Answer
It takes 9 rectangles to reach towards the length of $\sqrt 10$.
Work Step by Step
Let us consider triangle ABC is a right angled a B with legs of units $\sqrt x $ and 1 units.
Now by applying the Pythagoras theorem
$AC^{2}=1^{2} + \sqrt x^{2}$
=$1+x$
Now Ignoring the negative solution , we have
AC=$\sqrt x+1$
Hence, When the line segment from the center, that happens to be the hypotenuse of the right angle triangle increase in length as follows :
$\sqrt 2,\sqrt 3\sqrt 4, .............\sqrt 10$
