how to prove that the chord closer to the center is longer than the chord that is away from th center

or vise versa?

You can find this by finding the angle subtends  at the center .If the angle subtended  is large then it in means that the distance of the chord from the center is less. 

In second way ,you can draw a perpendicular from  the chord on the centre . Now we know the radius and  perpendicular distance. From this we can find the other side (by applying Pythagoras theorem) . Doubling the side you will get the length of the chord. So you can prove this from here also.

Hope this would have cleared your doubt .