my geometry book says that the apothem is a perpendicular bisector drawn to the center of a polygon; but it also says that the apothem of a triangle equals 1/3 the altitude (height). how is this? how can the center of something be only 1/3 of the height instead of 1/2? we’ve asked my geometry teacher this, and she can’t answer. (and no, this has nothing to do with homework; it’s just something that has been bothering the crap out of me for the past month or so.)

An equilateral triangle is base-heavy; if you picked a point halfway up, you’d have much more area on the bottom than on the top.

Your teacher should be able to answer this.

Eh, true, but perhaps not illustrative to every student. Try this, **Gypsymoth**; from each vertex of the triangle, draw a line toward the center with a length exactly one half of the height of the triangle. You see that the lines don’t touch? The center of an equilateral triangle is equidistant from the vertices, or from the centers of the sides, but the center can’t be equal distances from the sides *and* the vertices simultaneously – that would be a circle!

It isn’t a general rule for any polygon. A square, for example, is twice as high as the length of a line drawn from the center of one side to the center of the square. So, too, would a regular hexagon or a regular octogon be.

An equilateral triangle could be thought of as three congruent obtuse triangles put together. Try drawing this by drawing a line from the center of the triangle to each vertex. Now draw the perpendicular bisectors of the main triangle. You should have six right triangles now, with one leg of each triangle being the perpendicular bisector of the original triangle. Each leg of a right triangle must always be shorter than the hypotenuse. In this case, the perpendicular bisector of the main triangle must be shorter than the line drawn to the vertex.

Bisect all three sides of an equilateral triangle. The result is six triangles, all with the same size, shape, and angles. The angles are 30-60-90, with the hypothenuse along the height, and the short side on the inside of the original triangle. Because they are 30-60-90, the short side is half the hypothenuse. Viola.

hypothetically

Try drawing an equilateral triangle, its (vertical) altitude, and the bisector of the right base angle till it meets the altitude.

You will find 30-60-90 degree triangles, which I hope you also know as 1-2-square root of 3 triangles. Pick 2 for the length of the original triangle’s side and find other sides of the 30-60-90 triangles.

(rad 3 over 3 ) ÷ rad 3 = 1/3

```
[we've asked my geometry teacher this, and she can't answer.]
```

More likely she won’t answer, as this is something you can work out for yourself - the best (only?) way to learn mathematics. Try it and see if you can pick up some extra credit. BTW, you should also check out the definition of ‘center of a regular polygon’ and be able to show that the bisector mentioned in the first paragraph goes to the ‘center’ of the triangle.

Also, encourage that ‘bugging’ feeling - it’s the distinquishing difference between critical thinkers and the marching morons.

```
```

Wow! You’ve got to be quick around here.

In the time it took me to decide how to reply to the apothem question, 3 others jumped right in.

okay some of that stuff went way over my head; or maybe i just don’t have a long enough attention span. but would i be correct in saying the apothem goes to the circumcenter? what is the definition of “center” as opposed to ortho-, circum-, incenter, and - what’s the one i can’t remember?

In the case of a regular polygon, all the centers are the same.

Orthocenter - intersection of the altitudes

Centroid - intersection of the medians

Incenter - intersection of the angle bisectors

Circumcenter - intersection of the perpendicular bisectors of the sides

The ‘center’ is the circumcenter.

In an equilateral triangle, the altitudes, medians, angle bisectors, and perpendicular bisectors are all the same, so all these centers are the same, as erislover says.

Do you know the 30-60-90 triangle?

what do you mean? i know what it is, but no i have memorized all the facts and properties of it except that is has a 30 degree angle, a 60 degree one, and a right one.

You’ll get to it sooner or later, but 30-60-90 triangles have easy to remember properties. The side of the short leg is half the hypotenuse, and the side of the long leg is (short leg) X (square root of 3). When you learn some basic trigonometry you’ll find out all kinds of fun stuff about right triangles. The 45-45-90 is another good one.

This was Mentock’s and my development, by breaking down the equilateral triangle into smaller congruent 30-60-90 triangles, then showing that the height is given by the sum of one hypotenuse and the sum of one short leg.

The short story: As a general rule of thumb, the center is point at which you could balance the object. In other words, if you gently poke a pin at this point, the triangle would be perfectly balanced. Sometimes, an object’s center can fall *outside* the object! Consider a “U”-shaped object, the center can lie within the “U”, but it does not belong to the “U” itself…like the center of a circle does not belong to the circle! Think about that!

Longer story: If you get to take an Advanced Placement (AP) Physics class, you may learn about how we find the center of an various objects (i.e.: the centroid) which may or may coincide with the center of mass of that object (esp. when the mass is not evenly distributed about the object - like being top heavy, as a simple example.) Also, in AP Physics, you will learn how to find other characteristics related to bending and rotating of any object, especially “I” shapes, used to make I-beams, a common metal support for our buildings and bridges.

Just some food for thought…

- Jinx

Correction in bold-faced type. - Jinx