Question

if a diameter of a circle bisects each of two chords of a circle prove that the chords are parallel​

Answer 1

If diameter of a circle bisects each of the two chords of a circle. Then the chords are parallel.

Solution:

There are two chords AB and CD , which are bisected by the diameter

We have to prove that AB is parallel to CD

Since, ON bisects CD, therefore ON is perpendicular  to CD

(Perpendicular drawn from centre of the circle to a chord, bisect the chord in equal parts)  

Similarly, OM is perpendicular to AB

To prove that the two chords are parallel we need to show their alternative interior angles are equal

Since, For chord AB and CD , MN act as a transversal and also

Angle AMN = angle MND ( both are 90 degree)

Hence we can say both Chord AB and CD are parallel to each other

Answer 2

Answer:

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=90

o

Similarly, ∠CMO=90

o

Therefore, ∠ALO=∠CMO

But, these are corresponding angles.

So, AB∥CD.