10. (ROM 4) Consider two segments of length $a, b(a>b)$ and a segment of length $c=\sqrt{a b}$.
(a) For what values of $a / b$ can these segments be sides of a triangle?
(b) For what values of $a / b$ is this triangle right-angled, obtuse-angled, or acute-angled?
10. (a) Let us set $k=a / b>1$. Then $a=k b$ and $c=\sqrt{k} b$, and $a>c>b$. The segments $a, b, c$ form a triangle if and only if $k<\sqrt{k}+1$, which holds if and only if $1<k<\frac{3+\sqrt{5}}{2}$.
(b) The triangle is right-angled if and only if $a^{2}=b^{2}+c^{2} \Leftrightarrow k^{2}=k+1 \Leftrightarrow$ $k=\frac{1+\sqrt{6}}{2}$. Also, it is acute-angled if and only if $k^{2}<k+1 \Leftrightarrow 1<$ $k<\frac{1+\sqrt{5}}{2}$ and obtuse-angled if $\frac{1+\sqrt{5}}{2}<k<\frac{3+\sqrt{7}}{2}$.