Question

A chord of a circle has length 3n, where n is a positive integer. The segment cut off by the chord has height n, as shown. What is the smallest value of n for which the radius of the circle ia also a positive integer? Please do not add white space around the answe

Answer 1

where is diagram for the question

Answer 2

Thank you for asking this question, here is your answer:

We will consider the radius to be r

Chord length ,PQ = 3n

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

OB = OA - AB = r - n  

OQ = radius of circle = r

Here we will use the 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

Now we will chose the smallest positive integer for n and that would be 4

r = 1.625 × 4 = 6.5 ≠ integer

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

So we know that the smallest integer value is 8 now.

If there is any confusion please leave a comment below.