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How do you define "draw chords at random"?

Because different methods of selecting random chords yield different probabilities.

This problem is known as Bertrand's Paradox

en.wikipedia.org/wiki/Bertrand_paradox_(probability)

Assuming the circle is made up of 360 points.
A chord can be made by choosing any two points.
For any point we can choose upto 120 points on one side of it,
And 120 points on other side of it for the cord to be shorter.

=> 2/3 of the chords will be shorter

area of circle-area of eq/area of eq......=ans....

If R is radius of the circle

Length of a side of the equilateral triangle will be (sqrt(3))*R
Largest possible chord is the diameter - length 2R
so sqrt(3)/2 should be the answer

Draw a equilateral triange. Pick any one vertex and draw a tangent to it. Now if you notice, chords wth one end at this vertex have length more than side of triangle if they are passing through the inside portion of triangle in that case the angle made by the chord with the tangent is between 60 to 120 degrees whereas the chord can make any angle between 0 to 180 uniformly.

(120-60 )/180 = 1/3

Is it not a geometry question? lets say we have an equilateral triangle . Lets assume we remove the two sides of the equilateral triangle and left with one side. Now, the chords which will be lesser in length than this chord, will be on both sides of the diameter and lie symetrically. So the region behind the chord is the region we want. And then we can double this region. So the fraction can be found as follows
p*i*r*r (area of circle) == x
3*sqrt(3) / 4 *r*r (area of triangle) == y
x/2y is the answer.

Sine 30= Half