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 be2aand2band the distance apart bec.

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.

Givena,b, andcas defined in the above sketch. What is the condition ona,b, andcfor 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, given2aand2bas the length of the chords and the radiusrof the circle, find c, the distance between the parallel chords.

To the right is a CONSTRUCTED example with

a = 5cm

b = 4cm

r = 6cm

Chords DC and D'C' are each8 cmlong and parallel to chord EF that is10 cmlong. The radius isr = 6 cm.

By MEASUREMENT, the value ofcbetween EF and DC is approximately1.15 cm; the value ofcbetween 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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