Geometry: Exploring Midpoints
Recall that the midpoint of ¯AB is a point M on ¯AB that divides ¯AB into two congruent pieces. You can use this definition to prove that each piece has length 1/2 AB. That's certainly reasonable. If you divide a segment into two pieces of equal length, it makes sense that half of the original length will go to the first piece, the other half to the second piece. This is such a reasonable statement, it's just got to be a theorem. Consider this your first invitation to a formal proof. I'll go through each of the five steps in the process.
- Example 1: Prove that the midpoint of a segment divides the segment into two pieces, each of which has length equal to one-half the length of the original segment.
- Solution: Follow the steps outlined in how to write a formal proof.
- 1. Give a statement of the theorem:
- Theorem 9.1: The midpoint of a segment divides the segment into two pieces, each of which has length equal to one-half the length of the original segment.
- 2. Draw a picture. Theorem 9.1 talks only about a line segment and its midpoint. So Figure 9.1 only shows ¯AB with midpoint M.
- 3. State what is given in terms of our drawing. Given: a line segment ¯AB and a midpoint M.
- 4. State what you want to prove in terms of your drawing. Prove: AM = 1/2 AB.
- 5. Write the proof. You need a game plan. In proving this theorem, you will want to make use of any definitions, postulates, and theorems that you have at your disposal. The definition you will want to use is that of the midpoint.
- The postulate that will come in handy is the Segment Addition Postulate, which states that if X is a point on ¯AB, then AX + XB = AB. This theorem doesn't seem to have any special needs, so you will prove this theorem directly. Start with your given information, and don't stop until AM = 1/2AB.
|1.||M is the midpoint of ¯AB||Given|
|2.||¯~=¯MB||Definition of midpoint|
|3.||AM = MB||Definition of ~=|
|4.||AM + MB = AB||Segment Addition Postulate|
|5.||2AM = AB||Substitution (steps 3 and 4)|
|6.||AM = 1/2 AB||Algebra|
There is a little flexibility in the reasons given, especially when you are dealing with algebra. For example, the reason for step 6 was “algebra,” but it could also have been “the division property of equality.” When it comes to the geometrical parts of a proof, however, there is not much flexibility. The reason for step 2 could only have been the definition of a midpoint, and step 3 is valid only because of the Segment Addition Postulate. There are no other options in these cases.
Excerpted from The Complete Idiot's Guide to Geometry © 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.