24. The points $G$ and $I$ are on the circle with centre $H$, and $F I$ is tangent to the circle at $I$. The distances $F G$ and $H I$ are integers, and $F I=F G+6$. The point $G$ lies on the straight line through $F$ and $H$. How many possible values are there for $H I$ ?
A 0
B 2
C 4
D 6
E 8
24. D Let $x$ be the length $F G$ and let $r$ be the radius. Then $F I=x+6$ and $G H=H I=r$. Angle $F I H$ is a right angle (the tangent and radius are perpendicular) so $F I^{2}+H I^{2}+F H^{2}$, which gives $(x+6)^{2}+r^{2}=(x+r)^{2}$. Expanding this gives $x^{2}+12 x+36+r^{2}=x^{2}+2 r x+r^{2}$, which simplifies to $12 x+36=2 r x$. Halving this gives $6 x+18=r x$, which rearranges to $r=6+\frac{18}{x}$. Since $r$ is an integer, $x$ must be a (positive) factor of 18 , namely $1,2,3,6$, $9,18$; each of these six factors give a different value of $r$ (or $H I)$ as required.