Math Is Fun Forum

hodori0719guest

Guest

Geometry Circle Problem

Let

,

be twelve equally spaced points on a circle with radius 1.

Find

(The sum includes the square of the distance between any pair of points, so the sum includes

  terms.)

[latex fixed - bobbym]

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Geometry Circle Problem

hi hodori0719guest

Welcome to the forum.

Call the centre of the circle C. The triangle A1CA2 is isosceles so split it into two identical right angled triangles and

In the same way get expressions for A1A3, A1A4, A1A5, A1A6 and A1A7.

Square these and add up.

This will simplify as

and so on.

Distances from A1 to A8 ... A12 can be found by symmetry.

Then you can repeat for points A2, ...A12.

Every distance will have been calculated twice ( as A1A2 and again as A2A1 for example) so divide the final total by 2.

Hope that helps.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Enshrouded_

Member

Registered: 2015-07-31

Posts: 47

Re: Geometry Circle Problem

I see how to do it. Ugh, it's going to take so long

Can you please help even further? I got 114 + 1/2 + 10\sqrt{3}, which was wrong.

Last edited by Enshrouded_ (2015-09-13 10:18:37)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Geometry Circle Problem

hi Enshrouded_

Don't know how a root 3 crept in here.  This is how I started:

note:

and

So the sum of those is

After that you get lots more similar terms.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Enshrouded_

Member

Registered: 2015-07-31

Posts: 47

Re: Geometry Circle Problem

I've tried 15 different possibilities, but I still can't get it. I don't know what I'm doing wrong.

I just don't know what to do with this problem anymore.

Last edited by Enshrouded_ (2015-09-14 11:50:49)

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Geometry Circle Problem

hi Enshrouded_

Did you follow what I did in post 4 ?

After that I did a similar calculation for A1 to A7, A8, A9, A10, A11 and A12.

So then I had all the elements from a single point.

You can write down the same total for all the distances starting from A2, then A3 then A4, ...... then A12.

You'll get the total for them all by just multiplying the first result by 12.

But this will count every answer twice (eg A1A2 and A2A1).  So divide the total by 2.  That should give you the correct answer.

Bob

If ever you are stuck part way through a problem, it helps a lot if you say what you've done.  Then I know where to pick up from and also how difficult you are finding the problem.

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Enshrouded_

Member

Registered: 2015-07-31

Posts: 47

Re: Geometry Circle Problem

Yup, I got values in order: 2-\sqrt{3}, 1, 2, 3, 2+\sqrt{3}. Then A_1, A_7 = 2, because radius = 1. Then it just goes down 2+\sqrt{3}, 3, ... I summed those for 22. 22*12=264. 264/2 = 132. Is there anything I'm doing wrong?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Geometry Circle Problem

hi

This is my diagram for this. 

To calculate A1A2 I used trig on the yellow triangle.

The points are equally spaced at 30 degrees apart so the yellow triangle has an angle at the centre of 15 degrees.  The hypotenuse is 1 (radius) so the opposite is 1.sin(15)

The question wants A1A2^2 so this will be sin^2(15)

Here's a copy of my calculation from post 4.

note:

and

So the sum of those is

A8 to A12 will be a repeat of this so another 10.

A1A7 is 2 as you have said, so squaring that and adding to the others gives a total of 10 + 10 + 4 = 24.

Can you finish it from here ?

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Enshrouded_

Member

Registered: 2015-07-31

Posts: 47

Re: Geometry Circle Problem

Thanks so much! I just forgot to square A_1, A_7, so I ended up with the sum of 22, not 24. Thanks again bob!

reaganchoi

Member

Registered: 2015-12-18

Posts: 2

Re: Geometry Circle Problem

You could also use the fact that (A_1 A_4)^2 + (A_4 A_7)^2 = (A_1 A_7)^2 (pythagorean theorem) and same when it is A_2 instead of A_4, etc.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Geometry Circle Problem

hi reaganchoi

That's a neat suggestion.  :-)

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

dazzle1230

Member

Registered: 2016-05-17

Posts: 92

Re: Geometry Circle Problem

i got 197, but it's not correct.  I don't know what i did wrong

Bob

Administrator

Registered: 2010-06-20

Posts: 10,626

Re: Geometry Circle Problem

hi dazzle1230

If you post your working out, I'll have a look at it and see if I can spot any error.

Bob

---

Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

dazzle1230

Member

Registered: 2016-05-17

Posts: 92

Re: Geometry Circle Problem

I'm using the pythagorean theorem and thale's