Thread: Chords

The set of points created by the midpoints of all chords of length 4 centimeters in a circle of radius 8 centimeters is a what?

I'm having trouble visualizing this

Note that you can find OC where C is the center of a chord. It does not depend on the position of the chord.

And, by symmetry, all such points will be the same distance from the circle and the same distance from the O.

But the question was the midpoints create a circle not the endpoints of the chord so isnt the sketch off? Why is it the case too that the midpoints create a circle..it's still not so clear..

Originally Posted by mjoshua

But the question was the midpoints create a circle not the endpoints of the chord so isnt the sketch off?

I guess what you are saying is this: the problem was to prove that the midpoints of chords form a circle. Yes; this circle with radius OC is not shown in the picture. The circle that is shown is the original one with radius 8cm.

Why is it the case too that the midpoints create a circle..it's still not so clear..

Because the distance OC from the center of the circle to the midpoint is always the same for each chord. Conversely, for each point C at this distance from O there is a chord of length 4cm of which C is the center.