Polar Coordinates - Geometric Reasoning

Hi - I've attached an explanation from my Further Maths book which I'm unsure about.

Methods 1 & 2 I'm fine with, but as for Method 3 I'm not sure what exactly this proves? I'm fine with OPA being a right-angled triangle but from the line "Using trigonometry, r = 10cos(theta) as required...." onwards I'm not sure what's going on?

Can anyone help?

Edit: I've tried reuploading but the website keeps rotating the images

Hi, in method 2 you convert from polar form to Cartesian form and get the equation of the circle.

In method 3 you do the same thing from a geometrical perspective.

It's not very clear what they intend to do.

Yeah I'm not really sure either. I get the geometry but not sure what's happening after that.

Anyone able to help?

Yeah I'm not really sure either. I get the geometry but not sure what's happening after that.

Anyone able to help?

It's just a wordy geometrical argument rather than an actual algebraic proof.

They're just showing you that if you start off with a circle as described but choose your origin of (polar) coordinates to be the point on the left edge of the circumference, then as a point moves round the circle it is always true that $r = 10\cos\theta$ with the choice of r and $\theta$ that they have defined.