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  • If a line segment joining mid-points of two chords of a circle passes through the centre of the circle, prove that the two chords are parallel.

If a line segment joining mid-points of two chords of a circle passes through the centre of the circle, prove that the two chords are parallel.

Answers (1)

Given: M and N are midpoints of chords AB and CD respectively.

MN passes through the centre of the circle O

To prove: Chord AB and Chord CD are parallel

Construction: Join B and C.

Proof:

\angleM = \angleN = 90°                      (Segment joining the midpoint of the chord to the centre)

Consider, DOMB and DONC

\angleBOM = \angleCON                     (Vertically opposite angles)

OB = OC                                 (Radii of the same circle)

\angleOMB = \angleONC                     (90° each)

\therefore \triangleOMB\cong \angleONC                  (AAS congruence)

\therefore \angleOBM = \angleOCN               (CPCT)

\therefore \angleABC = \angleBCD                

We can see that \angleABC and \angleBCD are alternate angles of the chords AB and CD with BC as transversal.

\Rightarrow Hence, chord AB || chord CD.

Posted by

infoexpert24

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