How will we prove that diameter is the longest chord of the circle

Asked by  | 15th Jun, 2013, 06:14: PM

Expert Answer:

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.

Answered by  | 15th Jun, 2013, 09:26: PM

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