plane Geometry
High School Geometry

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