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Archive for the ‘Proofs’ Category

Invisible Misunderstandings: Square roots of 2 and 3




square root of 2 w/ "parent" triangle


FUN MATH PROBLEM — Circling the Square & Vice-Versa

From time to time I will post interesting math problems.



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and make your Subject: Fun Problem.






Have fun!

How to Use Definitions in Geometry

Illustration showing angle between curves.

Image via Wikipedia

Time for a post about geometry, which I tutor in addition to algebra and many other subjects.

I especially enjoy helping students learn how to do proofs, which I find is the hardest area of geometry for most kids.

Recently I came up with an analogy to help students understand the special usefulness of definitions in geometric proofs.

The analogy is:  Definitions are like reversible coats.

What?  … you say.

Coats. Reversible coats. As in two for the price of one.

Similarly with definitions:  you get two IF-THEN statements for the price of one when you work with a definition.

Here’s what I mean.

First consider a “standard theorem” in geometry, viewed in the IF-THEN format.

Theorem:  IF two angles are complements of the same angle, THEN they are congruent.

Notice that the converse of this statement doesn’t make much sense:

IF two angles are congruent, THEN they are complements of the same angle. (What other angle? We haven’t even mentioned another angle!)

But when it comes to definitions, you can:

a) First, turn the definition into and IF-THEN statement, and

b) Secondly, you can flip that IF-THEN statement around, and this new statement, called the “converse,” will always be true. You can bank on it!

Example of a definition:  A right angle is an angle that measures 90 degrees.

And here’s one IF-THEN statement that flows out of this definition:

1) IF an angle is a right angle, THEN it measures 90 degrees.

But notice that the converse is also true:

2)  IF an angle measures 90 degrees, THEN it is a right angle.

Let’s try this again, for the definition of perpendicular lines.

Definition: Two lines are perpendicular if they form four right angles.

First IF-THEN statement:

1)  IF two lines are perpendicular, THEN they form four right angles.

Second IF-THEN statement, the converse.

2)  IF two lines form four right angles, THEN the lines are perpendicular.

I am wondering if you are wondering why this is true. Why is it that, for definitions, both the statement and its converse are always true? The reason, I believe, has to do with the nature of a definition. With a definition, we are giving a name to some geometrical object, and stating what we consider to be the defining characteristic of that object.

To take a nonsensical example, suppose that you live in a world that has objects called “Snurfs,” which are measured in units called “Goobles.” Now imagine that some of the Snurfs are special because they have a measure of 100 Goobles. This fact makes these Snurfs so special that you wind up talking about them a lot. And because you talk about them a lot, it is helpful to give them a name. So you do give them a name; you decide to call them “Wombats.” What this means is that anytime a Snurf has a measure of 100 Goobles, you will call it a Wombat. And anytime you see the thing you call a Wombat, you can be sure that it will have a measure of 100 Goobles. For that is just what you have decided the word Wombat will mean. Based on this, you put forth the formal definition:

A Wombat is a Snurf with a measure of 100 Goobles.

Given this definition, notice that you can create two IF-THEN statements:

1)  IF a Snurf is a Wombat, THEN its measure is 100 Goobles.

And you can also state the converse, and it will be true:

2)  IF a Snurf has a measure of 100 Goobles, THEN it is a Wombat.

To me, this is how definitions work. They involve people noticing something they are talking about, and they decide to give it a name so they can talk about it more easily. When they define what the word means, they attach the word to the primary characteristic of this thing, and through this act, the word is born, and along with it, its definition.

Right angle.

Image via Wikipedia

Anyhow, in terms of doing geometry, the important thing to keep in mind is that all definitions can be used reversibly. So, going back to the example of the right angle, here’s what this means.

If, in the course of a proof, you establish that a particular angle is a right angle, you can conclude that the measure of this angle is 90 degrees. Reason:  Definition of a right angle.

And similarly, if in a proof you establish that a particular angle has a measure of 90 degrees, then you can conclude that this angle is a right angle. Reason:  Definition of a right angle.

This reversibility factor is why, when you read through geometric proofs, you will notice that “Definition of … ” is used quite often as a reason for steps. Because they are logically reversible, definitions are TWICE as useful as standard theorems.



The problem, once again:

For any polygon, a “diagonal” is defined as a line segment that runs from one vertex  to another, running  through the polygon’s interior. Find a formula that determines the number of diagonals in any convex polygon with n sides. Once you have the formula, use it to figure out the number of diagonals in a convex polygon with 1,000 sides (don’t try this by hand! — that’s why algebra was invented).

The winning answer was provided by Chris Mark. The formula, for a convex polygon with n sides, is this:  Number of Diagonals =  [n(n– 3)]/2. For n = 1000, the number of diagonals = 498,500.

The reasoning behind the formula. A polygon has as many vertices as sides. So a polygon with n sides also has n vertices. Now, consider any vertex of the n-gon. From that vertex the number of diagonals that can be drawn is (n – 3). That is because we cannot draw a diagonal to 3 vertices:  the vertex chosen, and the two adjacent vertices. So the expression (n – 3) = the number of diagonals that can be drawn from any vertex. We multiply (n – 3) by n to obtain the total number of diagonals that can be drawn from all n vertices. But if we simply multiply n by (n – 3), we’d be counting each diagonal twice. To eliminate that problem, we divide the product, [ n(n – 3)], by 2, and that provides the correct formula:

Diagonals  =  [n(n – 3)]/2

Applying this formula to a convex polygon with 1,000 sides, we see that the number of diagonals =  498,500.

In addition to providing the answer, Chris pointed out that the problem need not be restricted to regular polygons, as it was when posted. This formula works for all convex polygons, regular or not.

Thanks, Chris. And thanks to everyone who submitted answers.