Question

Prove that the line segments joining the mid-points of two parallel chords of a circle passes through the centre of the circle.

Answer 1

Given: A circle with centre O. AB and CD are two parallel chords of the circle. M and N are the mid-points of chords AB and CD respectively.

To Prove: The line segment MN joining the mid-points M and N passes through the centre O i.e., M, O, N are collinear.

Proof (Step-wise):

1. Join OM and ON, where O is the centre of the circle.

2. For any chord of a circle, the radius drawn to its midpoint is perpendicular to the chord.

Hence, OM ⟂ AB and ON ⟂ CD.

3. Given AB || CD, any line perpendicular to AB is also perpendicular to CD.

Therefore, the lines OM and ON are parallel OM || ON.

4. OM and ON are both lines through the same point O and they are parallel.

The only way two lines through the same point can be parallel is that they coincide.

Hence, OM and ON lie on the same straight line.

5. Since M and N lie on the same line with O, the points M, O, N are collinear.

Thus, the segment MN passes through O.

Therefore, the line segment joining the mid-points of two parallel chords of a circle passes through the centre of the circle.