Parallel Chords Problem

If we have a circle where we know the respective lengths of two parallel chords and we know the distance apart of the two chords, find the radius of the circle.

 

For convenience of notation, let the lengths of the two chords be 2a and 2b and the distance apart be c.

 

The problem probably requires discussion of two cases:

Case 1. Both parallel chords are on the same side of the center of the circle.

Case 2. The center of the circle is between the two parallel chords.

 

Given a, b, and c as defined in the above sketch. What is the condition on a, b, and c for determining whether the parallel chords are on the same side of the center, or one of them is the diameter, or they are on opposite sides of the center?

 

 

Extension.

The reverse problem: Given the lengths of two parallel chords and the radius of the circle, find the TWO possibilities for the distance apart of the parallel chords. That is, given 2a and 2b as the length of the chords and the radius r of the circle, find c, the distance between the parallel chords.

To the right is a CONSTRUCTED example with

a = 5 cm

b = 4 cm

r = 6 cm

Chords DC and D'C' are each 8 cm long and parallel to chord EF that is 10 cm long. The radius is r = 6 cm.

By MEASUREMENT, the value of c between EF and DC is approximately 1.15 cm; the value of c between EF and D'C' is approximately 7.74 cm.

 

COMMENT: This is probably an easier problem that the one for finding the length of the radius, given the lengths of the two chords and their distance apart.

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