Question

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.

Answer 1

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

$MN$ passes through the center of the circle $O$.

To prove: chord $AB\parallel$ chord $CD$

Solution:

Join points $B$ and $C$ as shown in the figure.

$\angle M=\angle N={90}^o$

(Segment joining the midpoint of the chord and center of the circle is perpendicular to the chord.)

Now, in $△OMB$ and $△ONC$,

$\angle BOM=\angle CON$ ....Vertically opposite angles

$\angle OMB=\angle ONC$ ...Each ${90}^o$

$OB=OC$ ...radii of the same circle

$\therefore △OMB\cong △ONC$ ...SAA test

$\therefore \angle OBM=\angle OCN$ ....C.A.C.T

$\therefore \angle ABC=\angle BCD$ ...Since $A-M-B$ and $C-N-D$

These are nothing but alternate angles for the chords $AB$ and $CD$ respectively.

Hence, chord $AB\parallel$ chord BC.