Chords and a Circle's Center - Problem 2

An application of what you know about chords and the center of the circle in this problem right here. Where I’m asking you, if I gave you an arc AB, so just part of a circle, how could you using geometry find out where the center of that circle is?

Now the easy answer would be, “Oh Mr. McCall, all I need to do is take my compass and extend it until I’ve got the radius and then I can see where the center of the circle is.” Not quite. The reason that this isn’t correct is because it doesn’t use anything that you can really prove.

So let’s think about, well, how could I find the center of that circle? Let’s look and see what I know about a perpendicular bisector of a coordinate. Well I see that if I bisect along the perpendicular, any chord in the circle, it will pass through the centre of that circle.

So if I could draw in a chord here, any chord and if I could construct its perpendicular bisector, so let’s say I constructed that perpendicular bisector, then I would know that somewhere along that line is the circle. But that’s not enough to determine where it is.

So what I would have to do is I would have to draw another chord and find the perpendicular bisector of that chord, and again I’m going to mark these as congruent, and where these two intersect has to be the centre of the circle.

So it’s kind of like taking what you know about perpendicular bisectors and the center of the circle and working backwards and saying if I found the perpendicular bisectors and I found where they crossed then I can determine where the centre of the circle is.