Question

prove that the diameter is the longest chord of a circle

Answer 1

Take any chord in a circle, say with endpoints AB. Let O be the center of the circle. Then segments AO and BO are radii of the circle. AOB is a triangle, and we know that the sum of the lengths of two sides of a triangle is always greater than or equal to the length of the third side. So:

|AB| ? |AO| + |BO|

(Here |AB| means length of AB.)

So, the maximum length of the chord AB can be equal to |AO| + |BO|. Now AO and BO are the radii of the circle, hence |AO| + |BO| = 2r = d (diamter of the circle).

So, maximum length of the chord AB = |AO| + |BO| = d
Hence diameter of the circle represents the longest chord in the circle.