Assertion (A): The line segment joining the mid-points of all parallel chords of a circle passes through the centre.

Reason (R): If the chords are on same side of centre then only the line through the mid-points of the chords passes through the centre.

1. A is true, R is false.
2. A is false, R is true.
3. Both A and R are true.
4. Both A and R are false.

A is true, R is false.

Explanation

Let AB and CD be two parallel chords of a circle with center O. P is mid-point of AB and Q is mid-point of CD.

We have to prove that the line joining the points P and Q passes through the center O, i.e., ∠ POQ = 180°.

Construction : Join OP and OQ. Also, draw OE parallel to AB and CD.

Proof : Since, line segment joining the mid-point of the chord with the center of the circle is perpendicular to the chord, therefore, OP is perpendicular to AB and OQ is perpendicular to CD.

⇒ ∠ OPA = 90° and ∠ OQC = 90°

Now, OE ∥ AB and OP is transversal.

∴ ∠ POE = ∠ OPA = 90° (Alternate angles)

Similarly, OE ∥ CD and OQ is transversal.

∴ ∠ QOE = ∠ OQC = 90° (Alternate angles)

∴ ∠ POQ = ∠ POE + ∠ QOE

= 90° + 90°

= 180°

⇒ POQ is a straight line.

i.e, the line joining the mid-points of two parallel chords passes through the center.

∴ Assertion (A) is true.

Let AB and CD be two parallel chords that are lying on the same side of a circle with center O. M is mid-point of AB and N is mid-point of CD.

We have to prove that the line joining the points M and N passes through the center O, i.e., ∠ MNO = 180°.

Construction : Join OM and ON. Also, draw OE parallel to AB and CD.

Proof : Since, line segment joining the mid-point of the chord with the center of the circle is perpendicular to the chord, therefore, OM is perpendicular to AB and ON is perpendicular to CD.

⇒ ∠ OMA = 90° and ∠ ONC = 90°

Now, OE ∥ AB and OM is transversal.

∴ ∠ MOE = ∠ OMA = 90° (Alternate angles)

Similarly, OE ∥ CD and ON is transversal.

∴ ∠ NOE = ∠ ONC = 90° (Alternate angles)

∴ ∠ MNO = ∠ MND + ∠ DNO

= ∠ MNC + ∠ CNO

= 90° + 90°

= 180°

⇒ MNO is a straight line.

i.e, The line passing through M and N also passes through O.

∴ Reason (R) is false.

Hence, Assertion (A) is true, Reason (R) is false.