Question

A chord of a circle has length 3n,n is a positive integer . the segment cutoff by the chard has height n,what is the smallest value of n for which the radius of circle is also a positive integer

Answer 1

Let radius of the circle is a positive integer e.g. r

Given, chord length ,PQ = 3n

so, PB = BQ = PQ/2 = 3n/2 = 1.5n

OB = OA - AB = r - n

OQ = radius of circle = r

now from Pythagoras theorem,

in ∆OBQ , OB² + BQ² = OQ²

now, (r - n)² + (1.5n)² = r²

r² + n² - 2rn + 2.25n² = r²

3.25n² - 2rn = 0

3.25n - 2r = 0

r = 3.25n/2 = 1.625n

because r is positive integer so, we have to choose a smallest positive integer of n.

let's choose n = 4

then, r = 1.625 × 4 = 6.5 ≠ integer

take n = 8 , r = 1.625 × 8 = 13 = a positive integer .

hence, smallest integer value of n = 8 by which we get positive integer of r e.g., 13 unit.