Question 17619: How can we determine the langth of the arc of a circle, cut by a cord in that circle? Can we do it if we just know the length of the cord (I don't think so? So what other data do we need to solve this?
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! draw a circle with P as centre and radius of r.draw any chord AB in the circle.mark Q the mid point of AB.join PQ.now in the triangles ,PAQ and PBQ we have
PA=PB=radius =r
AQ=QB.(since Q is the mid point of AB)
PQ=PQ..common side .Hence the 2 triangles are identical. so anglePQA = anglePQB..but sum of these 2 angles is a straight angle =180 .hence each angle is 90.So triangle PAQ is a right angled triangle.hence using pythogarus theorem,we have
PA^2=AQ^2+PQ^2..if we take length of chord AB as b ,since Q is mid point of AB,we have AQ=QB=b/2..If the distance of chord from centre of circle PQ is taken as d then the relation between length of chord (b),its distance from centre of circle (d) and the radius of the circle (r) is given by
r^2=d^2+(b/2)^2...
So your intution that "How can we determine the langth of the arc of a circle, cut by a cord in that circle? Can we do it if we just know the length of the cord (I don't think so? So what other data do we need to solve this?' is correct... YOU NEED TO KNOW RADIUS OF THE CIRCLE OR THE DISTANCE OF THE CHORD FROM THE CIRCLE TO SOLVE YOUR PROBLEM
now to determine the length of arc ,you can do it taking the 3 available data from above namely ,r,d and b.using trignometric ratios in the right angled triangle PAQ,find angle APQ which is same as angle BPQ,you can find angle APB,which is the angle subtended by the chord/arc AB at centre P.Now you know the full arc length is perimeter of the circle is 2*pie*r which subtends an angle of 360 at the centre P.so if this arc subtends an angle APB = x say then the arc length =(x/360)*2*pie*r
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