# Symmetry of a circle proof

1. Dec 22, 2007

### raintrek

1. The problem statement, all variables and given/known data

I'm trying to prove that the circle is symmetrical by showing that x² + y² = a² holds when the circle rotates.

I know that this is proved given the following two formulae:
x = x'cosθ - y'sinθ
y = x'sinθ + y'cosθ

but I don't know where those two equations have come from based on my diagram. Help!

2. Dec 22, 2007

### Shooting Star

Drop a perpendicular from where the x' axis cuts the circle to the x-axis and another perp from where the y' axis cuts the circle to the y-axis. Use some properties of similar triangles and right angled triangles.

3. Dec 22, 2007

### raintrek

I've got the x'cosθ part of the expression for x, but I just cannot see how the -y'sinθ is found...

4. Dec 22, 2007

### Shooting Star

My mistake for giving a hasty answer. Sorry.

Take a point P:(x,y) in the x-y system. Now draw x' and y' axes, rotated by some theta. If you drop perps from P on the x-axis and the x' axis, the first perp cuts the x-axis at a dist x from O and the 2nd perp cuts the x'-axis at a dist x' from O. Now, find x in terms of x' and y', using elementary geometry.

5. Dec 22, 2007

### HallsofIvy

Staff Emeritus

6. Dec 22, 2007

### raintrek

OK, i think I'm almost there,

I have the x'cosθ term, and I know I need to minus the purple section, which I trust is y'sinθ -- but I can't seem to show that it is, lol, it's the last stumbling block

7. Dec 22, 2007

### Shooting Star

(HallsofIvy has asked you a question. I am also curious.)

Have you drawn the diagram as I said in my 2nd post? You can show us, if possible.

8. Dec 22, 2007

### Shooting Star

EDIT: ignore

9. Dec 23, 2007

### Shooting Star

Hi raintrek,

I'm not able to see the pictures you posted initially. Have you removed them, or is something wrong with my browser settings? Please answer asap.