Points created by midpoints

[donnagirl]

The set of points created by the midpoints of all chords of length 4 cm in a circle of radius 8 cm is a: Circle. Why is this the case? How can we show this for all cases?

 

[pka]

The set of points created by the midpoints of all chords of length 4 cm in a circle of radius 8 cm is a: Circle. Why is this the case? How can we show this for all cases?

In a circle of radius $\displaystyle R$, the midpoint of a chord of length $\displaystyle a$ is $\displaystyle \frac{1}{2}\sqrt{4R^2-a^2}$ units from the center of the circle.

 

[Mrspi]

The set of points created by the midpoints of all chords of length 4 cm in a circle of radius 8 cm is a: Circle. Why is this the case? How can we show this for all cases?

Do these theorems sound familiar?

In a circle (or in congruent circles), congruent chords are the same distance from the center of the circle.

The distance from the center of a circle to a chord in the circle is measured along a perpendicular from the center to the chord.

A radius perpendicular to a chord bisects the chord (goes through the midpoint of the chord).

Now...ponder those ideas. You've got a bunch of congruent chords in a circle. You draw a perpendicular from the center of the circle to each of those chords and measure the distance from the center to each chord. In each case, you're looking at a segment joining the center of the circle to the MIDPOINT of a chord....can you visualize the figure formed by all of those midpoints?

Have you tried making a sketch? That might help.