A regular nonagon, whose center is (0,0)....

[leilsilver]

A regular nonagon, whose center is (0,0) has one vertex at (1,0). If this vertex is connected to each of the remaining 8 vertices, and the geometric mean of these segments is calculated, the result would be: a sqrt b (where a and b are both single digit natural numbers). Find a+b.

I know that it goes out cos 40 and sin 40 everytime, from the last coordinates, but the hard part is finding out how to get it all into a nice number...

 

[pka]

A regular nonagon, whose center is (0,0) has one vertex at (1,0). If this vertex is connected to each of the remaining 8 vertices, and the geometric mean of these segments is calculated, the result would be: a sqrt b (where a and b are both single digit natural numbers).

Are you sure that you have written this correctly?
As you can my computer algebra gets a geometric mean that would fit.

 

[leilsilver]

I found the answer to be 8 sqrt 9. So in effect a+b = 17. This was how I and my group found out about it:

1, cos 40, cos 60, cos 80, cos 120, cos 160, cos 200, cos 240, cos 280....
0, sin 40, sin 60, sin 80, sin 120, sin 160, sin 200, sin 240, sin 280....

And imagine the points are A,B,C,D,E,F,G,H,I,J

The Eight vertices would be at AB, AC, AD, AE, AF, AG, AH, AI, AJ, But they are reflections of each other after the (0,1) point. so AB=AJ, AC=AI, AD=AH, AF=AG.

And we did the law of sines on each one of them since there was always the point were It is a unit circle and its angle. And multiply it to the second power because they're the same segments and it's a geometric mean. Get 9 and that's it.

 

[pka]

I have two questions about what you and your group have done.

Why did you all use ten points and not nine? Is it not a nonagon?

The center is (0,0) and one vertex is (1,0), right?
That means that each side is about 0.684 units long, see the following page:
http://mathworld.wolfram.com/RegularPolygon.html
Each of the diagonals is less than 2. Thus the geometric mean is less that 2.
What definition of geometric mean are you using?
Please see this page: http://mathworld.wolfram.com/GeometricMean.html

 

[leilsilver]

A nonagon has nine sides but ten vertices right? We are not counting on all the nine sides of the nonagon. We are counting where one vertex connects to the other using imaginary lines.

As I was saying, label the vertices A,B,C,D,E,F,G,H,I,J and we want to find the length of from A to B. then A to C, then A to D. ... do you see where I am going?

Vertex AB = (1/sin70)= (a/sin40) <- where if you are wondering where sin70 came from: from units (1,0) and (cos 40,sin 40) it has a distance of 1 because of the unit circle concept. And in between those 2 points, is 40 degrees. Equilateral triange = equiangular degrees on both sides.


And since it wants the geometric mean of all of these, you are multiplying it instead of adding them. And there are 8 of these vertices that I am speaking of, not nine.

Sadly, I do not have a program to draw it on so I can show you what I'm talking of...I can only describe. It would be better if I showed it to you.

 

[skeeter]

A nonagon has nine sides but ten vertices right?

triangle ... 3 sides, 3 vertices

quadrilateral ... 4 sides, 4 vertices

pentagon ... 5 sides, 5 vertices

do you see a pattern here?

 

[Denis]

A nonagon has nine sides but ten vertices right? We are not counting on all the nine sides of the nonagon. We are counting where one vertex connects to the other using imaginary lines.

C'mon Silver; a square has 4 sides: does a square have 5 vertices?
A vertex is a point where two sides meet...a square has 4.

Regardless of what you're doing, WHERE is your point J ?

 

[pka]

A nonagon has nine sides but ten vertices right? We are not counting on all the nine sides of the nonagon.

“A nonagon has nine sides”, absolutely correct!
“but ten vertices right?”, absolutely wrong!
A nonagon has nine sides and nine vertices!

Please, please take a look at this webpage.
http://mathworld.wolfram.com/Nonagon.html

You did not answer my question about geometric mean!
How are you using that to get such a number?
What is the exact wording of the question?
Join the vertex (1,0) to each of the other eight vertices.
The sum of the squares of those lengths is 18.
\(\displaystyle \sqrt {18} = 3\sqrt 2\), but that is not a geometric mean!
But I suspect that is exactly what this problem is about.

 

[leilsilver]

Okay, sorry about the vertices thing. It is 9. It was finals week, done a lot of mistakes. My mistake.

But to answer pka's question, what we got was 8 sqrt 9. 8 because you are multiplying it 8 times. 9, is the result of multiplying all of the lenghths together. Isn't that what geometric mean means? if you multiply it eight times, you root to the eigth.

Here is what we got:

(sin 40/sin 70)^2 * (sin80/sin50)^2 * (sin 120/sin 30)^2* (sin160/sin 10)^2 = 9.

And unless my TI-89 calculator is wrong, that's right.