Question

If a diameter of a circle bisects each of the two chords of the circle, prove that
the chords are parallel.

Answer 1

Answer:

Let AB and CD be two chords of a circle whose center is O, and let PQ be a diameter bisecting chords AB and CD at L and M respectively. Since PQ is a diameter. So, it passes through the center O of the circle. Now,

L is the mid-point of AB.

OL⊥AB    [ Line joining the center of a circle to the mid-point of a chord is perpendicular to the chord]

∠ALO=90°

Similarly, ∠CMO=90°

Therefore, ∠ALO=∠CMO

But, these are corresponding angles.

So, AB∥CD.

Answer 2

Let AB and CD be two chords of a circle whose centre is O, and let PQ be a diameter bisecting chords AB and CD at L and M respectively. Since PQ is a diameter. So, it passes through the centre O of the circle. Now,

L is mid-point of AB.

OL⊥AB [ Line joining the centre of a circle to the mid-point of a chord is perpendicular to the chord]

∠ALO=90⁰

Similarly, ∠CMO=90⁰

Therefore, ∠ALO=∠CMO

But, these are corresponding angles.

So, AB∥CD.