P is any point in a circle with its centre O . If the length of the radius is 5 cm and OP = 3 cm the determine the least length of the chord passing though the point P .

Explanation

Let AB is a chord passing though P of the circle with centre at O. Let us join, O, P As per question , OA = 5 cm , OP = 3 cm
Now length of AB will be least if OP is a perpendiclular to AB.
`therefore Delta OAP` is a right angled triangle .
`therefore OA^(2) = AP^(2) + OP^(2) or (5)^(2) = AP^(2) + (3)^(2) or 25= AP^(2) + 9 or , AP^(2) = 25-9 or AP^(2) = 16`
`therefore AP = sqrt(6) = 4`
Again ` OP bot AB, therefore P` is the mid-point of AB.
`therefore AB = 2 xx AP = 2 xx 4 cm = 8 cm`
Hence the required least length = 8 cm