Getting Started with Two-Column ProofsDate: 03/19/2002 at 22:05:35 From: Kayla Subject: Getting started with Two-column Proofs Dear Dr. Math, I am really stuck with this Two-Column Proof thing. I am home schooled and I basically teach myself. I understand it a little bit but sometimes I can't get started on my own. I am really a numbers person and can't solve anything or prove anything without numbers. Here is an example of a question. Write a two-column proof of Theorem 8-1, which says: If the altitude is drawn from the vertex of the right triangle to its hypotenuse, then the two triangles formed are similar to the given triangle and to each other. How am I supposed to prove that? How do I get started? Kayla Date: 03/19/2002 at 22:53:13 From: Doctor Peterson Subject: Re: Getting started with Two-column Proofs Hi, Kayla. I'll be happy to help; we get a lot of questions from others in your position, because proofs are something new to most students. You'll want to read through our FAQ on the subject, if you haven't already: http://mathforum.org/dr.math/faq/faq.proof.html You call yourself a "numbers person"; do you feel comfortable with writing or reasoning as well? That is really closer to the idea of proofs than all the work with numbers that you've done in earlier math, and thinking of a proof as a kind of persuasive essay is a better image of what you have to do. Your job in a proof is to convince me that something is true, as long as I accept the postulates you're basing it on. Let's see how I would approach this particular proof: First, we have to draw a picture, so we can see what it's about: B + /| \ / | \ / | \ +---+-------+ A D C Here AC is the hypotenuse and B is the right angle, so we drew altitude BD, which is perpendicular at D. I've named all the points to help me talk about them; and on paper I would mark little squares to show that angles ABC, ADB, and CDB are right angles. That's what we're given, together with some immediate implications (such as which angles are right). Now, what is it we have to prove? The "two triangles formed" must be ABD and CBD; they are supposed to be similar to ABC. But before we can say this symbolically, we have to decide what orientation to use; that is, in what order should we name the vertices of each triangle, so we are matching up the right sides when we call them similar? Well, if they are similar, then corresponding angles will be congruent, and triangles ABC and ABD share one angle at A, so A must match in both. And both have a right angle, at B and D respectively. So one of our conclusions must be "ABC is similar to ADB." In the same way, the other conclusion must be "ABC is similar to BDC." Now, in just thinking enough to decide what it is we have to prove, I've actually discovered enough to go ahead and write the proof. See if you can do that, then write back and show me what you come up with. We can go through it together, and get you on the right track with proofs. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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